MA102: Single-Variable Calculus II « Saylor.org – Free Online Courses Built by Professors

Single-Variable Calculus II

Purpose of Course show close

This course is the second installment of Single-Variable Calculus. In Part I (MA101), we studied limits, derivatives, and basic integrals as a means to understand the behavior of functions. In this course (Part II), we will extend our differentiation and integration abilities and apply the techniques we have learned. Additional integration techniques, in particular, are a major part of the course. In Part I, we learned how to integrate by various formulas and by reversing the chain rule through the technique of substitution. In Part II, we will learn some clever uses of substitution, how to reverse the product rule for differentiation through a technique called integration by parts, and how to rewrite trigonometric and rational integrands that look impossible into simpler forms. Series, while a major topic in their own right, also serve to extend our integration reach: they culminate in an application that lets you integrate almost any function you’d like. Integration allows us to calculate physical quantities for complicated objects: the length of a squiggly line, the volume of clay used to make a decorative vase, or the center of mass of a tray with variable thickness. The techniques and applications in this course also set the stage for more complicated physics concepts related to flow, whether of liquid or energy, addressed in Multivariable Calculus (MA103). Part I covered several applications of differentiation, including related rates. In Part II, we introduce differential equations, wherein various rates of change have a relationship to each other given by an equation. Unlike with related rates, the rates of change in a differential equation are various-degree derivatives of a function, including the function itself. For example, acceleration is the derivative of velocity, but the effect of air resistance on acceleration is a function of velocity: the faster you move, the more the air pushes back to slow you down. That relationship is a differential equation.

Course Information show close

Welcome to MA102 Single-Variable Calculus II. Below, please find general information on this course and its requirements. Picking up where MA101 left off, this course looks at integration in more depth, before moving on to cover a variety of topics, including parametric equations, sequences and series, and the basics of ordinary differential equations. Calculus II can often feel like an ill-assorted grab-bag, but each of the disparate topics it introduces either extends the reach of our integration and differentiation techniques or addresses applications of those techniques.

Course Designer: Clare Wickman

Primary Resources: This course is comprised of a range of different free, online materials. However, the course makes primary use of the following materials:

University of Michigan: Scholarly Monograph Series: Wilfred Kaplan and Donald J. Lewis’s Calculus and Linear Algebra Vol. 1
University of Wisconsin: H. Jerome Keisler’s Elementary Calculus
Whitman College: Professor David Guichard’s Calculus
University of Houston: Dr. Selwyn Hollis’s “Video Calculus”
MIT: Professor Jerison’s Single Variable Calculus
Temple University: Gerardo Mendoza and Dan Reich’s Calculus on the Web
Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II”

Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. You will also need to complete:

Unit 1 Assessments
Unit 2 Assessments
Unit 3 Assessments
Unit 4 Assessments
Unit 5 Assessments
Unit 6 Assessments
The Final Exam

Note that you will only receive an official grade on your Final Exam. However, in order to adequately prepare for this exam, you will need to work through all of the resources and assessments in each unit.

In order to “pass” this course, you will need to earn a 70% or higher on the Final Exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.

Time Commitment: This course should take you approximately 125 hours to complete. At the beginning of each unit, there is a detailed list of time advisories for each subunit. These estimates factor in the time required to watch each lecture, work through each reading thoughtfully, and complete each assessment. However, these should be seen as guidelines, not goals; each learner is different, and you may find that your pace changes throughout the course. Mastery of the material, rather than strict adherence to the time estimates, is the measure of success in this course. It may be useful to take a look at these time advisories, to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself. For example, Unit 1 should take approximately 20.25 hours. Perhaps you can sit down with your calendar and decide to complete sub-subunit 1.1.1 (a total of 2 hours) on Monday night; sub-subunits 1.1.2 and 1.1.3 (a total of 3.5 hours) on Tuesday night; sub-subunit 1.1.4 (a total of 4 hours) on Wednesday night; etc.

Tips/Suggestions: If a lecture stops making sense to you, pause it – this is a luxury you only have in a course of this nature! – and return to the readings to get up-to-speed on the material. Remember to note down the time at which you paused the lecture, in case your browser times out. As noted in the “Course Requirements,” Single-variable Calculus Part I (MA101) is a pre-requisite for this course. If you are struggling with the mathematics as you progress through this course, consider taking a break to revisit MA101.

As you study the resources in each unit, take careful notes on a separate sheet of paper. Mark down any important equations, formulas, and definitions that stand out to you. These notes will serve as a useful review as you study for your Final Exam.

This course features a number of Khan Academy™ videos. Khan Academy™ has a library of over 3,000 videos covering a range of topics (math, physics, chemistry, finance, history and more), plus over 300 practice exercises. All Khan Academy™ materials are available for free at www.khanacademy.org.

Learning Outcomes show close

Upon successful completion of this course, the student will be able to:

Define and describe the indefinite integral.
Compute elementary definite and indefinite integrals.
Explain the relationship between the area problem and the indefinite integral.
Use the midpoint, trapezoidal, and Simpson’s rule to approximate the area under a curve.
State the fundamental theorem of calculus.
Use change of variables to compute more complicated integrals.
Integrate transcendental, logarithmic, hyperbolic, and trigonometric functions.
Find the area between two curves.
Find the volumes of solids using ideas from geometry.
Find the volumes of solids of revolution using disks, washers, and shells.
Find the surface area of a solid of revolution.
Compute the average value of a function.
Use integrals to compute displacement, total distance traveled, moments, centers of mass, and work.
Use integration by parts to compute definite and indefinite integrals.
Use trigonometric substitution to compute definite and indefinite integrals.
Use the natural logarithm in substitutions to compute integrals.
Integrate rational functions using the method of partial fractions.
Compute improper integrals of both types.
Graph and differentiate parametric equations.
Convert between Cartesian and polar coordinates.
Graph and differentiate equations in polar coordinates.
Write and interpret a parameterization for a curve.
Find the length of a curve described in Cartesian coordinates, described in polar coordinates, or described by a parameterization.
Compute areas under curves described by polar coordinates.
Define convergence and limits in the context of sequences and series.
Find the limits of sequences and series.
Discuss the convergence of the geometric and binomial series.
Show the convergence of positive series using the comparison, integral, limit comparison, ratio, and root tests.
Show the divergence of a positive series using the divergence test.
Show the convergence of alternating series.
Define absolute and conditional convergence.
Show the absolute convergence of a series using the comparison, integral, limit comparison, ratio, and root tests.
Manipulate power series algebraically.
Differentiate and integrate power series.
Compute Taylor and MacLaurin series.
Recognize a first order differential equation.
Recognize an initial value problem.
Solve a first order ODE/IVP using separation of variables.
Draw a slope field given an ODE.
Use Euler’s method to approximate solutions to basic ODE.
Apply basic solution techniques for linear, first order ODE to problems involving exponential growth and decay, logistic growth, radioactive decay, compound interest, epidemiology, and Newton’s Law of Cooling.

Course Requirements show close

In order to take this course, you must:

√ Have access to a computer.

√ Have continuous broadband Internet access.

√ Have the ability/permission to install plug-ins or software (e.g. Adobe Reader or Flash).

√ Have the ability to download and save files and documents to a computer.

√ Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).

√ Be competent in the English language.

√ Have access to a calculator.

√ Have read the Saylor Student Handbook.

√ Have completed MA101: Single-Variable Calculus I.

Unit Outline show close

1 The Integral
1.1 Review of Integration
1.1.1 The Indefinite Integral
1.1.2 The Area Problem and the Definite Integral
1.1.3 Approximating Integrals
1.1.4 The Fundamental Theorem of Calculus
1.1.5 Elementary Integrals
1.1.6 Integration by Substitution (Change of Variables)
1.2 Integration of Transcendental Functions
1.2.1 Exponential Functions
1.2.2 Natural Logarithmic Functions
1.2.3 Hyperbolic Functions

2 Applications of Integration
2.1 The Area between Curves
2.2 Volumes of Solids
2.3 Volume of Solids of Revolution
2.3.1 Disks and Washers
2.3.2 Cylindrical Shells
2.4 Lengths of Curves
2.5 Surface Areas of Solids
2.6 Average Value of Functions
2.7 Physical Applications
2.7.1 Distance
2.7.2 Mass and Density
2.7.3 Moments
2.7.4 Work

3 Techniques and Principles of Integration
3.1 Methods of Integration
3.1.1 Integration by Parts
3.1.2 Trigonometric Integration
3.1.3 Trigonometric Substitution
3.1.4 More Techniques Using the Natural Logarithm
3.1.5 Rational Functions and Integration with Partial Fractions
3.2 Integration with Tables & CAS (Computer Algebra Systems)
3.3 Numerical Integration
3.4 Improper Integrals
3.4.1 Type I
3.4.2 Type II

4 Parametric Equations and Polar Coordinates
4.1 Parametric Equations and Their Derivatives
4.2 Polar Coordinates
4.3 Derivatives and Curve Sketching in Polar Coordinates
4.4 Areas with Polar Coordinates
4.5 Arc Length with Polar Coordinates

5 Infinite Sequences and Series
5.1 Sequences
5.2 Series
5.2.1 Series and Basic Convergence
5.2.2 Properties of Infinite Series
5.2.3 Focus on the Geometric Series
5.2.4 Highlight: The Binomial Series
5.3 Test for Convergence of Positive Series
5.3.1 Divergence Test
5.3.2 Integral Test
5.3.3 Comparison Test
5.3.4 Ratio Test
5.3.5 Root Test
5.3.6 Limit Comparison Test
5.4 Tests for Absolute and Conditional Convergence
5.4.1 Alternating Series Test
5.4.2 Definition of Absolute and Conditional Convergence
5.4.3 Comparison Test for Absolute Convergence
5.4.4 Limit Comparison Test for Absolute Convergence
5.4.5 Ratio Test for Absolute Convergence
5.4.6 Root Test for Absolute Convergence
5.5 Series Representations of Functions
5.5.1 Power Series
5.5.2 Calculus with Power Series
5.5.3 Taylor and Maclaurin Series
5.5.4 Approximation by Power Series

6 Differential Equations
6.1 First-Order Differential Equations
6.2 Separation of Variables and Initial Value Problems
6.3 Slope Fields & Euler's Method
6.4 Exponential and Logistic Growth and Applications
6.4.1 Exponential and Logistic Growth
6.4.2 Radioactive Decay and Half-Lives:
6.4.3 Compound Interest
6.4.4 Epidemiology
6.4.5 Newton’s Law of Cooling

7 Final Exam

Expand All Resources Collapse All Resources

Unit 1: The Integral

We will begin by quickly reviewing the basics of integration, so integration is fresh in your mind before we extend its applications. Having completed MA101, you should be familiar with this material. We will then take a look at how integration applies to concepts like motion. Finally, we will discuss how logarithmic and exponential functions are integrated.

Unit 1 Time Advisory show close

Unit 1 Learning Outcomes show close

1.1 Review of Integration

1.1.1 The Indefinite Integral

Reading: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan’s and Donald J. Lewis’s Calculus and Linear Algebra Vol.1: “4-1 Introduction” and “4-2 The Indefinite Integral”
Link: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan and Donald J. Lewis’s Calculus and Linear Algebra Vol. 1: “4-1 Introduction” (HTML) and “4-2 The Indefinite Integral” (HTML)

Instructions: Please click on the links above, and read Sections 4-1 and 4-2 in their entirety. Note that for Section 4-2, you will need to click on the “next” link at the bottom of each page to continue the reading.

How does one “undo” differentiation? There are a few considerations. We need to loosely define “elementary function” to mean a function put together from rational and trigonometric functions, exponentials, radicals, and so forth. Although the derivative of an elementary function always is elementary, there are elementary functions with no elementary antiderivative. Since the derivative of a constant is 0, two functions that differ only by an added constant term will have the same derivative, so antiderivatives are never unique. However, these limitations do not forbid us from developing a strong theory of antidifferentiation, introduced in the readings linked above.

Studying these readings should take approximately 30 minutes.

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Reading: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan and Donald J. Lewis’s Calculus and Linear Algebra Vol. 1: “4- 5 Basic Properties of the Indefinite Integral” and “4-6 Applications of Rules of Integration”
Link: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan’s and Donald J. Lewis’s Calculus and Linear Algebra Vol. 1: “4-5 Basic Properties of the Indefinite Integral” (HTML) and “4-6 Applications of Rules of Integration” (HTML)

Instructions: Please click on the links above, and read Sections 4-5 and 4-6 in their entirety. Use the “previous” and “next” links at the bottom of the page to navigate through each reading. These readings discuss the “nuts and bolts” of finding antiderivatives.

Studying these reading should take approximately 30 minutes.

