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Calculus I

Purpose of Course  showclose

Calculus can be thought of as the mathematics of CHANGE.  Because everything in the world is changing, calculus helps us track those changes.  Algebra, by contrast, can be thought of as dealing with a large set of numbers that are inherently CONSTANT.  Solving an Algebra problem, like Y = 2X + 5, merely produces a pairing of two predetermined numbers, although an infinite set of pairs.  Algebra is even useful in rate problems, such as calculating how the money in your savings account increases because of the interest rate R, such as Y = X0+Rt where t is elapsed time and X0 is the initial deposit.  But with compounded interest, now things get complicated for algebra as the rate R is now itself a function of time with Y = X0 + R(t)t.  Now we have a rate of change which itself is changing.  Calculus “to the rescue,” as Isaac Newton introduced the world to mathematics specifically designed to handle “those things that change.” Calculus is among the most important and useful developments of human thought. Even though it is over 300 years old, it is still considered the beginning and cornerstone of modern mathematics.  It is a wonderful, beautiful, and useful set of ideas and techniques.  You will see the fundamental ideas of this course over and over again in future courses in mathematics as well as in all of the sciences, including physical, biological, social, economic, and engineering. However, calculus is an intellectual step up from your previous mathematics courses.  Many of the ideas you will learn in this course are more carefully defined and have both a functional and a graphical meaning.  Some of the algorithms are quite complicated, and in many cases, you will need to make a decision as to which appropriate algorithm to use.  Calculus offers a huge variety of applications and many of them will be saved for future courses you might take. This course is divided into four learning sections, or units, plus a reference section, or Appendix.  The course begins with a unit that provides a review of algebra specifically designed to help and prepare for the study of calculus.  The second unit discusses functions, graphs, limits, and continuity.  Understanding “limits” could not be more important as that topic really begins the study of calculus.  The third unit will introduce and explain derivatives.  With derivatives we are now ready to handle all those “things that change” mentioned above.  The fourth unit makes “visual sense” of derivatives by discussing derivatives and graphs.  Finally, the fifth unit provides a large collection of reference facts, geometry, and trigonometry that will assist in solving calculus problems long after the course is over.

Course Information  showclose

Welcome to MA005.  Below, please find general information on this course and its requirements.

Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Calculus I: “Welcome”
 
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Calculus I: “Welcome” (PDF)
 
Instructions:  Now that we made you feel at home, the real part of the course can begin.  Start by reading the general information for this course and its requirements, which follow immediately below.

Course Designer: Dale Hoffman

Primary Resources: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Calculus I

The text for the course has been carefully interwoven through the course subunits outlined below so that, while solving any mathematical problem, the theory you might need to solve the problem is only a page or two away.  This is the best way to proceed through the course.  However, many students have grown used to having a hard copy text for any college course, and use that to organize their timing and progress through a course.  Hence, you have the option to download the entire text for the course, if you prefer working that way.  Be advised that, depending upon your Internet speed, the file can take a couple of minutes to download as it contains 329 pages and more than 300 megabytes of file size.

Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Calculus I: “Complete Text”
 
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Calculus I (PDF)
 
Instructions: Please click on the link above to download the entire text.  Be patient during the download.  The reading for each chapter and section of the text to correspond to the topic in the course is indicated in the instructions section under each subunit of the course.  Consider saving this PDF file to your desktop as you will return to this textbook throughout the course.

Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials.  Pay special attention to Units 1 and 2, as these lay the groundwork for understanding the more advanced, exploratory material presented in the latter units.  You will also need to complete 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 many, if not all, of the problems presented for solution.

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: Both calculus teachers and students agree that calculus requires 2, or 3, hours every weeknight and 6 or 7 hours each weekend as a good pattern to follow if you expect to perform highly in this course.

This course should take you a total of approximately 85 hours to complete.  Each unit includes a “time advisory” that lists the amount of time you are expected to spend on each subunit.  These should help you plan your time accordingly.  It may be useful to take a look at these time advisories and 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 you 12 hours.  Perhaps you can sit down with your calendar and decide to complete subunits 1.1 and 1.2 (a total of 3 hours) on Monday night; subunit 1.3 (a total of 3 hours) on Tuesday night; etc.

Tips/Suggestions:  Calculus takes time.  Most people who fail a calculus course fail because they are unwilling, or unable, to devote the necessary time to the course.

  • Do Not Skip Topics.  The understanding of calculus is typically “sequential.”  It is very difficult to understand one topic after “lightly skipping over” a preceding topic. 
  • Consider Additional Resources.  When one of our sections seems to “not make sense” to you, consider one of the Additional Resources we have listed at the end of this page.  Sometimes an alternate approach to the same subject is “what you have been looking for.”
  • Test Yourself.  You are testing yourself when you follow the procedure of always solving a problem independently BEFORE looking at a solution of the same.

Finally, work on details.  Focus on the parts you missed.  Determine what you did not understand before moving on.



SBCTC
This course has been developed through a partnership with the Washington State Board for Community and Technical Colleges. Unless otherwise noted, all materials are licensed under a Creative Commons Attribution 3.0 Unported License. The Saylor Foundation has modified some materials created by the Washington State Board for Community and Technical Colleges in order to best serve our users.

 
A version of this course is also available in iTunes U.
Preview the course in your browser or view our entire suite of iTunes U courses.