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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s “Indefinite Integrals”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s “Indefinite Integrals” (HTML)

Instructions: Click on the above link. Then, click on the “Index” button. Scroll down to “1. Integration,” and click button 104 (Indefinite Integrals). Do problems 1-10. Use the buttons in the module to check your answer or to move on to subsequent problems. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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1.1.2 The Area Problem and the Definite Integral

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.1: The Definite Integral”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.1: The Definite Integral” (PDF)

Instructions: Please click on the link above, and read Section 4.1 in its entirety (pages 175 through 185).

This is a slightly different presentation of definite integrals and area from the one chosen for MA101. The author approaches limits using the idea of rigorously defined infinitesimals, which you may think of as infinitely small numbers. These ultimately give a fairly intuitive theory, but you will need some vocabulary, because it is not the typical presentation. An infinitesimal is a number strictly between −a and a, for every positive real a. The only real infinitesimal is 0; more appear when we use the hyperreals, which are a collection of numbers extending the reals. There are non-real hyperreals between any two reals: for example, if ε is a positive infinitesimal, 5+ε is a non-real hyperreal between 5 and 6 (or even 5 and 5.00000001). The standard part of a hyperreal is obtained by “rounding to the nearest real,” so the standard parts of 5+ε and 5−ε are both 5, and the standard part of ε by itself is 0. Division by a nonzero infinitesimal gives an infinite hyperreal, a hyperreal that is strictly larger than any real number. The author is careful in the text to confirm the calculations give finite hyperreals. Infinite hyperreals have no standard part. The sum of two infinitesimals is always infinitesimal, as is the product of an infinitesimal and any finite hyperreal number.

The Transfer Principle states that any mathematical statement true of a function on the reals is also true of that function’s natural extension to the hyperreals. For example, the fact that (x + y)² = x² + 2xy + y² holds for all real x and y means it also holds for all hyperreal x and y. The proof of this principle is not important for our purposes, but the principle itself is vital. Finally, a hyperinteger is the integer analog to a hyperreal. However, the only finite hyperintegers are the integers, so you may think of the hyperintegers as an extension of the integers into infinity.

Studying this reading should take approximately 1 hour.

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Lecture: MIT: David Jerison’s “Lecture 18: Definite Integrals” and University of Houston: Selwyn Hollis’s “Video Calculus: The Integral” Lecture
Link: MIT: David Jerison’s “Lecture 18: Definite Integrals” (YouTube)

Also Available in:
iTunes U

and University of Houston: Selwyn Hollis’s “Video Calculus: The Integral” (QuickTime)

Instructions: Please watch the first video lecture in its entirety (47:14 minutes). Note that lecture notes are available in PDF; the link is on the same page as the lecture.

If you desire a shorter presentation, choose the second video; click on the second link; then scroll down to Video 22: “The Integral.” Professor Jerison discusses the area problem and the cumulative sum problem and uses them to define the definite integral.

Studying one of these lectures should take approximately 1 hour.

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1.1.3 Approximating Integrals

Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: The Area under a Curve”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: The Area under a Curve” (QuickTime)

Instructions: Please click on the link; then scroll down to Video 21: “The Area under a Curve.” Please watch the entire lecture. We will return to the problem of approximating definite integrals numerically in a later unit using more complicated methods.

Viewing this lecture and pausing to take notes should take approximately 45 minutes.

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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: James Palermo and Matthias Beck’s “Midpoint Rule”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: James Palermo and Matthias Beck’s “Midpoint Rule” (HTML)

Instructions: Click on the above link. Then, click on the “Index” button. Scroll down to “1. Integration,” and click button 111 (Midpoint Rule). Do problems 1-5. If you do not fully understand the midpoint rule, click on the “Help” button for a quick refresher. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 45 minutes.

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1.1.4 The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a two part statement relating definite integrals to derivatives and antiderivatives. First, it says that integrating f(t) from a to x and then differentiating results in f(x). Second, it says that definite integrals may be calculated via antiderivatives. It is important, because first, it says the relationship between integration and differentiation is what it ought to be, and second, that we can evaluate definite integrals without always resorting to limits of Riemann sums.

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.2: The Fundamental Theorem of Calculus”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.2: The Fundamental Theorem of Calculus” (PDF)

Instructions: Please click on the link above, and read Section 4.2 in its entirety (pages 186 through 197). This is a recap of the fundamental theorem of calculus, which relates the area problem and the definite integral to antiderivatives.

Studying this reading should take approximately 1 hour.

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Lecture: MIT: David Jerison’s “Lecture 19: First Fundamental Theorem of Calculus”
Link: MIT: David Jerison’s “Lecture 19: First Fundamental Theorem of Calculus” (YouTube)

Instructions: Please watch this video lecture in its entirety. This lecture will cover sub-subunits 1.1.3-1.1.5. Professor Jerison states the first fundamental theorem of calculus and uses it to calculate some integrals. He then discusses some properties of the definite integral and the method of substitution for computing integrals.

Viewing this lecture and pausing to take notes should take approximately 1 hour.

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Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: The Fundamental Theorem of Calculus”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: The Fundamental Theorem of Calculus” (QuickTime)

Instructions: This video lecture is optional. Please click on the link; then scroll down to Video 23: “The Fundamental Theorem of Calculus.” This video explains the definition of a function via integration and the differentiation of such functions.

Viewing this lecture and pausing to take notes should take approximately 30 minutes.

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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Aaron Robertson’s “Differentiation and the Fundamental Theorem”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Aaron Robertson’s “Differentiation and the Fundamental Theorem” (HTML)

Instructions: Click on the above link. Then, click on the “Index” button. Scroll down to “1. Integration,” and click button 114 (Differentiation and the Fundamental Theorem). Do problems 4-20. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour and 30 minutes.

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1.1.5 Elementary Integrals

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.3: The Indefinite Integral”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.3: The Indefinite Integral” (PDF)

Instructions: Please click on the link above, and read Section 4.3 in its entirety (pages 198 through 207). In this text, the author chooses to present the indefinite integral after the definite integral; however, this should not interfere with your understanding of the chapter. The section is extremely well-written; pay close attention to the discussion of antiderivatives, theorem 3, the rules of integration, and example 9.

Studying this reading should take approximately 1 hour.

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Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Antidifferentiation and Indefinite Integrals”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: Antidifferentiation and Indefinite Integrals” (QuickTime)

Instructions: Please click on the link scroll down to Video 24: “Antidifferentiation and Indefinite Integrals,” and view the entire lecture.

Viewing this lecture and pausing to take notes should take approximately 45 minutes.

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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Russo’s “Definite Integrals”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Russo’s “Definite Integrals” (HTML)

Instructions: Click on the above link. Then, click on the “Index” button. Scroll down to “1. Integration,” and click button 108 (Definite Integrals). Do problems 5-15. These should be very easy. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 45 minutes.

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1.1.6 Integration by Substitution (Change of Variables)

The derivative of f(g(x)), by the chain rule, is f’(g(x))*g’(x). If presented with the latter expression as our integrand, we should get f(g(x)) back as the integral. However, it can be difficult to determine what f and g are at a glance. Substitution “cleans up” the integrand by hiding g(x) inside a new variable u, and combining g’(x) and dx into du. It may also be used to turn a root into a power: let u equal the radical, solve for x, then differentiate, and plug the result in for dx. We will later see other uses of substitution, but all are designed to rewrite the problem into a form where the method of integration is clear.

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.4: Integration by Change of Variables”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.4: Integration by Change of Variables” (PDF)

Instructions: Please click on the link above, and read Section 4.4 in its entirety (pages 209 through 215).

Studying this reading should take approximately 30 minutes.

Terms of Use: The above book is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Change of Variables (Substitution)”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: Change of Variables (Substitution)” (QuickTime)

Instructions: Please click on the link, scroll down to Video 25: “Change of Variables (Substitution),” and view the entire lecture.

Viewing this lecture and pausing to take notes should take approximately 30 minutes.

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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Substitution Methods”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Substitution Methods” (HTML)

Instructions: Click on the above link. Then, click on the “Index” button. Scroll down to “1. Integration,” and click button 109 (Substitution Methods). Do at least problems 1-10. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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1.2 Integration of Transcendental Functions

A transcendental number is a number that is not the root of any integer polynomial. A transcendental function, similarly, is a function that cannot be written using roots and the arithmetic found in polynomials. We address exponential, logarithmic, and hyperbolic functions here, having covered the integration and differentiation of trigonometric functions previously.

1.2.1 Exponential Functions

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.3: Derivatives of Exponential Functions and the Number e”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.3: Derivatives of Exponential Functions and the Number e” (PDF)

Instructions: Please click on the link above, and read Section 8.3 in its entirety (pages 441 through 447). This chapter recaps the definition of the number e and the exponential function and its behavior under differentiation and integration.

Studying this reading should take approximately 45 minutes.

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Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: The Natural Logarithmic Function” and “The Exponential Function”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: The Natural Logarithmic Function” (QuickTime) and “The Exponential Function” (QuickTime)

Instructions: These lectures will cover sub-subunits 1.2.1 and 1.2.2. Please watch these two video lectures AFTER doing the readings for 1.2.1 and 1.2.2. Click on the first link, scroll down to Video 31: “The Natural Logarithmic Function,” and watch the presentation through the 5^th slide (marked 5 of 8). Next, click on the second link above, and scroll down to Video 32: “The Exponential Function.” Choose the format that is most appropriate for your Internet connection, and listen to the entire 21 minute video lecture.

The first short video gives one definition of the natural logarithm and derives all the properties of the natural log from that definition. It does a number of examples of limits, curve sketching, differentiation, and integration using the natural log. We will return to this video later to watch the last three slides. The second video explains the number e, the exponential function and its derivative and antiderivative, curve sketching using the exponential function, and how to perform similar operations on power functions with other bases using the change of base formula.

Viewing these lectures and note-taking should take approximately 45 minutes.

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Assessment: University of California, Davis: Duane Kouba’s “The Integration of Exponential Functions: Problems 1-12”
Link: University of California, Davis: Duane Kouba’s “The Integration of Exponential Functions: Problems 1-12” (HTML)

Instructions: Click on the link above and work through all of the assigned problems. When you are done, check your solutions with the answers provided.

Completing this assessment should take approximately 1 hour.

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1.2.2 Natural Logarithmic Functions

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.5: Natural Logarithms”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.5: Natural Logarithms” (PDF)

Instructions: Please click on the link above, and read Section 8.5 in its entirety (pages 454 through 459). This chapter reintroduces the natural logarithm (the logarithm with base e) and discusses its derivative and antiderivative. Recall that you can use these properties of the natural log to extrapolate the same properties for logarithms with arbitrary bases by using the change of base formula.

Studying this reading should take approximately 30 minutes.

Terms of Use: The book above is released under a Creative Commons Attribution-NonCommercial-Share-Alike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck’s “Logarithm, Definite Integrals”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck’s “Logarithms, Definite Integrals” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “3. Transcendental Functions,” and click button 137 (Logarithm, Definite Integrals). Do problems 1-10. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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1.2.3 Hyperbolic Functions

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.4: Some Uses of Exponential Functions”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.4: Some Uses of Exponential Functions” (PDF)

Instructions: Please click on the link above and read Section 8.4 in its entirety (pages 449 through 453). In this chapter, you will learn the definitions of the hyperbolic trig functions and how to differentiate and integrate them. The chapter also introduces the concept of capital accumulation.

Studying this reading should take approximately 15-20 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: Gaussian Technologies: GaussianMath.com’s “Hyperbolic Functions” and “Hyperbolic Functions – Derivatives”
Link: YouTube: Gaussian Technologies: GaussianMath.com’s “Hyperbolic Functions” (YouTube) and “Hyperbolic Functions – Derivatives” (YouTube)

Instructions: Click on the links above and watch the videos. The creator of the video pronounces “sinh” as “chingk.” The more usual pronunciation is “sinch.”

Viewing these lectures and pausing to take notes should take approximately 30 minutes.

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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Hyperbolic Functions”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Hyperbolic Functions” (PDF)

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Instructions: Please click on the link above, and work through each of the sixteen examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 1 hour and 30 minutes.

Terms of Use: The material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Unit 2: Applications of Integration

In this unit, we will take a first look at how integration can and has been used to solve various types of problems. Now that you have conceptualized the relationship between integration and areas and distances, you are ready to take a closer look at various applications; these range from basic geometric identities to more advanced situations in Physics and Engineering.

Unit 2 Time Advisory show close

Unit 2 Learning Outcomes show close

2.1 The Area between Curves

Suppose you want to find the area between two concentric circles. How would you do this? Logic dictates that you subtract the area of the smaller circle from that of the larger circle. As this subunit will demonstrate, this method also works when you are trying to determine the area between curves.

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.5: Areas between Two Curves”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.5: Areas between Two Curves” (PDF)

Instructions: Please click on the link above, and read Section 4.5 in its entirety (pages 218 through 222).

Studying this reading should take approximately 30 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: MIT: David Jerison’s “Lecture 21: Applications to Logarithms and Geometry”
Link: YouTube: MIT: David Jerison’s “Lecture 21: Applications of Logarithms and Geometry” (YouTube)

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Instructions: Please watch the segment of this video lecture from time 21:30 minutes through the end. Note that lecture notes are available in PDF; the link is on the same page as the lecture. In this lecture, Dr. Jerison will explain how to calculate the area between two curves.

Viewing this lecture and pausing to take notes should take approximately 45 minutes.

Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to MIT and the original version can be found here (Flash or MP4).
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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Aaron Robertson’s “Area between Curves I” and “Area between Curves II”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Aaron Robertson’s “Area between Curves I” (HTML) and “Area between Curves II” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “2. Applications of Integration,” and click button 115 (Area between Curves I). Do problems 6-13. Next, choose button 116 (Area between Curves II), and do problems 4-10. If at any time a problem set seems too easy for you, feel free to move on.

Completing these assessments should take approximately 1 hour.

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Assessment: Indiana University Southeast: Margaret Ehringe’s “Section 5.3 Area between Two Curves”
Link: Indiana University Southeast: Margaret Ehringe’s “Section 5.3 Area between Two Curves” (HTML)

Instructions: Click on the link above, and do problems 1-3 and 6-9. When you have finished, scroll down to the bottom of the page—the answers to the problems are printed upside-down at the end of the exercises.

The point of this third assessment is for you to practice setting up and completing these problems without the graphical aids provided by the Temple University media; you will have to graph these curves for yourself in order to begin the problems.

Completing this assessment should take approximately 30 minutes.

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2.2 Volumes of Solids

We often take basic geometric formulas for granted. (Have you ever asked yourself why the volume of a right cylinder is V=πhr²?) In this subunit, we will explore how some of these formulas were developed. The key lies in viewing solids as functions that revolve around certain lines. Consider, for example, a constant, horizontal line, and then imagine that line revolving around the x-axis (or any parallel line). The resulting shape is a right cylinder. We can find the volume of this figure by looking at infinitesimally thin “slices” and adding them all together. This concept enables us to calculate the volume of some extremely complex figures. In this subunit, we will learn how to do this in general; in the next, we will now take a look at two conventional methods for doing so when the figure has rotational symmetry.

Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Volumes I”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: Volumes I” (QuickTime)

Instructions: Please click on the link, scroll down to Video 27: “Volumes I,” and view the entire video. This video explains how to use integral calculus to calculate the volumes of general solids.

Viewing this lecture and pausing to take notes should take approximately 30 minutes.

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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Finding Volumes by Slicing”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Finding Volumes by Slicing” (PDF)

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Instructions: Please click on the link above, and work through each of the three examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 30 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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2.3 Volume of Solids of Revolution

When we are presented with a solid that was produced by rotating a curve around an axis, there are two sensible ways to take that solid apart: slice it thinly perpendicularly to the axis, into disks (or washers, if the solid had a hole in the middle), or peel layers from around the outside like the paper wrapper of a crayon. The latter method is known as the shell method and produces thin cylinders. In both cases, we find the area of the thin segments and add them up to find the volume; as usual, when we have infinitely many pieces, this “addition” is really integration.

2.3.1 Disks and Washers

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 6: Applications of the Integral: “Section 6.2: Volumes of Solids of Revolution”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 6: Applications of the Integral: “Section 6.2: Volumes of Solids of Revolution” (PDF)

Instructions: Please click on the link above, and read Section 6.2 in its entirety (pages 308 through 318). This reading will cover sub-subunits 2.3.1-2.3.2.

Studying this reading should take approximately 1 hour.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: MIT: David Jerison’s “Lecture 22: Volumes by Disks and Shells”
Link: YouTube: MIT: David Jerison’s “Lecture 22: Volumes by Disks and Shells” (YouTube)

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Instructions: Please click on the link above, and watch the entirety of this video. Note that lecture notes are available in PDF; the link is on the same page as the lecture. Dr. Jerison elaborates on some tangential material for a few minutes in the middle, but returns to the essential material very quickly. This lecture will cover the topics outlined for sub-subunits 2.3.1 and 2.3.2.

Viewing this lecture and pausing to take notes should take approximately 1 hour.

Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to MIT and the original version can be found here (Flash or MP4).
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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Aaron Robertson and Dan Birmajer’s “Solid of Revolution – Washers”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Aaron Robertson and Dan Birmajer’s “Solid of Revolution – Washers” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “2. Applications of Integration,” and click button 119 (Solid of Revolution – Washers). Do problems 1-12. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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2.3.2 Cylindrical Shells

Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Aaron Robertson and Dan Birmajer’s “Solid of Revolution – Shells”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Aaron Robertson and Dan Birmajer’s “Solid of Revolution – Shells” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “2. Applications of Integration,” and click button 120 (Solid of Revolution – Shells). Do problems 5-17. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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Assessment: The Saylor Foundation: Math Centre’s “Volumes: Exercises”
Link: The Saylor Foundation: Math Centre’s “Volumes: Exercises” (Flash)

Instructions: This assessment is for subunits 2.2 and 2.3; do not complete this assessment until you have worked through these subunits in their entirety. Click on the link above, and work through the exercises using the method you feel is most appropriate.

Completing this assessment should take approximately 30 minutes.

Terms of Use: The resource above is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 2.0 UK: England & Wales License (HTML). It is attributed to Math Centre, and the original version can be found here (Flash).
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2.4 Lengths of Curves

In this subunit, we will make use of another concept that you have known and understood for quite some time: the distance formula. If you want to estimate the length of a curve on a certain interval, you can simply calculate the distance between the initial point and terminal point using the traditional formula. If you want to increase the accuracy of this measurement, you can identify a third point in the middle and calculate the sum of the two resulting distances. As we add more points to the formula, our accuracy increases: the exact length of the curve will be the sum (i.e. the integral) of the infinitesimally small distances.

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 6: Applications of the Integral: “Section 6.3: Length of a Curve”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 6: Applications of the Integral: “Section 6.3: Length of a Curve” (PDF)

Instructions: Please click on the link above, and read Section 6.3 in its entirety (pages 319 through 325). This reading discusses how to calculate the length of a curve, also known as arc length. This includes calculating arc length for parametrically-defined curves.

Studying this reading should take approximately 45 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: MIT: David Jerison’s “Lecture 31: Parametric Equations, Arclength, Surface Area”
Link: YouTube: MIT: David Jerison’s “Lecture 31: Parametric Equations, Arclength, Surface Area” (YouTube)

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Instructions: Please watch this video lecture from the beginning up to time 26:10 minutes. Note that lecture notes are available in PDF; the link is on the same page as the lecture.

Viewing this lecture and pausing to take notes should take approximately 45 minutes.

Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to MIT and the original version can be found here (Flash or MP4).
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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Russo’s “Arc Length”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Russo’s “Arc Length” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “2. Applications of Integration,” and click button 125 (Arc Length). Do all problems (1-9). If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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2.5 Surface Areas of Solids

In this subunit, we will combine what we learned earlier in this unit. Though you might expect that calculating the surface area of a solid will be as easy as finding its volume, it actually requires a number of additional steps. You will need to find the curve-length for each of the “slices” we identified earlier and then add them together.

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 6: Applications of the Integral: “Section 6.4: Area of a Surface of Revolution”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 6: Applications of the Integral: “Section 6.4: Area of a Surface of Revolution” (PDF)

Instructions: Please click on the link above, and read Section 6.4 in its entirety (pages 327 through 335). In this beautiful presentation of areas of surfaces of revolution, the author again makes use of rigorously-defined infinitesimals, as opposed to limits. Recall that the approaches are equivalent; using an infinitesimal is the same as using a variable and then taking the limit as that variable tends to zero.

Studying this reading should take approximately 1 hour.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: MIT: David Jerison’s “Lecture 31: Parametric Equations, Arclength, Surface Area”
Link: YouTube: MIT: David Jerison’s “Lecture 31: Parametric Equations, Arclength, Surface Area” (YouTube)

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Instructions: Please watch this video lecture from time 26:10 minutes to time 40:35. Note that lecture notes are available in PDF; the link is on the same page as the lecture.

Viewing this lecture and pausing to take notes should take approximately 15-20 minutes.

Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to MIT and the original version can be found here (Flash or MP4).
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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Areas of Surfaces of Revolution”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Areas of Surfaces of Revolution” (PDF)

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Instructions: Please click on the link above, and work through each of the three examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 15-20 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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2.6 Average Value of Functions

Note: You probably learned about averages (or mean values) quite some time ago. When you have a finite number of numerical values, you add them together and divide by the number of values you have added. There is nothing preventing us from seeking the average of an infinite number of values (i.e. a function over a given interval). In fact, the formula is intuitive: we add the numbers using an integral and divide by the range.

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 6: Applications of the Integral: “Section 6.5: Averages”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 6: Applications of the Integral: “Section 6.5: Averages” (PDF)

Instructions: Please click on the link above, and read Section 6.5 in its entirety (pages 336 through 340).

Studying this reading should take approximately 15-20 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: MIT: David Jerison’s “Lecture 23: Work, Average Value, Probability”
Link: YouTube: MIT: David Jerison’s “Lecture 21: Applications of Logarithms and Geometry” (YouTube)

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Instructions: Please watch this video lecture from the beginning up to time 30:00 minutes. Note that lecture notes are available in PDF; the link is on the same page as the lecture. In this lecture, Professor Jerison will explain how to calculate average values and weighted average values.

Viewing this lecture and pausing to take notes should take approximately 45 minutes.

Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to MIT and the original version can be found here (Flash or MP4).
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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Russo’s “Average Value of a Function”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Russo’s “Average Value of a Function” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “4. Assorted Application,” and click button 124 (Average Value). Do problems 3-11. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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2.7 Physical Applications

We will now apply what we have learned about integration to various aspects of science. You may know that in physics, we calculate “work” by multiplying the force of the work by the distance over which it is exerted. You may also know that density is related to mass and volume. But we now know that distance and volume are very much related to integration. In this subunit, we will explore these and other connections.

2.7.1 Distance

Reading: The Saylor Foundation: Whitman College: David Guichard’s Calculus: Chapter 9: Applications of Integration: “Section 9.2: Distance, Velocity, Acceleration”
Link: The Saylor Foundation: Whitman College: David Guichard’s Calculus: Chapter 9: Applications of Integration: “Section 9.2: Distance, Velocity, Acceleration” (PDF)

Instructions: Please click on the link above, and read the Section 9.2 in its entirety (pages 192 through 194).

Studying this reading should take approximately 15-20 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of David Guichard, and can be viewed in its original form here (PDF). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Web Media: The Saylor Foundation: UC College Prep’s Calculus BC II for AP: “Applications of Integrals: Displacement Versus Total Distance”
Link: The Saylor Foundation: UC College Prep’s Calculus BC II for AP: “Application of Antiderivatives & Definite Integrals” (YouTube)

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Instructions: Click on the link above, and watch the interactive lecture. You may want to have a pencil and paper close by, as you will be prompted to work on related problems during the lecture.

Viewing this lecture should take approximately 45 minutes.

Terms of Use: The resource above is released under a Creative Commons Attribution-NonCommercial-NoDerivs License (HTML). It is attributed to the University of California College Prep, and the original version can be found here (Java).
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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Displacement vs. Distance Traveled”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Displacement vs. Distance Traveled” (PDF)

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Instructions: Please click on the link above, and work through each of the three examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 30 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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2.7.2 Mass and Density

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Kiesler’s Elementary Calculus 6.6 “Some Applications to Physics”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Kiesler’s Elementary Calculus 6.6 “Some Applications to Physics” (PDF)

Instructions: Please click on the above link and read the indicated section (pages 341-351).

Studying this reading should take approximately 1 hour.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Web Media: The Saylor Foundation: UC College Prep’s Calculus BC II for AP: “Applications of Integrals: Center of Mass and Density”
Link: The Saylor Foundation: UC College Prep’s Calculus BC II for AP: “Applications of Integrals: Center of Mass and Density” (YouTube)

Also Available in:
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Instructions: Click on the link above and watch the interactive lecture. You may want to have a pencil and paper close by, as you will be prompted to work on related problems during the lecture.

Viewing this lecture should take approximately 45 minutes.

Terms of Use: The resource above is released under a Creative Commons Attribution-NonCommercial-NoDerivs License 3.0 (HTML). It is attributed to The Regents of the University of California and the original version can be found here (Java).
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2.7.3 Moments

Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Moments and Centers of Mass”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Moments and Centers of Mass” (PDF)

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Instructions: This assessment will test you on what you learned in sub-subunits 2.7.2 and 2.7.3. Please click on the link above, and work through each of the four examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 30 minutes.