Learning Outcomes  showclose

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

  • Calculate or estimate limits of functions given by formulas, graphs, or tables by using properties of limits and L’hopital’s Rule.
  • State whether a function given by a graph or formula is continuous or differentiable at a given point or on a given interval and justify the answer.
  • Calculate average and instantaneous rates of change in context, and state the meaning and units of the derivative for functions given graphically.
  • Calculate derivatives of polynomial, rational, common transcendental functions, and implicitly defined functions.
  • Apply the ideas and techniques of derivatives to solve maximum and minimum problems and related rate problems, and calculate slopes and rates for function given as parametric equations.
  • Find extreme values of modeling functions given by formulas or graphs.
  • Predict, construct, and interpret the shapes of graphs.
  • Solve equations using Newton’s Method.
  • Find linear approximations to functions using differentials.
  • Restate in words the meanings of the solutions to applied problems, attaching the appropriate units to an answer.
  • State which parts of a mathematical statement are assumptions, such as hypotheses, and which parts are conclusions.

Course Requirements  showclose

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.).

√     Have competency in the English language.

√    Have read the Saylor Student Handbook.

√     Have completed the equivalent course in Intermediate College Algebra.

Unit Outline show close


Expand All Resources Collapse All Resources
  • Unit 1: Preview and Review  

    While a first course in calculus can strike you as “something new” to learn, it is not comparable to learning a foreign language where everything “seems different.”  Calculus still depends on most of the things you learned in algebra and the true genius of Isaac Newton was to realize that he could get answers for this “something new” by relying on simple and “known” things like graphs, geometry, and algebra. 

    So there is a need to review those concepts here, where a graph can reinforce the adage that “one picture worth thousand words.”  This unit starts right off with one of the most important steps in being good at problem solving: have a clear and precise statement of what the problem really is about.

    Unit 1 Time Advisory   show close
    Unit 1 Learning Outcomes   show close
  • 1.1 Preview of Calculus  
  • 1.1.1 Two Basic Problems of Calculus  

    Note: This topic is covered by the reading assigned beneath subunit 1.1.  Please focus on pages 1-4 for an example of two basic problems in calculus.

  • 1.1.2 The Slope of a Tangent Line  

    Note: This topic is covered by the reading assigned beneath subunit 1.1.  Please focus on page 3 for information about slope of the tangent line.

  • 1.1.3 The Area of a Shape  

    Note: This topic is covered by the reading assigned beneath subunit 1.1.  Please focus on pages 3 and 4 to learn how calculus helps us determine the areas of shapes.

  • 1.1.4 Limits  

    Note: This topic is covered by the reading assigned beneath subunit 1.1.  Please focus on page 4 of the reading to learn about the unifying process of limits.

  • 1.1.5 Differentiation and Integration  

    Note: This topic is covered by the reading assigned beneath subunit 1.1.  Please focus on page 4 of the reading to learn how differentiation and integration are related.

  • 1.1.6 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 1.1.  Please make sure to attempt the problem set (questions 1-7) at the end of the reading. 

  • 1.2 Lines in the Plane  
  • 1.2.1 The Real Number Line  

    Note: This topic is covered by the reading assigned beneath subunit 1.2.  Please focus on page 1 of the reading to learn about the real number line.

  • 1.2.2 The Cartesian Plane  

    Note: This topic is covered by the reading assigned beneath subunit 1.2.  Please focus on page 2 of the reading to learn about the Cartesian plane.

  • 1.2.3 Increments and Distance  

    Note: This topic is covered by the reading assigned beneath subunit 1.2.  Please focus on page 2 of the reading to learn about increments and distance.

  • 1.2.4 Slope Between Points  

    Note: This topic is covered by the reading assigned beneath subunit 1.2.  Please focus on page 3 of the reading to learn about the slope between points.

  • 1.2.5 Equations of Lines  

    Note: This topic is covered by the reading assigned beneath subunit 1.2.  Please focus on page 5 of the reading to learn about the equations of lines.

  • 1.2.6 Two-Point and Slope-Intercept Equations  

    Note: This topic is covered by the reading assigned beneath subunit 1.2.  Please focus on page 6 of the reading to learn about intercept equations.

  • 1.2.7 Angles between Lines  

    Note: This topic is covered by the reading assigned beneath subunit 1.2.  Please focus on page 7 of the reading to learn about angles between lines.

  • 1.2.8 Parallel and Perpendicular Lines  

    Note: This topic is covered by the reading assigned beneath subunit 1.2.  Please focus on page 7 of the reading to learn about Parallel and Perpendicular Lines.

  • 1.2.9 Angles and Intersecting Lines  

    Note: This topic is covered by the reading assigned beneath subunit 1.2.  Please focus on page 9 of the reading to learn about angles and intersecting lines.

  • 1.2.10 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 1.2.  Please make sure to attempt the problem set (questions 1-30) at the end of the reading.

  • 1.3 Functions and Their Graphs  
  • 1.3.1 What Is a Function?  

    Note: This topic is covered by the reading assigned beneath subunit 1.3.  Please focus on page 1 of the reading to learn about characteristics of a function.

  • 1.3.2 Function Machines  

    Note: This topic is covered by the reading assigned beneath subunit 1.3.  Please focus on pages 1 and 2 of the reading to learn about function machines.

  • 1.3.3 Functions Defined by Equations  

    Note: This topic is covered by the reading assigned beneath subunit 1.3.  Please focus on page 2 of the reading to learn about functions defined by equations.