Terms of Use: The material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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2.7.4 Work

Reading: The Saylor Foundation: Whitman College: David Guichard’s Calculus: Chapter 9: Applications of Integration: “Section 9.5: Work”
Link: The Saylor Foundation: Whitman College: David Guichard’s Calculus: Chapter 9: Applications of Integration: “Section 9.5: Work” (PDF)

Instructions: Please click on the link above, and read Section 9.5 in its entirety (pages 205 through 208). Work is a fundamental concept from physics roughly corresponding to the distance travelled by an object multiplied by the force required to move it that distance.

Studying this reading should take approximately 30 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of David Guichard, and can be viewed in its original form here (PDF). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Web Media: The Saylor Foundation: UC College Prep’s Calculus BC II for AP: “Applications of Integrals: Work Done Moving an Object”
The Saylor Foundation: UC College Prep’s Calculus BC II for AP: “Applications of Integrals: Work Done Moving an Object” (Youtube)

Also Available in:
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Instructions: Click on the link above, and watch the interactive lecture. You may want to have a pencil and paper close by, as you will be prompted to work on related problems during the lecture.

Completing this resource should take approximately 30 minutes.

Terms of Use: This video is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives License (HTML). It is attributed to University of California College Prep, and the original version can be found here (Java).
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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Work, Fluid Pressures, and Forces”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Work, Fluid Pressures, and Forces” (PDF)

Also Available in:
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Instructions: Please click on the link above, and work through each of the seven examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 1 hour.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Unit 3: Techniques and Principles of Integration

Until now, we have been spending the majority of our time on the integration of relatively simple functions (at least in comparison to some of the functions we discussed in Part I). In this unit, we will learn how to analyze more complex functions using more sophisticated machinery. This includes clever methods of substitution, guides to algebraic simplification, and integration by parts, as well as using tables of integration or approximating the integral numerically. We will also address how to manage integrals where either the integrand is discontinuous in the domain of integration, or the domain of integration is infinite.

Unit 3 Time Advisory show close

Unit 3 Learning Outcomes show close

3.1 Methods of Integration

So far, we have seen two ideas for computing integrals: directly apply a formula (adjusting for coefficients and using the sum rule to break apart polynomials and similar); or rewrite the integral in some way, either by algebraic manipulation or by substitution. This subunit expands the number of rewriting techniques at our disposal and adds a new technique entirely: integration by parts.

Unlike differentiation, integration requires a little more forethought or “creativity” in certain situations, as the correct implementation of integration methods is less obvious. In fact, there are often multiple correct methods to solve a complicated integral problem, though they may vary in difficulty. It may not be clear whether the approach you are using is correct until you are partway through the problem, so stick out your attempt until you either succeed or hit an obvious wall.

3.1.1 Integration by Parts

Though this formula may at first seem arbitrary, integration by parts is merely the product rule in reverse. With it, we use the information we have to determine what the initial functions were. Integration by parts is useful for integrands that are the product of two functions from different “families,” such as an exponential with a polynomial.

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.4: Integrals by Parts”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.4: Integration by Parts” (PDF)

Instructions: Please click on the link above, and read Section 7.4 in its entirety (pages 391 through 395). Integration by parts is a technique used to integrate more complicated combinations of functions. It is easy to derive – simply rearrange the product rule! Careful bookkeeping is essential for mastering this technique, so keep plenty of scrap paper on hand, use different variables if you have to perform integration by parts a second time in the same problem, and be neat. It will save you time in the end.

Studying this reading should take approximately 30 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: MIT: David Jerison’s “Lecture 30: Integration by Parts, Reduction Formulae”
Link: YouTube: MIT: David Jerison’s “Lecture 30: Integration by Parts, Reduction Formulae” (YouTube)

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Instructions: Please watch the segment of Lecture 30 from time 18:20 minutes through the end. Note that lecture notes are available in PDF; the links are on the same pages as the lectures.

Viewing this lecture and pausing to take notes should take approximately 45 minutes.

Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to MIT and the original version can be found here (Flash or MP4)
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Assessment: University of California, Davis: Duane Kouba’s “The Method of Integration by Parts: Problems 1-23”
Link: University of California, Davis: Duane Kouba’s “The Method of Integration by Parts: Problems 1-23” (HTML)

Instructions: Click on the link above and work through all of the assigned problems. When you are done, check your solutions with the answers provided.

Completing this assessment should take approximately 2 hours.

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3.1.2 Trigonometric Integration

Trigonometric identities will be used heavily in this and the next sub-subunit. If you wish to review trigonometric identities, they are covered in the first three sections of Unit 3 of Precalculus II (MA003). Pay particular attention to sin² x + cos² x = 1 and its counterparts for tan/sec and cot/csc, the half-angle formulas, and the double-angle formulas.

Trigonometric integration is a simplification method: if you are asked to integrate sin⁸ x cos x, you can substitute for sin x and be on your way. However, the situation is different if cos x also has a larger exponent. This sub-subunit covers methods to “whittle down” the exponents of such a problem until substitution applies.

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.5: Integrals of Powers of Trigonometric Functions”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.5: Integrals of Powers of Trigonometric Functions” (PDF)

Instructions: Please click on the link above, and read Section 7.5 in its entirety (pages 397 through 401).

Studying this reading should take approximately 30 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: MIT: Haynes Miller’s “Lecture 27: Trigonometric Integrals and Substitution” and “Lecture 28: Integration by Inverse Substitution; Completing the Square”
Link: YouTube: MIT: Haynes Miller’s “Lecture 27: Trigonometric Integrals and Substitution” (YouTube)

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and “Lecture 28: Integration by Inverse Substitution; Completing the Square” (YouTube)

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Instructions: Please click on the first link above, and watch the entirety of Lecture 27. Please watch Lecture 28 from the beginning up to time 15:50 minutes. Note that lecture notes are available in PDF; the links are on the same pages as the lectures. In these lectures, Dr. Miller will discuss how to integrate powers of trigonometric functions. At the end of Lecture 27, he will do an example related to trigonometric substitution, which will be the focus of the next section.

Viewing these lectures and pausing to take notes should take approximately 1 hour and 15 minutes.

Terms of Use: The videos above are released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). They are attributed to MIT and the original versions can be found here (Flash or MP4) and here (Flash or MP4).
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Assessment: University of California, Davis: Duane Kouba’s “The Integration of Trigonometric Integrals: Problems 1-27”
Link: University of California, Davis: Duane Kouba’s “The Integration of Trigonometric Integrals: Problems 1-27” (HTML)

Instructions: Click on the link above, and work through all of the assigned problems. When you are done, check your solutions with the answers provided.

Completing this assessment should take approximately 2 hours and 15 minutes.

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3.1.3 Trigonometric Substitution

Trigonometric substitution is a particular “inverse substitution” technique. In substitution as we have most commonly seen so far, the new variable is given as a function of the old variable; in inverse substitution, this relationship is reversed. In trigonometric substitution, we let our old variable be a trigonometric function of the new variable, chosen so the Pythagorean Theorem applies to simplify the original integrand. This technique is useful when you have a binomial you wish were a monomial, typically because your binomial is on the bottom of a fraction or under a radical.

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.6: Trigonometric Substitutions”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.6: Trigonometric Substitutions” (PDF)

Instructions: Please click on the link above, and read Section 7.6 in its entirety (pages 402 through 405).

Studying this reading should take approximately 15-20 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: MIT: Haynes Miller’s “Lecture 28: Integration by Inverse Substitution; Completing the Square”
Link: YouTube: MIT: Haynes Miller’s “Lecture 28: Integration by Inverse Substitution; Completing the Square” (YouTube)

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Instructions: Please watch Lecture 28 from time 15:50 minutes to the end. Note that lecture notes are available in PDF; the link is on the same page as the lecture.

Viewing this video and pausing to take notes should take approximately 45 minutes.

Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to MIT and the original version can be found here (Flash or MP4).
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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Methods of Integration”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Methods of Integration” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “4. Methods of Integration,” and click button 164 (Trigonometric Substitution). Do problems 1-11. Note that the first problem has been done for you as an example; just click through the example in order to get a sense of the setup of the module. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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3.1.4 More Techniques Using the Natural Logarithm

If y = f(x), then ln y = ln (f(x)). If f(x) involves powers, products, or quotients, taking the natural logarithm may allow us to simplify, making differentiation easier. On the other hand, the fact that the integral of du/u is ln u seems only narrowly applicable. However, certain functions that do not originally appear to be of the form du/u may be manipulated into that form. In particular, this approach allows us to integrate powers of tangent and secant more easily.

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.7: Derivatives and Integrals Involving ln(x)”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.7: Derivatives and Integrals Involving ln(x)” (PDF)

Instructions: Please click on the link above, and read Section 8.7 in its entirety (pages 469 through 473).

Studying this reading should take approximately 30 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: The Natural Logarithmic Function”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: The Natural Logarithmic Function” (QuickTime)

Instructions: Please click on the link, scroll down to Video 31: “The Natural Logarithmic Function,” and watch from the 6th slide (marked 6 of 8) to the end. Feel free to watch the entire video if you would like a refresher on some earlier concepts. This short video gives examples of how to use the properties of the natural log to compute some more complicated integrals.

Studying this lecture should take approximately 30 minutes.

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Assessment: University of California, Davis: Duane Kouba’s “The Integration of Rational Functions, Resulting in Logarithmic of Arctangent Functions: Problems 1-22”
Link: University of California, Davis: Duane Kouba’s “The Integration of Rational Functions, Resulting in Logarithmic or Arctangent Functions: Problems 1-22” (HTML)

Instructions: This assessment will cover sub-subunits 3.1.3 and 3.1.4. Click on the link above, and work through all of the assigned problems. When you are done, check your solutions with the answers provided.

Completing this assessment should take approximately 2 hours and 15 minutes.

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3.1.5 Rational Functions and Integration with Partial Fractions

You first learned about partial fractions in precalculus, when you learned to re-write the quotient of two rational functions when the denominator function can be written as the product of smaller factors. Here, we use these methods to split apart a complicated fraction into the sum of simpler fractions – in particular, fractions we know how to integrate.

Polynomial factoring and long division will be used in this sub-subunit. For a review of those topics, see Beginning Algebra (MA001), Unit 4 and subunit 3.4, respectively.

Reading: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan’s and Donald J. Lewis’s Calculus and Linear Algebra Vol. 1: “4-10 Partial Fractions Expansions of Rational Functions”
Link: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan’s and Donald J. Lewis’s Calculus and Linear Algebra Vol.1: “4-10 Partial Fractions Expansions of Rational Functions” (HTML)

Instructions: Please click on the link above, and read the assigned section. Use the “previous” and “next” links at the bottom of the webpage to navigate through this reading. This reading explains how to expand a rational function in terms of fractions of simpler rational functions. If you are not familiar with this method, this reading will be essential for understanding the rest of this section.

Studying this reading should take approximately 30 minutes.

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Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.8: Integration of Rational Functions”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.8: Integration of Rational Functions” (PDF)

Instructions: Please click on the link above, and read Section 8.8 in its entirety (pages 474 through 480). Once you have mastered partial fractions, there are established methods for integrating rational functions. Careful, neat work will help you enormously in this section.

Studying this reading should take approximately 45 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: MIT: David Jerison’s “Lecture 29: Partial Fractions” and “Lecture 30: Integration by Parts, Reduction Formulae”
Link: YouTube: MIT: David Jerison’s “Lecture 29: Partial Fractions” (YouTube)

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and “Lecture 30: Integration by Parts, Reduction Formulae” (YouTube)

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Instructions: Please click on the first link above, and watch the entirety of Lecture 29. Please watch the segment of Lecture 30 from the beginning to time 18:20. Note that lecture notes are available in PDF; the links are on the same pages as the lectures.

Viewing these lectures and pausing to take notes should take approximately 1 hour and 30 minutes.

Terms of Use: The videos above are released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). They are attributed to MIT and the original versions can be found here (Flash or MP4) and here (Flash or MP4).
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Assessment: University of California, Davis: Duane Kouba’s “The Method of Integration by Partial Fractions: Problems 1-20”
Link: University of California, Davis: Duane Kouba’s “The Method of Integration by Partial Fractions: Problems 1-20” (HTML)

Instructions: Click on the link above, and work through all of the assigned problems. You will need to scroll down the page a bit to get to the problems. When you are done, check your solutions with the answers provided.

Completing this assessment should take approximately 2 hours.

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3.2 Integration with Tables & CAS (Computer Algebra Systems)

In theory, one could use the techniques we have learned so far to integrate any integrable function. However, using tables allows us to avoid “reinventing the wheel” and re-deriving the techniques for more complicated but standard integrands by summarizing the results of the process.

Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Integration Techniques: Using Integral Tables”
Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Integration Techniques: Using Integral Tables” (HTML)

Instructions: Please click the link above and read this entire section. In this section, Professor Dawkins gives some helpful hints about using integral tables to compute integrals quickly; this involves reducing whatever problem you are faced with to a problem in the tables. He bases his discussion on the tables in the textbook used by his classes. However, it is easy to find tables of integrals on the Internet; we list several at the end of this page.

Studying this reading should take approximately 1 hour.

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3.3 Numerical Integration

With numerical integration, we abandon the antiderivative and work with modifications of the Riemann sum. This is possible to do for any integrand, whereas antiderivatives do not exist for every integrand. In most cases, we may also compute an error bound, allowing us to approximate the definite integral to any degree of accuracy we like (provided we have time to carry out all of the computations).

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.6 “Numerical Integration”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.6: Numerical Integration” (PDF)

Instructions: Please click on the link above, and read Section 4.6 in its entirety (pages 224 through 233). In MA101, we used this text to introduce the trapezoidal rule. You will review that method for numerical integration and also learn about Simpson’s Rule in this reading. These approximation methods are used by mathematical software to calculate the values of definite integrals to very high degrees of accuracy.

Studying this reading should take approximately 1 hour.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: MIT: David Jerison’s “Lecture 24: Numerical Integration” and “Lecture 25: Exam Review”
Link: YouTube: MIT: David Jerison’s “Lecture 24: Numerical Integration” (YouTube)

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and “Lecture 25: Exam Review” (YouTube)

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Instructions: Please click on the first link above, and watch the segment of Lecture 24 from time 33:50 minutes through the end. Then, click on the second link above, and watch Lecture 25 from the beginning up to time 14:04 minutes. Note that lecture notes are available in PDF; the links are on the same pages as the lectures. In these lectures, Dr. Jerison will discuss motivation and methods for numerical integration, including the trapezoidal rule and Simpson’s rule.

Viewing these lectures and pausing to take notes should take approximately 45 minutes.

Terms of Use: The videos above are released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). They are attributed to MIT and the original versions can be found here (Flash or MP4) and here (Flash or MP4).
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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Numerical Integration”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Numerical Integration” (PDF)

Also Available in:
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Instructions: Please click on the link above, and work through each of the six examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 1 hour.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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3.4 Improper Integrals

In previous integration examples, we have either ignored the domain of the function (i.e. antiderivatives) or integrated over an interval without any discontinuities. But what if we want to integrate over an interval tending toward infinity, or what if we want to find the area under a curve on an interval with a vertical asymptote? This subunit will introduce you to these integrals, which we refer to as “improper integrals.”

3.4.1 Type I

Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Integration Techniques: Improper Integrals”
Link: Lamar University: Paul Dawkin’s Paul’s Online Math Notes: Calculus II: “Integration Techniques: Improper Integrals” (HTML)

Instructions: This reading will cover sub-subunits 3.4.1 and 3.4.2. Please click the link above, and read this entire section. The two types of improper integrals are Type I, those with “infinite” limits of integration, and Type II, those where the integrand has infinite discontinuities somewhere in the interval of integration. Professor Dawkins does not use this terminology, although it is common, but he does discuss how to deal with each type of improper integral.

Studying this reading should take approximately 1 hour.

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Lecture: YouTube: MIT: David Jerison’s “Lecture 36: Improper Integrals” and “Lecture 37: Infinite Series and Convergence Tests”
YouTube: MIT: David Jerison’s “Lecture 36: Improper Integrals” (YouTube)

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and “Lecture 37: Infinite Series and Convergence Tests” (YouTube)

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Instructions: These lectures will cover sub-subunits 3.4.1 and 3.4.2. Please click on the first link above, and watch Lecture 36 from time 3:22 minutes to the end. Please click on the second link above, and watch Lecture 37 from the beginning up to time 17:35 minutes. Note that lecture notes are available in PDF; the link is on the same page as the lecture. In these lectures, Professor Jerison will discuss how to estimate as well as evaluate improper integrals of both types.

Viewing these lectures and pausing to take notes should take approximately 1 hour and 15 minutes.

Terms of Use: The videos above are released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). They are attributed to MIT and the original versions can be found here (Flash or MP4) and here (Flash or MP4).
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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s “Improper Integrals over Unbounded Intervals”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s “Improper Integrals over Unbounded Intervals” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “6. Improper Integrals,” and click button 172 (Improper Integrals over Unbounded Intervals). Do problems 1-18. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 2 hours.

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3.4.2 Type II

Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s “Improper Integrals of Unbounded Functions”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s “Improper Integrals of Unbounded Functions” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “6. Improper Integrals,” and click button 173 (Improper Integrals of Unbounded Functions). Do all 16 problems. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour and 30 minutes.

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Unit 4: Parametric Equations and Polar Coordinates

So far we have worked in Cartesian (rectangular) coordinates where there has been one dependent variable, say x, and one dependent variable y=f(x). At times this has been inconvenient. Think about the equation describing the graph of a circle, x²+ y² = r²: here, y cannot be given as an explicit function of x. For situations like this one there are other ways of describing graphs which make calculations much simpler.

You studied parametric equations and polar coordinates in subunits 4.2 and 4.5 of Precalculus II (MA003), so for an alternate approach, you may review those subunits. Make sure to come back to this unit, because MA003 does not cover as much material.

Unit 4 Time Advisory show close

Unit 4 Learning Outcomes show close

4.1 Parametric Equations and Their Derivatives

Parametric equations treat x and y each as functions of a third variable, typically t. It is helpful to think of t as time, and the equations as instructing how a curve is to be drawn, giving the pen’s coordinates for each point in time. This easily extends to curves in three-dimensional space by adding an equation for z as a function of t.

Reading: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan’s and Donald J. Lewis’s Calculus and Linear Algebra Vol. 1: “3-9 Parametric Equations”
Link: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan’s and Donald J. Lewis’s Calculus and Linear Algebra Vol.1: “3-9 Parametric Equations” (HTML)

Instructions: Please click on the link above, and read the assigned section. Use the “previous” and “next” links at the bottom of each webpage to navigate through the reading. This reading discusses parametric equations for curves and how to differentiate them. Note the similarity to related rates. Ignore the reference in the first paragraph to vector equations; that is Calculus III material.

Studying this reading should take approximately 30 minutes.

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Lecture: Khan Academy's “Parametric Equations I” and “Parametric Equations II”; MIT: David Jerison’s “Lecture 31: Parametric Equations, Arclength, Surface Area” and “Lecture 32: Polar Coordinates; Area in Polar Coordinates”
Khan Academy's “Parametric Equations I” (YouTube) and “Parametric Equations II” (YouTube); MIT: David Jerison’s “Lecture 31: Parametric Equations, Arclength, Surface Area” (YouTube)

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and “Lecture 32: Polar Coordinates; Area in Polar Coordinates” (YouTube)

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Instructions: Please click on the links above to watch Salman Khan’s “Parametric Equation I” and “Parametric Equations II.” Then, watch Professor Jerison’s Lecture 31 from time 40:35 minutes to the end and Lecture 32 from the beginning up to time 22:50 minutes. Note that lecture notes are available in PDF; the link is on the same page as the lecture.

Salman Khan’s first video gives a very intuitive example of the concept of parametric curves—two-dimensional motion with a time parameter. The second video shows how to eliminate the parameter and gives a second example. In the lectures from MIT, Professor Jerison will work through a more advanced example and discuss how to calculate arc length for curves expressed by parametric equations.

Viewing these lectures and pausing to take notes should take approximately 1 hour.

Terms of Use: The Khan videos above are released under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. They are attributed to Khan Academy.

The MIT videos above are released under a Creative Commons Attribution-Share-Alike License 3.0. They are attributed to MIT.
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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Parameterizations of Plane Curves” and “Calculus with Parameterized Curves”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Parameterization of Plane Curves” (PDF) and “Calculus with Parameterized Curves” (PDF)

Also Available in:
HTML (“Parameterization of Plane Curves”)
HTML (“Calculus with Parameterized Curves”)

Instructions: Please click on the first link above, and work through each of the nine examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example. Do the same with the second link; work through each of the eight examples on the page on your own before checking them with the given solutions.

Completing these assessments should take approximately 2 hours.

Terms of Use: The material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML) and here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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4.2 Polar Coordinates

Note: Polar Coordinates are both a different coordinate system to describe two-dimensional space and, when related back to the “x-y plane,” a different parameterization for curves in that system. Instead of representing location by horizontal and vertical distance from the origin, we represent it by straight-line distance from the origin and angle from the positive x-axis (measured counter-clockwise).

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.7: Polar Coordinates”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.7: Polar Coordinates” (PDF)

Instructions: Please click on the link above, and read Section 7.7 in its entirety (pages 406 through 411). Polar coordinates use two parameters, angle and radius, to describe the graphs of curves.

Studying this reading should take approximately 45 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Polar Coordinates and Graphs”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: Polar Coordinates and Graphs” (QuickTime)

Instructions: This lecture will cover subunits 4.2 and 4.3. Please click on the link, scroll down to Video 41: “Polar Coordinates and Graphs,” and watch the entire lecture. In this video, you will learn how to graph a number of well-known figures in polar coordinates, such as cardioids, roses, and limaçons. You will also learn more about derivatives and tangent lines in polar coordinates.

Viewing this lecture and pausing to take notes should take approximately 45 minutes.

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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Plotting Points in Polar Coordinates”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Plotting Points in Polar Coordinates” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “5. Geometry, Curves, and Polar Coordinates,” and click button 181 (Sketching Polar Curves). Do five of the problems in the module as they are presented to you. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 30 minutes.

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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Polar Coordinates”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Polar Coordinates” (PDF)

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Instructions: Please click on the link above, and work through each of the eighteen examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 2 hours.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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4.3 Derivatives and Curve Sketching in Polar Coordinates

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.8: Slopes and Curve Sketching in Polar Coordinates”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.8: Slopes and Curve Sketching Polar Coordinates” (PDF)

Instructions: Please click on the link above, and read Section 7.8 in its entirety (pages 412 through 419). Here, you will learn tips and tricks for graphing equations in polar coordinates and discover how to take derivatives of such functions.

Studying this reading should take approximately 45 minutes.

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Activity: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Sketching Curves in Polar Coordinates”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Sketching Curves in Polar Coordinates” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “5. Geometry, Curves, and Polar Coordinates,” and click button 183 (Sketching Polar Curves). Do each of the problems (six in total). This is an exploratory graphing assessment which is more like an applet. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 45 minutes.

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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Graphing in Polar Coordinates”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Graphing in Polar Coordinates” (PDF)

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Instructions: Please click on the link above, and work through each of the eight examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 1 hour.

Terms of Use: The material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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4.4 Areas with Polar Coordinates

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.9: Area in Polar Coordinates”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.9 Area in Polar Coordinates” (PDF)

Instructions: Please click on the link above, and read Section 7.9 in its entirety (pages 420 through 424).

Studying this reading should take approximately 30 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: MIT: David Jerison’s “Lecture 33: Exam 4 Review”
Link: YouTube: MIT: David Jerison’s “Lecture 33: Exam 4 Review” (YouTube)

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Instructions: Please watch Lecture 33 from the beginning to time 34:58. Note that lecture notes in PDF are available for this video; the link is on the same page as the lecture. In this lecture, Professor Jerison will touch on computing area under curves described by polar coordinates and also do several more examples of curve sketching in polar coordinates.

Viewing this lecture and pausing to take notes should take approximately 45 minutes.

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Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Areas and Length Using Polar Coordinates”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: Areas and Lengths Using Polar Coordinates” (QuickTime)

Instructions: This lecture will cover subunits 4.4 and 4.5. Please click on the link, scroll down to Video 42: “Areas and Lengths Using Polar Coordinates,” and watch the entire video. This video discusses area (slides 1-5) and arc length (slides 7-9) in polar coordinates.

Viewing this video lecture should take approximately 30 minutes.

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4.5 Arc Length with Polar Coordinates

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.10: Length of a Curve in Polar Coordinates”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 7: Trigonometric Functions: “Section 7.10: Length of a Curve in Polar Coordinates” (PDF)

Instructions: Please click on the link above, and read Section 7.10 in its entirety (pages 425 through 427).

Studying this reading should take approximately 15-20 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Integration in Polar Coordinates”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Integration in Polar Coordinates” (PDF)

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Instructions: Please click on the link above, and work through each of the nine examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example. The examples cover both Arc Length and Area in Polar Coordinates.

Completing this assessment should take approximately 1 hour.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Unit 5: Infinite Sequences and Series

In this unit, you will become acquainted with the infinite lists called sequences and infinite sums called series. The main question for each is whether it converges: do the terms of the sequence have a finite limit? Do the series terms have a finite sum? You will learn ways to test for convergence or divergence. After learning a number of such tests, we will look at Taylor series, which are infinite polynomials. Any function that may be differentiated an unlimited number of times gives rise to a Taylor series, whose partial sums are approximations to the function using ever higher-order derivatives. We will consider questions like: for which values of the variable does the series converge? For those values, is it equal to the function from which it was defined?