  • 1.3.4 Functions Defined by Graphs and Values  

    Note: This topic is covered by the reading assigned beneath subunit 1.3.  Please focus on page 3 of the reading to learn about functions defined by graphs and values.

  • 1.3.5 Creating Graphs of Functions  

    Note: This topic is covered by the reading assigned beneath subunit 1.3.  Please focus on page 4 of the reading for information on how to create graphs of functions.

  • 1.3.6 Reading Graphs  

    Note: This topic is covered by the reading assigned beneath subunit 1.3.  Please focus on page 5 of the reading for practice on reading graphs.

  • 1.3.7 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 1.3.  Please make sure to attempt the problem set (questions 1-23) at the end of the reading.

  • 1.4 Combinations of Functions  
  • 1.4.1 Multiline Functions  

    Note: This topic is covered by the reading assigned beneath subunit 1.4.  Please focus on page 1 of the reading to learn about multiline functions.

  • 1.4.2 Chill Index Sample  

    Note: This topic is covered by the reading assigned beneath subunit 1.4.  Please focus on page 1 of the reading to see wind chill being used as an example.

  • 1.4.3 Composition of Functions - Functions of Functions  

    Note: This topic is covered by the reading assigned beneath subunit 1.4.  Please focus on page 3 of the reading to learn about composition of functions.

  • 1.4.4 Shifting and Stretching Graphs  

    Note: This topic is covered by the reading assigned beneath subunit 1.4.  Please focus on page 4 of the reading to learn about shifting and stretching graphs.

  • 1.4.5 Iteration of Functions  

    Note: This topic is covered by the reading assigned beneath subunit 1.4.  Please focus on page 5 of the reading to study iteration of functions.

  • 1.4.6 Absolute Value and Greatest Integer  

    Note: This topic is covered by the reading assigned beneath subunit 1.4. Please focus on page 6 of the reading to study absolute and greater values.

  • 1.4.7 Broken Graphs and Graphs with Holes  

    Note: This topic is covered by the reading assigned beneath subunit 1.4.  Please focus on page 8 of the reading to learn about graphs with holes.

  • 1.4.8 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 1.4.  Please make sure to attempt the problem set (questions 1-32) at the end of the reading.

  • 1.5 Mathematical Language  
  • 1.5.1 Equivalent Statements  

    Note: This topic is covered by the reading assigned beneath subunit 1.5.  Please focus on page 1 of the reading to explore equivalent statements.

  • 1.5.2 The Logic of "And" and "Or"  

    Note: This topic is covered by the reading assigned beneath subunit 1.5.  Please focus on page 1 of the reading to learn about “and/or” logic.

  • 1.5.3 Negation of a Statement  

    Note: This topic is covered by the reading assigned beneath subunit 1.5.  Please focus on page 2 of the reading to gain an understanding of negating.

  • 1.5.4 "If-then" Statements  

    Note: This topic is covered by the reading assigned beneath subunit 1.5.  Please focus on pages 2 to 3 of the reading to learn about “if-then” logic.

  • 1.5.5 Contrapositive of "If-then" Statements  

    Note: This topic is covered by the reading assigned beneath subunit 1.5.  Please focus on page 4 of the reading to learn about “if-then” contrapositive.

  • 1.5.6 Converse of "If-then" Statements  

    Note: This topic is covered by the reading assigned beneath subunit 1.5.  Please focus on pages 4 to 5 of the reading to study the converse of an “if-then” statement.

  • 1.5.7 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 1.5.  Please make sure to attempt the problem set (questions 1-26) at the end of the reading.

  • 1.6 Solutions to Odd Numbered Problems  
  • Unit 2: Functions, Graphs, Limits and Continuity  

    In this unit, the concepts of “continuity” and the meaning of a “limit” really form the foundation for all of calculus.  Not only must you understand both of these concepts individually, but you must understand how they relate to each other.  They are a kind of “Siamese Twins” in calculus problems as we always hope they show up together.

    A student taking a calculus course during a winter term came up with the best analogy for tying these concepts together that I have ever heard.  The weather was raining ice—the kind of weather that no human being in his right mind would be driving a car.  When he stepped out on the front porch to see whether the ice-rain had stopped, he could not believe his eyes when he saw the headlights of an automobile heading down his road, which ended in a dead end at a brick house.  When the car hit the brakes, the students intuitive mind concluded that at the rate at which the velocity was decreasing, (assuming continuity) there was no way the car could stop in time and it would hit the house (the limiting value).  Oops.  He forgot that there was a gravel stretch at the end of the road and the car stopped many feet from the brick house.  The gravel represented a “discontinuity” in his “calculations” and so his “limiting value” was not correct.

    Unit 2 Time Advisory   show close
    Unit 2 Learning Outcomes   show close
  • 2.1 Tangent Lines, Velocities, and Growth  
  • 2.1.1 The Slope of a Tangent Line  

    Note: This topic is covered by the reading assigned beneath subunit 2.1.  Please focus on pages 1 and 2 of the reading to learn about slopes and tangent lines.

  • 2.1.2 A Falling Tomato Example  

    Note: This topic is covered by the reading assigned beneath subunit 2.1.  Please focus on page 3 of the reading to see previous topics applied to a simple free-fall example.

  • 2.1.3 Growing Bacteria Example  

    Note: This topic is covered by the reading assigned beneath subunit 2.1.  Please focus on page 4 of the reading to see previous topics applied to a biology example.

  • 2.1.4 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 2.1.  Please make sure to attempt the problem set (questions 1-10) at the end of the reading.