Many students find series the most difficult of the topics in Calculus II. There are multiple expositions of each topic included in this unit, so be patient with yourself and study each resource carefully.

Unit 5 Time Advisory show close

Unit 5 Learning Outcomes show close

5.1 Sequences

A sequence is merely a list of terms (usually numbers) that are arranged in a particular order. In this subunit, we will look at a sequence of numbers ordered by some rule or function.

Reading: The Saylor Foundation: Whitman College: David Guichard’s Calculus: Chapter 11: Sequences and Series: “Section 11.1 Sequences”
Link: The Saylor Foundation: Whitman College: David Guichard’s Calculus: Chapter 11: Sequences and Series: “Section 11.1: Sequences” (PDF)

Instructions: Please click on the link above, and read the brief introduction to chapter 11 and Section 11.1 (pages 253 through 260).

Studying this reading should take approximately 1 hour.

Terms of Use: The linked material above has been reposted by the kind permission of David Guichard, and can be viewed in its original form here (PDF). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Sequences I” and “Sequences II”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: Sequences I” (QuickTime) Lecture and “Sequences II” (QuickTime)

Instructions: Please click on the link, scroll down to Video 47: “Sequences I,” and watch the entire video. Next, scroll down to Video 48: “Sequences II,” and watch it from the beginning through the 7^th slide (the end of the slide marked 7 of 12). If you are interested, feel free to watch the rest of the video.

In the first video, Dr. Hollis discusses sequences and limits, goes over several important limits, and explains growth rates and order comparisons. In the second video, he gives a precise definition of limits, shows how to do an epsilon-N proof for limits of sequences, and discusses boundedness and monotonicity. In the optional slides (8-12), he discusses recursively-defined sequences, fixed points, and cobweb plots.

Viewing these lectures and note-taking should take approximately 1 hour.

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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “Limits of Sequences”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “Limits of Sequences” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “1. Sequences,” and click button 184 (Limits of Sequences). Do problems 1-22. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 2 hours.

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5.2 Series

A series is the sum of the terms in an infinite sequence. You are likely familiar with series arranged in an arithmetic or geometric progression; this subunit will take a look at terms defined by more intricate functions. You can also view a series as another type of sequence – a sequence of partial sums. In the following readings, you will learn what it means for a series to converge and study some important types of series.

5.2.1 Series and Basic Convergence

Reading: The Saylor Foundation: Whitman College: David Guichard’s Calculus: Chapter 11: Sequences and Series: “Section 11.2 Series”
Link: The Saylor Foundation: Whitman College: David Guichard’s Calculus: Chapter 11: Sequences and Series: “Section 11.2: Series” (PDF)

Instructions: Please click on the link above, scroll down and read Section 11.2 in its entirety (pages 260 through 263). This section will introduce you to infinite series and make mention of the geometric series, which will be discussed in more detail below. It also explains what it means for a series to converge.

Studying this reading should take approximately 15-20 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of David Guichard, and can be viewed in its original form here (PDF). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Series”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: Series” (QuickTime)

Instructions: This lecture will cover sub-subunits 5.2.1-5.3.1. Please click on the link, scroll down to Video 49: “Series,” and watch the entire video. You are welcome to break it into parts as you go along. Slides 1-4 correspond roughly to 5.2.1; slides 5-7 correspond roughly to 5.2.3; slides 8-11 are an exposition of 5.2.2; and slide 12 corresponds to 5.3.1.

Viewing this lecture and note-taking should take approximately 30 minutes.

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5.2.2 Properties of Infinite Series

Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II “Special Series”
Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Special Series” (HTML)

Instructions: Please click on the link above, and read the information on this webpage. You may skip the section on telescoping series; you are not responsible for that material. However, pay attention to the beginning through the discussion following Example 2, pick up again at the paragraph before Example 5, and read to the end. The notion that any finite number of terms has no effect on the convergence behavior of a series is important and can save you a lot of effort.

Studying this reading should take approximately 30 minutes.

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5.2.3 Focus on the Geometric Series

A series is geometric if every successive term is the product of the previous term with a fixed value called the ratio of the series. For example, 3 + 6 + 12 + 24 + ... is a geometric series with ratio 2. Geometric series are unusual in that not only can we easily determine convergence or divergence for them, but in the case of convergence, we can also find the sum of the series exactly.

Reading: The Saylor Foundation: MIT: Gilbert Strang’s Calculus: Chapter 10: Infinite Series: “Section 10.1: The Geometric Series”
Link: The Saylor Foundation: MIT: Gilbert Strang’s Calculus: Chapter 10: Infinite Series: “Section 10.1: The Geometric Series” (PDF)

Instructions: Read the several paragraphs at the beginning of chapter 10 (the discussion of the geometric series begins here) and section 10.1 (pages 366-372).

Studying this reading should take approximately 1 hour.

Terms of Use: The article above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML). It is attributed to Professor Gilbert Strang (MIT) and the original version can be found here (PDF).
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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Russo’s “Geometric Series”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Russo’s “Geometric Series” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “1. Integration,” and click button 101 (Geometric Series). Do problems 2-10. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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5.2.4 Highlight: The Binomial Series

If you wish to expand (1 + x)² or (1 + x)³, you may simply multiply them out. But what are the coefficients of the fifteen terms in the expansion of (1 + x)¹⁴? The binomial theorem provides an answer in terms of combinations, also known as binomial coefficients. The binomial series takes this even a step further, allowing the expansion of expressions such as 1/(1 + x) and (1 + x)^1/2 to infinite polynomials.

Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Sequences and Series: Binomial Series”
Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Sequences and Series: Binomial Series” (HTML)

Instructions: Please click the link above, and read this entire section. Recall that combinations arise from the problem of counting subsets of a collection of objects: the number of ways to choose two of the four letters ABCD is the combination 4 choose 2, which is 6. If you have not seen it before, the connection to powers of binomials may not be clear. To expand (1 + x)², you write it as two copies of 1 + x and multiply one term from the first copy by one term from the second copy, using every possible pairing exactly once. Likewise, to expand (1 + x)⁴, you must multiply across the four copies of (1 + x), taking every possible quartet exactly once: all the 1’s, the first two 1’s and the last two x’s, the first two x’s and the last two 1’s, etc. Combinations allow you to find the coefficients, because, for example, the coefficient of x² in the expansion of (1 + x)⁴ is 6: the number of quartets that contain exactly two x’s, or in other words the number of ways to select the two copies of (1 + x) that provide the x’s.

The combination k choose n is typically written as the fraction (k!)/(n!(k-n)!). Professor Dawkins writes it the way he does so it will generalize to negative and non-integer values of k.

Studying this reading should take approximately 30 minutes.

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5.3 Test for Convergence of Positive Series

You will first learn how to check when a series with only positive terms converges – i.e. the limit of its sequence of partial sums exists and is finite. The theory begins here with positive series, because it is the simplest problem to consider; these tests often work by “squeezing” the partial sums between zero and some larger series which are known to converge.

It is important to notice many of the tests related to series are one-directional implications: for example, if the series terms do not limit to zero, you can conclude the series diverges. However, if it lacks that property, then you need more information to conclude either convergence or divergence. Be careful not to assume a lack of one conclusion implies the opposite conclusion.

5.3.1 Divergence Test

Convergence of a series is convergence of its sequence of partial sums. That is, for the series to converge, the partial sums must settle down and overall get closer and closer to a fixed finite value. In order for that to happen, the amount being added to each partial sum to produce the next one must gradually shrink away to nothing. That is the idea of the divergence test, which applies to any series (not just those with all positive terms): if the limit of the terms of the series is not zero, the series diverges.

This is not an equivalence, however! Many divergent series have terms that limit to zero. The terms must shrink to zero rapidly to give convergence. However, whether the terms shrink to zero at all is straightforward to check and may save you work making more complicated tests on a divergent series.

Reading: The Saylor Foundation: MIT: Gilbert Strang’s Calculus: Chapter 10: Infinite Series: “Section 10.2: Convergence Tests: Positive Series”
Link: The Saylor Foundation: MIT: Gilbert Strang’s Calculus: Chapter 10: Infinite Series: “Section 10.2 Convergence Tests: Positive Series” (PDF)

Instructions: This reading will cover sub-subunits 5.3.1 – 5.3.6. Please click on the link above, and read Section 10.2 in its entirety (pages 374 through 379).

The section presents a criterion for divergence, the integral test, the comparison test, and the ratio and root tests. We will revisit all these tests later in a slightly different context and give a more thorough justification of the last two tests.

Studying this reading should take approximately 45 minutes.

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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Divergence Test”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Divergence Test” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “2. Series,” and click button 187 (The Divergence Test). Do problems 1-10. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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5.3.2 Integral Test

There is an imprecise correspondence between sequences and functions as well as between series and integrals. The integral test shows this correspondence; though this relationship is not perfect, it is close enough to be useful.

Lecture: YouTube: MIT: David Jerison’s “Lecture 37: Infinite Series and Convergence Tests”
Link: YouTube: MIT: David Jerison’s “Lecture 37: Infinite Series and Convergence Tests” (YouTube)

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Instructions: This lecture will cover sub-subunits 5.3.2 - 5.3.4. Please watch Lecture 37 from time 17:35 minutes to the end. Note that lecture notes are available in PDF; the link is on the same page as the lecture. Professor Jerison will begin his discussion of infinite series with the series [Sum of 1 divided by (n squared)]. He introduces important terminology and then proves the convergence of the above series by comparison with the integral of the summand. He extends this argument to state the integral test. Finally, he goes over the limit comparison test for positive sequences.

Viewing this lecture and pausing to take notes should take approximately 45 minutes.

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Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: The Integral Test”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: The Integral Test” (QuickTime)

Instructions: Please click on the link above, scroll down to Video 50: “The Integral Test,” and watch this entire lecture. This short video restates the integral test in a more concise way and provides several other important applications of the integral test, such as proving the convergence of the p-series and estimating remainders of partial sums.

Viewing this lecture should take approximately 15 minutes.

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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Integral Test”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Integral Test” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “2. Series,” and click button 188 (The Integral Test). Do problems 1-10. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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5.3.3 Comparison Test

Series with no negative terms either have terms that get small enough or fast enough for the series to converge, or terms that remain too large and hence cause the series to diverge. It would seem logical that if Series A converges, and the n^th term of Series B is less than or equal to the n^th term of Series A for all n (at least after a finite number of terms), then Series B should converge: if A’s terms are small enough, B’s should also be. Likewise, a series with terms that are larger than the corresponding terms of a divergent series should diverge. This is true and is known as the (direct) comparison test.

Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Comparison Tests”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: Comparison Tests” (QuickTime)

Instructions: This lecture will cover sub-subunits 5.3.3-5.3.6. Please click on the link above, scroll down to Video 51: “Comparison Tests,” and watch the video in its entirety. This video states, proves, and applies the comparison, limit comparison, ratio, and root tests for positive series.

Viewing this lecture and pausing to take notes should take approximately 30 minutes.

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Assessment: The Saylor Foundation: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “5.4 Infinite Series: The Comparison Test”
Link: The Saylor Foundation: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “5.4 Infinite Series: The Comparison Test” (PDF)

Instructions: Please click the link above, and do problems 1 (a, c, e, g), 2 (a, c, e, g), and 4. When finished, click here for solutions (courtesy of the author’s blog).

Completing this assessment should take approximately 30 minutes.

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5.3.4 Ratio Test

The ratio and root tests are two ways of checking whether a series is “geometric in the limit” and thereby drawing conclusions about its behavior. The geometric series with terms arⁿ, a and r positive, has the properties that the ratio of the n+1^st term to the nth term is always r, and the n^th root of the n^th term is always r times the n^th root of a. The limit of each of those values as n tends to infinity is r. The ratio test and root test check the limits of those values as computed from other series. Although we lose some precision – i.e., a geometric series with ratio 1 diverges, but a limit of 1 in the ratio or root test is inconclusive – these tests greatly increase the number of series for which we can determine convergence or divergence. Typically, only one of these tests is algebraically feasible for a given series, but in the event both are, note that if one is inconclusive, the other will be as well.

Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Ratio Test”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Ratio Test” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “2. Series,” and click button 189 (The Ratio Test). Do problems 4-17. (See the navigation buttons below the problem.) If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour and 30 minutes.

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5.3.5 Root Test

Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The nth Root Test” Module
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The nth Root Test” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “2. Series,” and click button 190 (The nth Root Test). Do problems 3-17. (See the navigation buttons below the problem.) If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour and 30 minutes.