  • 2.2 The Limit of a Function  
  • 2.2.1 Informal Notion of a Limit  

    Note: This topic is covered by the reading assigned beneath subunit 2.2.  Please focus on pages 1 to 5 of the reading to learn an informal way to approach the concept of a “limit.”

  • 2.2.2 One Sided Limits  

    Note: This topic is covered by the reading assigned beneath subunit 2.2.  Please focus on page 6 of the reading to learn how to approach a limit from a direction and to learn the definition of left and right limits.

  • 2.2.3 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 2.2.  Please make sure to attempt the problem set (questions 1-20) at the end of the reading.

  • 2.3 Properties of Limits  
  • 2.3.1 Main Limit Theorem  

    Note: This topic is covered by the reading assigned beneath subunit 2.3.  Please focus on page 1 of the reading to understand the Main Limit Theorem.

  • 2.3.2 Limits by Substitution  

    Note: This topic is covered by the reading assigned beneath subunit 2.3.  Please focus on page 2 of the reading to learn how to calculate a limit by using substitution.

  • 2.3.3 Limits of Combined or Composed Functions  

    Note: This topic is covered by the reading assigned beneath subunit 2.3.  Please focus on pages 2 to 3 of the reading to learn how to calculate a limit when the functions are combined or composed.

  • 2.3.4 Tangent Lines as Limits  

    Note: This topic is covered by the reading assigned beneath subunit 2.3.  Please focus on page 4 of the reading to learn how to use a tangent line to calculate a limit.

  • 2.3.5 Comparing the Limits of Functions  

    Note: This topic is covered by the reading assigned beneath subunit 2.3.  Please focus on pages 4 and 5 of the reading to learn how to calculate a limit by comparing functions.

  • 2.3.6 Showing that a Limit Does Not Exist  

    Note: This topic is covered by the reading assigned beneath subunit 2.3.  Please focus on page 6 of the reading to learn how to show that limit does not even exist, so there is no need to calculate the limit.

  • 2.3.7 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 2.3.  Please make sure to attempt the problem set (questions 1-21) at the end of the reading.

    • Assessment: The Saylor Foundation’s “Problem Set 2”

      Link: The Saylor Foundation’s “Problem Set 2” (HTML)
       
      Instructions: You are now ready to complete Problem Set 2.  Please click on the link above.  If you have not already done so, create a free account on the Moodle website in order to access the quiz then work on answering the ten multiple choice questions.  When have you completed, click “submit all and finish” and your score will be tabulated.

      See a broken link? Please let us know!

  • 2.4 Continuous Functions  
  • 2.4.1 Definition and Meaning of Continuous  

    Note: This topic is covered by the reading assigned beneath subunit 2.4.  Please focus on page 1 of the reading to understand the definition and meaning of the concept continuity.

  • 2.4.2 Graphic Meaning of Continuity  

    Note: This topic is covered by the reading assigned beneath subunit 2.4.  Please focus on pages 2 and 3 of the reading to see the concept of continuity demonstrated graphically.

  • 2.4.3 The Importance of Continuity  

    Note: This topic is covered by the reading assigned beneath subunit 2.4.  Please focus on page 4 the reading to see why continuity is so important in mathematics.

  • 2.4.4 Combination of Continuous Functions  

    Note: This topic is covered by the reading assigned beneath subunit 2.4.  Please focus on pages 4 and 5 the reading to learn how to determine continuity when the function is a combination of functions.

  • 2.4.5 Which Functions are Continuous  

    Note: This topic is covered by the reading assigned beneath subunit 2.4.  Please focus on pages 5 and 6 the reading to learn how to determine quickly whether any function is continuous.

  • 2.4.6 Intermediate Value Property  

    Note: This topic is covered by the reading assigned beneath subunit 2.4.  Please focus on pages 6 and 7 the reading to learn why continuous functions also have an intermediate value.

  • 2.4.7 Bisection Algorithm  

    Note: This topic is covered by the reading assigned beneath subunit 2.4.  Please focus on pages 8 and 9 the reading to learn to use the Bisection Algorithm to determine whether a function is continuous.

  • 2.4.8 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 2.4.  Please make sure to attempt the problem set (questions 1-23) at the end of the reading.

  • 2.5 Definition of a Limit  
  • 2.5.1 Intuitive Approach  

    Note: This topic is covered by the reading assigned beneath subunit 2.5.  Please focus on page 2 to 3 of the reading to see an intuitive approach to the concept of a limit.

  • 2.5.2 The Formal Definition of a Limit  

    Note: This topic is covered by the reading assigned beneath subunit 2.5.  Please focus on pages 6 to 8 of the reading to now learn the formal definition of a limit.

  • 2.5.3 Two Limit Theorems  

    Note: This topic is covered by the reading assigned beneath subunit 2.5.  Please focus on page 9 of the reading to see two theorems used in the calculation of limits.

  • 2.5.4 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 2.5.  Please make sure to attempt the problem set (questions 1-23) at the end of the reading.