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5.3.6 Limit Comparison Test

Just as the ratio and root tests check whether a series is “geometric in the limit,” the limit comparison test checks whether two series are “equal in the limit,” up to a non-zero constant multiple. If so, we can conclude the series have the same convergence behavior.

Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: “The Limit Comparison Test”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “The Limit Comparison Test” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “2. Series,” and click button 191 (The Limit Comparison Test). Do problems 1-12. (See the navigation buttons below the problem.) If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s Supplemental Notes for Calculus II: “Ratio and Root Test for Series of Nonnegative Terms” and “Infinite Series”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s Supplemental Notes for Calculus II: “Ratio and Root Test for Series of Nonnegative Terms” (PDF) and “Infinite Series” (PDF)

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Instructions: This assessment will cover subunits 5.2-5.3. Please click on the first link above and work through each of the seven examples on the page. Next, please click on the second link and work through each of the thirteen examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 2 hours.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML) and here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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5.4 Tests for Absolute and Conditional Convergence

Thus far, we have worked with series with all non-negative terms. When series with negative terms are allowed, the picture changes slightly. If all terms are negative, of course, the series is simply -1 times a series of all positive terms and has the same convergence behavior as the positive series. If the terms are mixed sign, however, the positive and negative terms cancel each other out to some degree. For such series, we have essentially two options: the alternating series test, which requires the signs alternate, or to take the absolute value of each term and test that series for convergence.

5.4.1 Alternating Series Test

An alternating series is one in which the terms alternate between positive and negative. You may view it as a sequence of partial sums that alternately increase and decrease. The alternating series test says that if the magnitudes of the terms decrease to zero in the limit, then the series converges. For example, if every increase or decrease of the partial sums is smaller than the previous, and they limit to zero, the partial sums themselves have a finite limit. Alternating series that are easy to write down tend either to diverge by the divergence test or converge by the alternating series test, but be aware that it is easy to define an alternating series with terms limiting to zero (but not decreasing to zero) that diverges. For example, 1 −1/2 + 2/3 −1/3 + 1/2 −1/4 + …. This is the harmonic series in disguise, as you will see if you pair off each positive term with the negative term following it.

Reading: The Saylor Foundation: Whitman College: David Guichard’s Calculus: Chapter 11: Sequences and Series: “Section 11.4: Alternating Series”
Link: The Saylor Foundation: Whitman College: David Guichard’s Calculus: Chapter 11: Sequences and Series: “Section 11.4: Alternating Series” (PDF)

Instructions: Please click on the link above, locate Chapter 11, and read Section 11.4 in its entirety (pages 269 through 273).

Studying this reading should take approximately 30 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of David Guichard, and can be viewed in its original form here (PDF). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Alternating Series and Absolute Convergence”
Link: University of Houston: Selwyn Hollis’s “Video Calculus: Alternating Series and Absolute Convergence” (QuickTime)

Instructions: This lecture will cover the topics outlined in sub-subunits 5.4.1 and 5.4.2 as well as 5.4.4 and 5.4.5. Please click on the link above, scroll down to Video 52: “Alternating Series and Absolute Convergence,” and watch the video lecture in its entirety. This video explains alternating series, conditional and absolute convergence, and the ratio and root tests for absolute convergence.

Viewing this lecture and pausing to take notes should take approximately 30 minutes.

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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Alternating Series”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Alternating Series” (PDF)

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Instructions: Please click on the link above, and work through each of the five examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 30 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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5.4.2 Definition of Absolute and Conditional Convergence

As we have seen with the harmonic and alternating harmonic series, the cancellation effect of a mixture of positive and negative terms can be vital for the convergence of a series. The alternating harmonic is called a conditionally convergent series: its convergence is conditional on the cancellation effect. If it is possible to eliminate the cancellation (by taking the absolute value of each term) and still have convergence, the series is called absolutely convergent. When series have a mixture of positive and negative terms but do not alternate sign, taking the absolute value of the terms and testing the resulting series for convergence is often a good method. It cannot tell you if the original series diverges, but it can show if the original series converges, absolutely.

Reading: The Saylor Foundation: MIT: Gilbert Strang’s Calculus: Chapter 10: Infinite Series: “Section 10.3: Convergence Tests: All Series”
Link: The Saylor Foundation: MIT: Gilbert Strang’s Calculus: Chapter 10: Infinite Series: “Section 10.3: Convergence Tests: All Series” (PDF)

Instructions: This reading will cover sub-subunits 5.4.2 and 5.4.3. Please click on the link above, and read Section 10.3 in its entirety (pages 381 through 384). It is worth pointing out that if a series converges conditionally, it does not have a well-defined sum. By rearranging the terms, the value of the sum can be changed. This is an example of the fact that infinity is a strange place.

Studying this reading should take approximately 30 minutes.

Terms of Use: The article above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML). It is attributed to Professor Gilbert Strang (MIT) and the original version can be found here (PDF).
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5.4.3 Comparison Test for Absolute Convergence

Note: This topic is covered by the reading assigned below sub-subunit 5.4.1.

5.4.4 Limit Comparison Test for Absolute Convergence

Note: This topic is covered by the lecture assigned below sub-subunit 5.4.1.

5.4.5 Ratio Test for Absolute Convergence

Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Sequences and Series: Ratio Test”
Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Sequences and Series: Ratio Test” (HTML)

Instructions: Please click the link above, and read this entire section. This reading defines, applies, and proves the ratio test.

Studying this reading should take approximately 30 minutes.

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5.4.6 Root Test for Absolute Convergence

Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Sequences and Series: Root Test”
Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Calculus II: “Sequences and Series: Root Test” (HTML)

Instructions: Please click the link above, and read this entire section. This reading defines, applies, and proves the root test.

Studying this reading should take approximately 30 minutes.

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Assessment: The Saylor Foundation: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “5.6 Infinite Series: Absolute Convergence”
Link: The Saylor Foundation: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “5.6 Infinite Series: Absolute Convergence” (PDF)

Instructions: This assessment will cover sub-subunits 5.4.1-5.4.6. Please click the link above, and do problems 1 (a, c, e, g), 2 (a, c, e, g), and 3 (a, b, c). When finished, click h ere for solutions (courtesy of the author’s blog).

Completing this assessment should take approximately 45 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0. It is attributed to Dan Sloughter and the original version can be found here (HTML). Please respect the copyright and terms of use displayed on the solutions guide above.
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Assessment: Millersville University: Bruce Ikenaga’s “Absolute Convergence and Conditional Convergence”
Link: Millersville University: Bruce Ikenaga’s “Absolute Convergence and Conditional Convergence” (HTML)

Instructions: This assessment will cover sub-subunits 5.4.1-5.4.6. Please click on the link above and scroll down the page. Work through the last four examples before looking at their solutions.

Completing this assessment should take approximately 30 minutes.

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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Absolute and Conditional Convergence”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Absolute and Conditional Convergence” (PDF)

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Instructions: This assessment will cover sub-subunits 5.4.1-5.4.6. Please click on the link above, and work through each of the six examples on the page. As in any assessment, solve the problem on your own first. Detailed solutions are given beneath each example.

Completing this assessment should take approximately 45 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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5.5 Series Representations of Functions

In Single-Variable Calculus I (MA101), we learned that we can approximate a function about a point when we have information about the function’s value and its slope at that point. In this subunit, you will learn how to be even more accurate by gathering additional information about the function at a particular point (i.e. by using the second derivative, third derivative, fourth derivative, etc.). The more information you collect, the closer you will get to the function itself. The series representation of a function is the infinite series about a point, taking into consideration all of the derivatives about that point and in the form of a polynomial. This will enable us to look at functions, derivatives, and integrals in new and rather intuitive ways.

5.5.1 Power Series

A power series is any series where the n^th term contains xⁿ (where x as the name of the variable and the exact matching of the term index with the exponent are not essential). Essentially, it is an infinite polynomial in x.

Reading: The Saylor Foundation: Whitman College: David Guichard’s Calculus: “Chapter 11: Sequences and Series: Section 11.8: Power Series”
Link: The Saylor Foundation: Whitman College: David Guichard’s Calculus: “Chapter 11: Sequences and Series: Section 11.8: Power Series” (PDF)

Instructions: Please click on the link above and read Section 11.8 in its entirety (pages 278 through 281). A key term to understand in this section is radius of convergence.

Studying this reading should take approximately 15-20 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of David Guichard, and can be viewed in its original form here (PDF). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Lecture: YouTube: MIT: David Jerison’s “Lecture 38: Taylor Series” and Haynes Miller’s “Lecture 39: Final Review”
Link: YouTube: MIT: David Jerison’s “Lecture 38: Taylor Series” (YouTube)

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and Haynes Miller’s “Lecture 39: Final Review” (YouTube)

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Instructions: This lecture covers sub-subunits 5.4.1-5.4.3. After completing the readings for these sections, please click on the first link above, and watch Lecture 38 from time 22:45 minutes to the end. Please click on the second link above, and watch the entirety of Lecture 39. Note that lecture notes are available in PDF; the links are on the same pages as the lectures. In Lecture 38, Professor Jerison will discuss general power series before introducing Taylor Series. Professor Miller will continue this exposition; he will go over the derivations for the power series for the exponential, the sine, and the cosine before moving on to other examples.

Viewing these lectures and pausing to take notes should take approximately 1 hour and 15 minutes.

Terms of Use: The videos above are released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). They are attributed to MIT and the original versions can be found here (Flash or MP4) and here (Flash or MP4).
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Reading: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Power Series and the Uses of Power Series”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Power Series and the Uses of Power Series” (PDF)

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Instructions: Please click on the link above, and work through each of the six examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 45 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “Power Series”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “Power Series” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “3. Power and Taylor Series,” and click button 192 (Power Series). Do the problems in the module as they are presented to you (18 total). These problems all deal with computing the radius of convergence for a power series. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 2 hours.

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5.5.2 Calculus with Power Series

Happily, our infinite polynomials interact with integration and differentiation in the same way as finite polynomials: the integral of a sum is the sum of the integrals of each term, and likewise for derivatives. The bookkeeping aspects of this topic will be easier if you think of the “point of view” of the mathematical operators: the integral sign and derivative operator d/dx see x as a variable and n as a constant (a different fixed value for each series term). On the other hand, the summation operator sees n as the variable and x as the constant (a value that will be the same for every series term).

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 9: Infinite Series: “Section 9.8: Derivatives and Integrals of Power Series”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 9: Infinite Series: “Section 9.8: Derivatives and Integrals of Power Series” (PDF)

Instructions: Please click on the link above, and read Section 9.8 in its entirety (pages 533 through 539).

Under certain conditions, power series can be differentiated and integrated. Certain characteristics of the series may change, however, such as the interval of convergence. We have not been using Keisler’s text so far this unit, because hyperreals are not as helpful to the intuition for series as they are for integrals. In the last several sub-subunits, they do not appear except in the proof of Theorem 1, part (iii), on pages 537-538 of this section. That statement, that the radius of convergence remains the same when a power series is integrated or differentiated, is typically given in calculus texts without proof. Do not fret too much over the proof.

Studying this reading should take approximately 45 minutes to complete.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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5.5.3 Taylor and Maclaurin Series

Taylor series are a particular kind of power series, defined from a function. Maclaurin series are a particular kind of Taylor series. The definition allows you to expand any function into an infinite polynomial, which in many cases will be provably equal to the original function, and may be much easier to compute with.

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 9: Infinite Series: “Section 9.10: Taylor’s Theorem” and “Section 9.11: Taylor Series”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 9: Infinite Series: “Section 9.10: Taylor’s Theorem” (PDF) and “Section 9.11: Taylor Series” (PDF)

Instructions: Please click on the links above, and read Sections 9.10 and 9.11 (pages 547 through 560).

Taylor Series use the information provided by the derivatives of a function (slope of the tangent line, concavity, etc.) to approximate the function by a sequence of polynomials. Taylor’s Theorem tells how good this approximation is. Taylor Series are used extensively in higher mathematics, especially in numerical analysis.

Studying these readings should take approximately 2 hours.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Taylor and MacLaurin Series”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Taylor and MacLaurin Series” (PDF)

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Instructions: Please click on the link above, and work through each of the seven examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 1 hour.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “Taylor Series”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book III: Gerardo Mendoza’s “Taylor Series” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “3. Power and Taylor Series,” and click button 193 (Taylor Series). Choose at least 5 of the problems to do. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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5.5.4 Approximation by Power Series

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 9: Infinite Series: “Section 9.9: Approximations by Power Series”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 9: Infinite Series: “Section 9.9: Approximations by Power Series” (PDF)

Instructions: Please click on the link above, and read Section 9.9 in its entirety (pages 540 through 546). Approximation by power series is a very important topic; for instance, it is how calculators compute sines and cosines!