  • 2.6 Odd Numbered Solutions  
  • Unit 3: Derivatives  

    In this unit, we start to see calculus become more “visible,” when abstract ideas such as a Derivative and a Limit appear as parts of slopes, lines, and curves.  Then, there are circles, ellipses, and parabolas that are even more geometric and so, what was previously an abstract concept, can now be something “we can see.” And nothing makes calculus more “tangible” than to recognize the first derivative of an automobile’s position is its velocity and the second derivative of that position is its acceleration.  Then, we are at the very point that started Isaac Newton on his quest to master this mathematics, we now call calculus, when he recognized that the second derivative was precisely what he needed to formulate his Second Law of Motion “F = MA,” where F is the force on any object, M its mass, and A the second derivative of its position.  Now he could connect all the variables of a moving object mathematically, including its acceleration, velocity, and position and explain what really makes motion “happen.”

    Unit 3 Time Advisory   show close
    Unit 3 Learning Outcomes   show close
  • 3.1 Introduction to Derivatives  
  • 3.1.1 Slopes of Lines  

    Note: This topic is covered by the reading assigned beneath subunit 3.1.  Please focus on pages 1 and 2 to see how slopes lead us to the concept of a derivative.

  • 3.1.2 y=x2 Sample  

    Note: This topic is covered by the reading assigned beneath subunit 3.1.  Please focus on page 3 to see a simple example of what slopes can tell us.

  • 3.1.3 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 3.1.  Please make sure to attempt the problem set (questions 1-18) at the end of the reading.

  • 3.2 The Definition of Derivative  
  • 3.2.1 Formal Definition  

    Note: This topic is covered by the reading assigned beneath subunit 3.2.  Please focus on page 1 to learn the formal definition of a derivative.

  • 3.2.2 Calculations Using the Definition  

    Note: This topic is covered by the reading assigned beneath subunit 3.2.  Please focus on page 2 to observe calculations using the definition of a derivative.

  • 3.2.3 Tangent Line Formua  

    Note: This topic is covered by the reading assigned beneath subunit 3.2.  Please focus on pages 3 to 4 to learn how the tangent line formula can be used to determine a derivative.

  • 3.2.4 Sin and Cos Examples  

    Note: This topic is covered by the reading assigned beneath subunit 3.2.  Please focus on pages 4 and 5 to learn the derivatives of two familiar functions from trigonometry.

  • 3.2.5 Interpretations of the Derivative  

    Note: This topic is covered by the reading assigned beneath subunit 3.2.  Please focus on pages 5 to 6 to learn useful interpretations of the derivative.

  • 3.2.6 A Useful Formula  

    Note: This topic is covered by the reading assigned beneath subunit 3.2.  Please focus on page 7 to learn a very useful formula for calculating a derivative.

  • 3.2.7 Important Definitions and Results  

    Note: This topic is covered by the reading assigned beneath subunit 3.2.  Please focus on page 9 to learn very useful definitions and results from what we have learned for far.

  • 3.2.8 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 3.2.  Please make sure to attempt the problem set (questions 1-37) at the end of the reading.

  • 3.3 Derivatives, Their Properties, and Formulas  
  • 3.3.1 Which Functions Have Derivatives?  

    Note: This topic is covered by the reading assigned beneath subunit 3.3.  Please focus on pages 1 and 2 to learn which functions have derivatives.

  • 3.3.2 Derivatives of Elementary Combination of Functions  

    Note: This topic is covered by the reading assigned beneath subunit 3.3.  Please focus on pages 3 to 5 to learn how to find the derivative of combined functions.

  • 3.3.3 Using the Differentiation Rules  

    Note: This topic is covered by the reading assigned beneath subunit 3.3.  Please focus on pages 6 and 7 to learn rules for calculating derivatives.

  • 3.3.4 Evaluating a Derivative at a Point  

    Note: This topic is covered by the reading assigned beneath subunit 3.3.  Please focus on page 7 to learn how to evaluate a derivative at a point.

  • 3.3.5 Important Results  

    Note: This topic is covered by the reading assigned beneath subunit 3.3.  Please focus on page 8 for a summary of what we have learned so far in subunit 3.3.

  • 3.3.6 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 3.3.  Please make sure to attempt the problem set (questions 1-55) at the end of the reading.

  • 3.4 More Differentiation Probems  
  • 3.4.1 A Power Rule for Functions  

    Note: This topic is covered by the reading assigned beneath subunit 3.4.  Please focus on pages 1 and 2 to learn the Power Rule for Functions.

  • 3.4.2 Derivatives of Trigonometric and Exponential Functions  

    Note: This topic is covered by the reading assigned beneath subunit 3.4.  Please focus on pages 2 and 3 to see derivatives applied to trigonometric functions.

  • 3.4.3 Higher Derivatives – Derivatives of Derivatives  

    Note: This topic is covered by the reading assigned beneath subunit 3.4.  Please focus on page 5 to learn how higher derivatives are formed.

  • 3.4.4 Bent and Twisted Functions  

    Note: This topic is covered by the reading assigned beneath subunit 3.4.  Please focus on page 6 to learn how we can even handle some very strange functions indeed.

  • 3.4.5 Important Results  

    Note: This topic is covered by the reading assigned beneath subunit 3.4.  Please focus on page 7 to review some important results we have discovered so far in subunit 3.4.

  • 3.4.6 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 3.4.  Please make sure to attempt the problem set (questions 1-48) at the end of the reading.

  • 3.5 The Chain Rule  
  • 3.5.1 The Chain Rule Using Leibnitz Notation Form  

    Note: This topic is covered by the reading assigned beneath subunit 3.5.  Please focus on page 1 to learn the definition of the Chain Rule.

  • 3.5.2 The Chain Rule Composition Form  

    Note: This topic is covered by the reading assigned beneath subunit 3.5.  Please focus on pages 2 and 3 to learn how to use the Chain Rule with composed functions.

  • 3.5.3 The Chain Rule and Tables of Derivatives  

    Note: This topic is covered by the reading assigned beneath subunit 3.5.  Please focus on page 5 to learn how to form tables of derivatives using the Chain Rule.

  • 3.5.4 The Power Rule for Functions  

    Note: This topic is covered by the reading assigned beneath subunit 3.5.  Please focus on page 5 to learn how the Chain Rule leads to a Power Rule.

  • 3.5.5 Derivatives of Families of Functions  

    Note: This topic is covered by the reading assigned beneath subunit 3.5.  Please focus on page 8 and 9 to see the Chain Rule applied to a family of functions.

  • 3.5.6 Practice Problems  

    Note: This topic is covered by the reading assigned beneath subunit 3.5.  Please make sure to attempt the problem set (questions 1-83) at the end of the reading

    • Assessment: The Saylor Foundation’s “Problem Set 4”

      Link: The Saylor Foundation’s “Problem Set 4” (HTML)
       
      Instructions: You are now ready to complete Problem Set 4.  Please click on the link above.  If you have not already done so, create a free account on the Moodle website in order to access the quizthen work on answering the ten multiple choice questions.  When have you completed, click “submit all and finish” and your score will be tabulated.

      See a broken link? Please let us know!

  • 3.6 Some Applications of the Chain Rule  
  • 3.6.1 Derivatives of Logarithms  

    Note: This topic is covered by the reading assigned beneath subunit 3.6.  Please focus on pages 1 and 2 to learn how to the Chain Rule helps to determine the derivatives of logarithms.

  • 3.6.2 Derivative of ax  

    Note: This topic is covered by the reading assigned beneath subunit 3.6.  Please focus on page 2 to see how the Chain Rule is used to determine the derivative of this strange function.

  • 3.6.3 Applied Problems  

    Note: This topic is covered by the reading assigned beneath subunit 3.6.  Please focus on pages 3 and 4 to study more applications of the Chain Rule.

  • 3.6.4 Parametric Equations  

    Note: This topic is covered by the reading assigned beneath subunit 3.6.  Please focus on pages 4 and 5 to see the Chain Rule applied to Parametric Equations.

  • 3.6.5 Speed  

    Note: This topic is covered by the reading assigned beneath subunit 3.6.  Please focus on page 6 to see the Chain Rule applied to something as simple as the concept of speed.

  • 3.6.6 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 3.6.  Please make sure to attempt the problem set (questions 1-50) at the end of the reading.

  • 3.7 Related Rates  
  • 3.7.1 The Derivative as a Rate of Change  

    Note: This topic is covered by the reading assigned beneath subunit 3.7.  Please focus on pages 1 to 6 to learn how derivatives represent the rate of change of functions as well as moving objects.

  • 3.7.2 Practice Problems  

    Note: This topic is covered by the reading assigned beneath subunit 3.7.  Please focus on pages 4 to 6 to observe the derivative used in practice problems.

  • 3.7.3 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 3.7.  Please make sure to attempt the problem set (questions 1-21) at the end of the reading.

  • 3.8 Newton’s Method for Finding Roots  
  • 3.8.1 Off on a Tangent  

    Note: This topic is covered by the reading assigned beneath subunit 3.8.  Please focus on pages 1 and 2 to learn what roots have to do with tangents and derivatives.

  • 3.8.2 The Algorithm for Newton’s Method  

    Note: This topic is covered by the reading assigned beneath subunit 3.8.  Please focus on pages 3 and 4 to see Newton’s Method expressed as an algorithm.

  • 3.8.3 What Can Go Wrong?  

    Note: This topic is covered by the reading assigned beneath subunit 3.8.  Please focus on page 5 to see mistakes that can occur when using Newton’s Method.

  • 3.8.4 Chaotic Behavior and Newton’s Method  

    Note: This topic is covered by the reading assigned beneath subunit 3.8.  Please focus on page 6 to learn when Newton’s Method becomes chaotic.

  • 3.8.5 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 3.8.  Please make sure to attempt the problem set (questions 1-22) at the end of the reading.

  • 3.9 Linear Approximation and Differentials  
  • 3.9.1 Linear Approximation  

    Note: This topic is covered by the reading assigned beneath subunit 3.9.  Please focus on page 1 to see the connection between linear approximation and differentials.

  • 3.9.2 The Linear Approximation Process  

    Note: This topic is covered by the reading assigned beneath subunit 3.9.  Please focus on page 2 to study the linear approximation process.

  • 3.9.3 Linear Approximation Error  

    Note: This topic is covered by the reading assigned beneath subunit 3.9.  Please focus on pages 3 and 4 to see how the linear approximation process can be used to measure errors.

  • 3.9.4 Relative Error and Percentage Error  

    Note: This topic is covered by the reading assigned beneath subunit 3.9.  Please focus on pages 5 to 6 to learn about relative and percentage errors expressed as differentials.

  • 3.9.5 The Differential of a Function  

    Note: This topic is covered by the reading assigned beneath subunit 3.9.  Please focus on page 6 to understand the formal definition of the differential of a function.

  • 3.9.6 The Linear Approximation Error  

    Note: This topic is covered by the reading assigned beneath subunit 3.9.  Please focus on page 7 to applications of the linear approximation error.

  • 3.9.7 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 3.9.  Please make sure to attempt the problem set (questions 1-19) at the end of the reading.