Studying this reading should take approximately 1 hour.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Web Media: The Saylor Foundation: UC College Prep’s Calculus BC II for AP: “Infinite Sequences and Series: Approximating Functions Using Polynomials” and “Infinite Sequences and Series: Applications of Taylor Series”
Link: The Saylor Foundation: UC College Prep’s Calculus BC II for AP: “Infinite Sequences and Series: Approximating Functions Using Polynomials” (Youtube) and “Infinite Sequences and Series: Applications of Taylor Series” (Youtube)

Also Available in:
Java (Approximating Functions Using Polynomials)
Java (Applications of Taylor Series)

Instructions: Click on the links above, and watch the interactive lectures. You may want to have a pencil and paper close by, as you will be prompted to work on related problems during the lecture.

Studying these lectures should take approximately 1 hour.

Terms of Use: The videos above are licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives License (HTML). They are attributed to University of California College Prep, and the original versions can be found here and here (Java).
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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Other Topics Related to Taylor Series”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Other Topics Related to Taylor Series” (PDF)

Also Available in:
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Instructions: Please click on the link above, and work through each of the five examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 45 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Unit 6: Differential Equations

This final unit will introduce the relationship between the mathematical machinery we have been developing and mathematical modeling. In practical situations, you will rarely have all of the information or data needed to represent an initial function. You will likely only have information about how the data changes. In this unit, you will learn how to apply what we know about functions and how they behave in order to model and interpret data.

Unit 6 Time Advisory show close

Unit 6 Learning Outcomes show close

6.1 First-Order Differential Equations

A differential equation represents the relationship between an unknown function and its various higher-order derivatives. The order of the relationship is defined by the highest-ordered derivative in the equation. In this subunit, we will only study equations involving an unknown function and its first derivative. We will leave higher-ordered differential equations for later courses.

Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Differential Equations: “Basic Concepts: Definitions”
Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Differential Equations: “Basic Concepts: Definitions” (HTML)

Instructions: Please click the link above, and read this entire section. This reading introduces you to the fundamental concepts of differential equations. Key words to remember are: order, linear differential equation, initial condition, and initial value problem.

Studying this reading should take approximately 45 minutes.

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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Differential Equations and Initial Value Problems”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Differential Equations and Initial Value Problems” (PDF)

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Instructions: Please click on the link above, and work through examples 1-5 on the page. As in any assessment, solve the problem on your own first. Detailed solutions are given beneath each example.

Completing this assessment should take approximately 45 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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6.2 Separation of Variables and Initial Value Problems

Note: Separation of variables is a method for solving certain types of differential equations. It is based on the assumption that we can “separate” our equation into two pieces: a function of the independent variable and a function of the dependent variable, with no occurrences of one in the other.

Reading: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.6: Some Differential Equations”
Link: The Saylor Foundation: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.6: Some Differential Equations” (PDF)

Instructions: Please click on the link above, and read Section 8.6 in its entirety (pages 461 through 468).The most basic differential equation is the separable equation; in this section, you will learn how to solve such equations.

Studying this reading should take approximately 1 hour.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).
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Lecture: YouTube: MIT: David Jerison’s “Lecture 16: Differential Equations, Separation of Variables”
Link: YouTube: MIT: David Jerison’s “Lecture 16: Differential Equations, Separation of Variables” (YouTube)

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Instructions: Please click on the link above, and watch the segment of this video lecture beginning at 1:50 minutes and ending at 43:20 minutes. Note that lecture notes are available in PDF; the link is on the same page as the lecture. Professor Jerison discusses a number of problems involving separation of variables. His first example is good, but somewhat more complicated than later examples.

Viewing this lecture and note-taking should take approximately 45 minutes.

Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). It is attributed to MIT and the original version can be found here (Flash or MP4).
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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: First Order Differential Equations”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: First Order Differential Equations” (PDF)

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Instructions: This assessment will cover subunits 6.1-6.2. Please click on the link above, and work through each of the six examples on the page. As in any assessment, solve the problem on your own first. Solutions are given beneath each example.

Completing this assessment should take approximately 45 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Hartenstine’s “Differential Equations”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Hartenstine’s “Differential Equations” (HTML)

Instructions: This assessment will cover subunits 6.1-6.2. Click on the link above. Then, click on the “Index” button. Scroll down to “2. Applications of Integration,” and click button 127 (Differential Equations). Do all problems (1-10). If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

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6.3 Slope Fields & Euler's Method

Recall that antiderivatives of functions are not unique; they differ from one another by constants. Thus, initial values are very important in determining solutions of differential equations. Slope fields, also called direction fields, are a way to capture the behavior of a whole family of solutions to a particular differential equation with different initial conditions. Euler’s Method is a way to approximate solutions to differential equations numerically; it is similar in flavor to Newton’s Method.

Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Differential Equations: “Basic Concepts: Direction Fields” and “First Order DEs: Euler’s Method”
Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Differential Equations: “Basic Concepts: Direction Fields” (HTML) and “First Order DEs: Euler’s Method” (HTML)

Instructions: Please click the links above, and read these webpages in their entirety.

Studying these webpages should take approximately 2 hours.

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Lecture: YouTube: MIT: Arthur Mattuck’s “Lecture 1: The Geometrical View of y’=f(x,y): Direction Fields, Integral Curves” and “Lecture 2: Euler’s Method for y’=f(x,y) and Its Generalizations”
Link: YouTube: MIT: Arthur Mattuck’s “Lecture 1: The Geometrical View of y’=f(x,y): Direction Fields, Integral Curves” (YouTube) and “Lecture 2: Euler’s Method for y’=f(x,y) and Its Generalizations” (YouTube)

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Instructions: Please click on the links above, and watch both lectures. In these videos, Professor Mattuck will explain the concept of direction fields and do several examples. He will state several important principles to keep in mind when sketching slope fields and integral curves and will outline Euler’s Method and discuss its error. These are the first and second lectures for a differential equations class.

Viewing these lectures and pausing to take notes should take approximately 2 hours.

Terms of Use: The videos above are released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). They attributed to MIT and the original versions can be found here (YouTube) and here (YouTube).
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Interactive Lab: MIT’s d’Arbeloff Interactive Math Project: “Euler’s Method” Applet
Link: MIT’s d’Arbeloff Interactive Math Project: “Euler’s Method” Applet (Java)

Instructions: This is an optional resource. Click on the link above to explore Euler’s method with this applet. If you need more guidance, click on the “Help” link in the upper right-hand corner of the applet.

You should dedicate approximately 30 minutes to exploring this resource.

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Assessment: MIT: Haynes Miller and Arthur Mattuck’s Spring 2004 Differential Equations Class: “Problem Set 1”
Link: MIT: Haynes Miller and Professor Arthur Mattuck’s Spring 2004 Differential Equations Class: “Problem Set 1” (PDF)

Instructions: Click on the link above, and then find the link to the problem set, marked “Problem Set 1.” Do the parts of problems 1 and 2 that are intended for pencil and paper, but ignore references to the “Mathlet.” When you are finished, click on the link above again, and find the link to the PDF with solutions to problem set 1.

Completing this assessment should take approximately 15-20 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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6.4 Exponential and Logistic Growth and Applications

A kind of bacteria might have the property where, given sufficient space, every individual produces one additional individual once an hour, so the population of bacteria doubles every hour. If the bacteria are in a petri dish, however, there are space limitations, and the bacteria may reproduce once per hour at first but taper off dramatically as the population grows close to filling the available space. An exponential function describes the former situation, and a logistic function describes the latter situation; such functions may also describe population decreases. The list of possible applications is long, including the decay of radioactive material, the temperature of melting ice or cooling coffee, the number of people who have heard a rumor, and the accrual of interest in a bank account.

6.4.1 Exponential and Logistic Growth

Web Media: The Saylor Foundation: UC College Prep’s Calculus BC II for AP: “Applications of Integrals: Exponential, Bounded Growth and Decay” and “Applications of Integrals: Logistic Equation and Population Growth”
Link: The Saylor Foundation: UC College Prep’s Calculus BC II for AP: “Applications of Integrals: Exponential, Bounded Growth and Decay” (YouTube) and “Applications of Integrals: Logistic Equation and Population Growth” (YouTube)

Also Available in:
Java (Exponential, Bounded Growth and Decay)
Java (Logistic Equations and Population Growth)

Instructions: Click on the links above, and watch the interactive lectures. You may want to have a pencil and paper close by, as you will be prompted to work on related problems during the lecture.

Studying these resources should take approximately 1 hour.

Terms of Use: The videos above are licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives License (HTML). They are attributed to University of California College Prep, and the original versions can be found here and here (Java).
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Reading: The Saylor Foundation: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “6.3 Models of Growth and Decay”
Link: The Saylor Foundation: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “6.3 Models of Growth and Decay” (PDF)

Instructions: This reading covers the topics outlined in sub-subunits 6.4.1-6.4.4. Please click the link above, and read the entirety of the section.

Completing this reading should take approximately 30 minutes.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to Dan Sloughter and the original version can be found here (HTML). Please respect the copyright and terms of use displayed on the solutions guide above.
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Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck’s “Differential Equation of Proportional Growth”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck’s “Differential Equation of Proportional Growth” (HTML)

Instructions: Click on the link above. Then, click on the “Index” button. Scroll down to “4. Logarithms and Exponentials, Applications,” and click button 146 (Differential Equation of Proportional Growth). Do problems 1-4. If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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6.4.2 Radioactive Decay and Half-Lives:

Lecture: Khan Academy's “Introduction to Exponential Decay” and “Exponential Growth”
Khan Academy's “Introduction to Exponential Decay” (YouTube) and “Exponential Growth” (YouTube)

Instructions: Please watch these two videos. In the first, you will learn that the half-life of a substance is the amount of time it takes for half of the original amount of the substance to decay through natural processes. (Think of certain radioactive substances which are very unstable or the carbon measured in carbon-dating.) The second video goes over an example from biology: exponential growth of bacteria.

Watching these lectures and pausing to take notes should take approximately 30 minutes.

Terms of Use: The videos above are released under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. They are attributed to the Khan Academy.
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6.4.3 Compound Interest

Lecture: Khan Academy's “Compound Interest and e (part 2)” and “Compound Interest and e (part 3)”
Khan Academy's “Compound Interest and e (part 2)” (YouTube) and “Compound Interest and e (part 3)” (YouTube)

Instructions: Please click on the links above and watch Salman “Compound Interest and e (part 2)” and “Compound Interest and e (part 3).” These videos derive the formula for continuously compounded interest and apply it.

Watching these video lectures and pausing to take notes should take approximately 30 minutes.

Terms of Use: The videos above are released under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. They are attributed to the Khan Academy.
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6.4.4 Epidemiology

Web Media: UC College Prep’s Calculus BC II for AP: “Applications of Integrals: Other Examples: Spread of Disease, Rumor”
Link: UC College Prep’s Calculus BC II for AP: “Applications of Integrals: Other Examples: Spread of Disease, Rumor” (HTML)

Instructions: Click on the link above and watch the interactive lecture.

Studying this resource should take approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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Assessment: The Saylor Foundation: Furman University: Dan Sloughter’s Difference Equations to Differential Equations “6.3 Models of Growth and Decay”
Link: The Saylor Foundation: Furman University: Dan Sloughter’s Difference Equations to Differential Equations “6.3 Models of Growth and Decay” (PDF)

Instructions: This assessment covers subunits 6.4.1-6.4.4. Please click the link above and do problems 1 (a, c, e, f), 4, 6-11, 13, and 14. When finished, click here (HTML) for solutions (courtesy of the author’s blog).

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML). It is attributed to Dan Sloughter and the original version can be found here (HTML). Please respect the copyright and terms of use displayed on the solutions guide above.
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6.4.5 Newton’s Law of Cooling

Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s “Population Growth” and Dan Reich’s “Newton’s Law of Cooling”
Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s “Population Growth” (HTML) and Dan Reich’s “Newton’s Law of Cooling” (HTML)

Instructions: Click on the above link. Then, click on the “Index” button. Scroll down to “4. Logarithms and Exponentials, Applications,” and click button 147 (Population Growth). Do problems 1-4. Next, click button 148 (Newton’s Law of Cooling), and do problems 1-4. If at any time a problem set seems too easy for you, feel free to move on.

Completing these assessments should take approximately 1 hour.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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Assessment: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Growth and Decay”
Link: The Saylor Foundation: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: Growth and Decay” (PDF)

Also Available in:
HTML

Instructions: Please click on the link above, and work through each of the five examples on the page. As in any assessment, solve the problem on your own first. Detailed solutions are given beneath each example.

Completing this assessment should take approximately 30 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here (HTML). Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.
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Final Exam

Final Exam: The Saylor Foundation's “MA102 Final Exam”
Link: The Saylor Foundation's “MA102 Final Exam”

Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.
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