  • 3.10 Implicit and Logarithmic Differentiation  
  • 3.10.1 Implicit Differentiation  

    Note: This topic is covered by the reading assigned beneath subunit 3.10.  Please focus on page 1 to learn what implicit differentiation is and how it is applied.

  • 3.10.2 Logarithmic Differentiation  

    Note: This topic is covered by the reading assigned beneath subunit 3.10.  Please focus on pages 3 and 4 to learn what logarithmic differentiation is and how it is applied.

  • 3.10.3 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 3.10.  Please make sure to attempt the problem set (questions 1-55) at the end of the reading.

  • 3.11 Odd Numbered Solutions  
  • Unit 4: Derivatives and Graphs  

    A “visual person” should find this section extremely helpful in understanding the concepts of calculus as a major emphasis in this section is to display those concepts graphically.  That allows us to see what, so far, we could only imagine.  And it works both ways.  Graphs help us to visualize ideas that are hard enough to conceptualize—like limits going to infinity but still have a finite meaning, or asymptotes—lines that approach each other but never quite get there.

    Graphs can also be used in a kind of reverse—display something for which we should take another mathematical look.  It is hard enough to imagine a limit going to infinity, and therefore never quite getting there, but the graph can tell us that it has a finite value, when it finally does get there, so we had better take a serious look at it mathematically.

    Unit 4 Time Advisory   show close
    Unit 4 Learning Outcomes   show close
  • 4.1 Finding Maximums and Minimums  
  • 4.1.1 Methods for Finding Maximums and Minimums  

    Note: This topic is covered by the reading assigned beneath subunit 4.1.  Please focus on page 1 to study methods for finding maximums and minimums.

  • 4.1.2 Terminology  

    Note: This topic is covered by the reading assigned beneath subunit 4.1.  Please focus on pages 1 and 2 to learn the terminology applied to maximums and minimums of functions.

  • 4.1.3 Finding Maximums and Minimums of a Function  

    Note: This topic is covered by the reading assigned beneath subunit 4.1.  Please focus on page 2 to learn the procedures for finding the maximums and minimums of functions.

  • 4.1.4 Is f(a) a Maximum, Minimum, or Neither?  

    Note: This topic is covered by the reading assigned beneath subunit 4.1.  Please focus on pages 2 and 3 to learn how to determine whether a function has a maximum, minimum, or neither.

  • 4.1.5 Endpoint Extremes  

    Note: This topic is covered by the reading assigned beneath subunit 4.1.  Please focus on page 4 to learn how to determine whether a function has a maximum, minimum, or neither.

  • 4.1.6 Critical Numbers  

    Note: This topic is covered by the reading assigned beneath subunit 4.1.  Please focus on page 6 to learn about critical numbers of functions.

  • 4.1.7 Which Functions Have Extremes?  

    Note: This topic is covered by the reading assigned beneath subunit 4.1.  Please focus on pages 6 and 7 to learn which functions have extremes.

  • 4.1.8 Extreme Value Theorem  

    Note: This topic is covered by the reading assigned beneath subunit 4.1.  Please focus on page 7 to learn to apply the Extreme Value Theorem.

  • 4.1.9 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 4.1.  Please make sure to attempt the problem set (questions 1-43) at the end of the reading.

  • 4.2 The Mean Value Theorem and Its Consequences  
  • 4.2.1 Rolle’s Theorem  

    Note: This topic is covered by the reading assigned beneath subunit 4.2.  Please focus on page 1 to see a proof of Rolle’s Theorem.

  • 4.2.2 The Mean Value Theorem  

    Note: This topic is covered by the reading assigned beneath subunit 4.2.  Please focus on pages 2 and 3 to see a proof of the Mean Value Theorem.

  • 4.2.3 Consequences of the Mean Value Theorem  

    Note: This topic is covered by the reading assigned beneath subunit 4.2.  Please focus on pages 4 and 5 to study the consequences of the Mean Value Theorem.

  • 4.2.4 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 4.2.  Please make sure to attempt the problem set (questions 1-36) at the end of the reading.

  • 4.3 The First Derivative and the Shape of a Function f(x)  
  • 4.3.1 First Shape Theorem  

    Note: This topic is covered by the reading assigned beneath subunit 4.3.  Please focus on page 2 to read a proof of the First Shape Theorem.

  • 4.3.2 Second Shape Theorem  

    Note: This topic is covered by the reading assigned beneath subunit 4.3.  Please focus on page 4 to read a proof of the Second Shape Theorem.

  • 4.3.3 Using the Derivative to Test for Extremes  

    Note: This topic is covered by the reading assigned beneath subunit 4.3.  Please focus on page 6 to learn how to use derivatives to test for extremes.

  • 4.3.4 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 4.3.  Please make sure to attempt the problem set (questions 1-30) at the end of the reading.

  • 4.4 The Second Derivative and the Shape of a Function f(x)  
  • 4.4.1 Concavity  

    Note: This topic is covered by the reading assigned beneath subunit 4.4.  Please focus on page 1 to learn how the second derivative is used to determine concavity.

  • 4.4.2 The Second Derivative Condition for Concavity  

    Note: This topic is covered by the reading assigned beneath subunit 4.4.  Please focus on page 2 to learn under what conditions the second derivative determines concavity.

  • 4.4.3 Feeling the Second Derivative  

    Note: This topic is covered by the reading assigned beneath subunit 4.4.  Please focus on pages 3 and 4 to study intuitive approaches to the second derivative.

  • 4.4.4 The Second Derivative and Extreme Values  

    Note: This topic is covered by the reading assigned beneath subunit 4.4.  Please focus on page 4 to learn how to use the second derivative to determine extreme values.

  • 4.4.5 Inflection Points  

    Note: This topic is covered by the reading assigned beneath subunit 4.4.  Please focus on page 5 to learn how to use the second derivative to determine inflection points.

  • 4.4.6 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 4.4.  Please make sure to attempt the problem set (questions 1-18) at the end of the reading.

    • Assessment: The Saylor Foundation’s “Problem Set 6”

      Link: The Saylor Foundation’s “Problem Set 6” (HTML)
       
      Instructions: You are now ready to complete Problem Set 6.  Please click on the link above.  If you have not already done so, create a free account on the Moodle website in order to access the quiz then work on answering the ten multiple choice questions.  When have you completed, click “submit all and finish” and your score will be tabulated.

      See a broken link? Please let us know!

  • 4.5 Applied Maximum and Minimum Problems  
  • 4.5.1 Sample Problems  

    Note: This topic is covered by the reading assigned beneath subunit 4.5.  Please focus on pages 1 to 4 to learn from sample problems presented.

  • 4.5.2 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 4.5.  Please make sure to attempt the problem set (questions 1-34) at the end of the reading.

  • 4.6 Infinite Limits and Asymptotes  
  • 4.6.1 Limits as X Goes to Infinity  

    Note: This topic is covered by the reading assigned beneath subunit 4.6.  Please focus on pages 1 and 2 to learn how to handle functions when the variable X becomes arbitrarily large.

  • 4.6.2 Using Calculators to Find Limits as X Goes to Infinity  

    Note: This topic is covered by the reading assigned beneath subunit 4.6.  Please focus on page 4 to learn how to use calculators when the variable X becomes arbitrarily large.

  • 4.6.3 When the Limit is Infinite  

    Note: This topic is covered by the reading assigned beneath subunit 4.6.  Please focus on page 5 to learn how to handle functions when the limit itself is infinite.

  • 4.6.4 Horizontal Asymptotes  

    Note: This topic is covered by the reading assigned beneath subunit 4.6.  Please focus on page 6 to understand the definition and use of horizontal asymptotes.

  • 4.6.5 Vertical Asymptotes  

    Note: This topic is covered by the reading assigned beneath subunit 4.6.  Please focus on page 6 to study the definition and use of vertical asymptotes.

  • 4.6.6 Other Asymptotes as X goes to Infinity  

    Note: This topic is covered by the reading assigned beneath subunit 4.6.  Please focus on page 7 to learn the definition and use of asymptotes that are neither vertical nor horizontal.

  • 4.6.7 Definition of limx--?f(x) = k  

    Note: This topic is covered by the reading assigned beneath subunit 4.6.  Please focus on page 8 to learn a precise definition of limx--∞f(x) = k

  • 4.6.8 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 4.6.  Please make sure to attempt the problem set (questions 1-59) at the end of the reading.

  • 4.7 L’Hopital’s Rule  
  • 4.7.1 A Linear Example  

    Note: This topic is covered by the reading assigned beneath subunit 4.7.  Please focus on page 1 to see L’Hopital’s Rule applied to a linear example.

  • 4.7.2 “0/0” Form of L’Hopital’s Rule  

    Note: This topic is covered by the reading assigned beneath subunit 4.7.  Please focus on page 2 to learn to use L’Hopital’s Rule even when dealing with 0/0.

  • 4.7.3 Strong Version of L’Hopital’s Rule  

    Note: This topic is covered by the reading assigned beneath subunit 4.7.  Please focus on page 3 to learn to use the strong version of L’Hopital’s Rule even when dealing with 0/0.

  • 4.7.4 Which Function Grows Faster?  

    Note: This topic is covered by the reading assigned beneath subunit 4.7.  Please focus on page 4 to learn to use L’Hopital’s Rule to determine which of two functions grows faster.

  • 4.7.5 Other Indeterminate Forms  

    Note: This topic is covered by the reading assigned beneath subunit 4.7.  Please focus on pages 4 to 6 to observe other indeterminate forms

  • 4.7.6 Problems for Solution  

    Note: This topic is covered by the reading assigned beneath subunit 4.7.  Please make sure to attempt the problem set (questions 1-30) at the end of the reading.

  • 4.8 Odd Numbered Solutions  
  • Unit 5: Appendix  

    By reviewing and having access to this unit, you will have an invaluable list of references to assist you in solving future calculus problems after this course has ended.  It is a standard experience, when solving calculus problems “on your own,” to react to the new problem with “We did not solve that kind of problem in the course.”  Ah, but we did, in that the new problem is often a combination, or composition, of two problem types that were covered. 

    The course could not cover all possible trigonometric functions you will encounter.  If you encounter a need for the derivative of Tan(x), it is sufficient to recall that Tan(x) = Sin(x)/Cos(x) and that Sine and Cosine were covered.  You can eventually become “so good at this” that future calculus problems can almost seem to be little more than “plugging into formulas.”

    Engineering students, who have to take several courses that involve the use of calculus, are noted for having a Table of Integrals “on their hip” wherever they go, such as this one posted on Wikipedia.*
     
    * Terms of Use: Please respect the copyright and terms of use on the webpage displayed above.

    Unit 5 Time Advisory   show close
  • Final Exam  

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