Calculus 1
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 = X_{0}+Rt, where t is elapsed time and X_{0 }is the initial deposit. With compound interest, things get complicated for algebra, as the rate R is itself a function of time with Y = X_{0 }+ R(t)t. Now we have a rate of change which itself is changing. Calculus came 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 (e.g., physical, biological, social, economic, and engineering). However, calculus is an intellectual step up from your previous mathematics courses. Many of the ideas you will gain 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 courses you might take in the future.
This course is divided into five 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 you 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 introduces and explains derivatives. With derivatives, we are now ready to handle all of those things that change mentioned above. The fourth unit makes visual sense of derivatives by discussing derivatives and graphs. The fifth unit introduces and explains antiderivatives and definite integrals. Finally, the reference section provides a large collection of reference facts, geometry, and trigonometry that will assist you in solving calculus problems long after the course is over.
This course provides students the opportunity to earn actual college credit. It has been reviewed and recommended for 4 credit hours by The National College Credit Recommendation Service (NCCRS). While credit is not guaranteed at all schools, we have partnered with a number of schools who have expressed their willingness to accept transfer of credits earned through Saylor. You can read more about our NCCRS program here.
Course Information showclose
Course Designer: Lenny Tevlin
Primary Resources: While this course comprises a range of different free, online materials, the primary source used for this course is:
 Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus
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 problem sets in each unit and 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 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: We recommend that you dedicate approximately 2 or 3 hours of work every weeknight and 6 or 7 hours each weekend if you expect to perform highly in this course.
This course should take you a total of approximately 130.75 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, determine how much time you have over the next few weeks to complete each unit, and then set goals for yourself. For example, Unit 1 should take you 7.75 hours. Perhaps you can sit down with your calendar and decide to complete Subunit 1.1 and Subunit 1.2 (a total of 2.5 hours) on Monday night; Subunit 1.3 and Subunit 1.4 (a total of 3.5 hours) on Tuesday night; Subunit 1.5 and the unit assessment (a total of 1.75 hours) on Wednesday night; and so forth.
Tips/Suggestions: Calculus takes time. Most people who fail a calculus course do so 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.
 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.
 Work on details. Focus on the parts you missed. Determine what you did not understand before moving on.
 While taking the final exam, you are welcome to make use of this Formula Sheet (PDF).
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.

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.

A version of this course is also available in iTunes U.
Preview the course in your browser or view all our iTunes U courses. 
Learning Outcomes showclose
 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, and common transcendental functions, compositions thereof, 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 functions 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;
 find antiderivatives by changing variables and using tables; and
 calculate definite integrals.
Course Requirements showclose
√ have access to a computer;
√ have continuous broadband Internet access;
√ have the ability/permission to install plugins 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; and
√ have completed MA004, or the equivalent course in Intermediate College Algebra.
Unit Outline show close

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. There is a need to review those concepts in this unit, where a graph can reinforce the adage that a picture is worth one thousand words. This unit starts right off with one of the most important steps in mastering 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
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.1: Preview of Calculus”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.1: Preview of Calculus” (PDF)
Instructions: Read Section 1.1 on pages 1  4 for an introduction to calculus.
Reading this section should take approximately 15 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.1: Preview of Calculus”

1.1.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.1: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.1: Problems for Solution” (PDF)
Instructions: Read the brief text after the heading “Problems for Solution” on page 4, and then work through the oddnumbered problems 1  7 on page 5 and page 6. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing this problem set should take approximately 15 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.1: Problems for Solution”

1.1.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ The Slope of a Tangent Line
To review this topic, focus on page 1 and page 2 of the reading.
√ The Area of a Shape
To review this topic, focus on page 3 and page 4 of the reading.
√ Limits
To review this topic, focus on page 4 of the reading.
√ Differentiation and Integration
To review this topic, focus on page 4 of the reading. 
1.2 Lines in the Plane
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.2: Lines in the Plane”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.2: Lines in the Plane” (PDF)
Instructions: Read Section 1.2 on pages 1  10 for an introduction to lines in the plane. Work through practice problems 1  9. For solutions to the practice problems, see page 15.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.2: Lines in the Plane”

1.2.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.2: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.2: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  29 on pages 10  14. Once you have completed the problem set, check your answers here.
Completing this problem set should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.2: Problems for Solution”

1.2.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ The Real Number Line
To review this topic, focus on page 1 of the reading.
√ The Cartesian Plane
To review this topic, focus on page 2 of the reading.
√ Increments and Distance between Points in the Plane
To review this topic, focus on page 2 and page 3 of the reading.
√ Slope between Points in the Plane
To review this topic, focus on pages 4  6 of the reading.
√ Equations of Lines
To review this topic, focus on page 6 of the reading.
√ TwoPoint and SlopeIntercept Equations
To review this topic, focus on page 6 and page 7 of the reading.
√ Angles between Lines
To review this topic, focus on page 8 of the reading.
√ Parallel and Perpendicular Lines
To review this topic, focus on page 8 and page 9 of the reading.
√ Angles and Intersecting Lines
To review this topic, focus on page 10 of the reading. 
1.3 Functions and Their Graphs
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.3: Functions and Their Graphs”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.3: Functions and Their Graphs” (PDF)
Instructions: Read Section 1.3 on pages 1  8 for an introduction to functions and their graphs. Work through practice problems 1  5. For solutions to the practice problems, see page 13 and page 14.
Reading this section and completing the practice problems should take approximately 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.3: Functions and Their Graphs”

1.3.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.3: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.3: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  23 on pages 8  13. Once you have completed the problem set, check your answers here.
Completing this problem set should take approximately 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.3: Problems for Solution”

1.3.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Definition of a Function
To review this topic, focus on page 1 of the reading.
√ Function Machines
To review this topic, focus on page 2 of the reading.
√ Functions Defined by Equations
To review this topic, focus on page 2 and page 3 of the reading.
√ Functions Defined by Graphs and Tables of Values
To review this topic, focus on page 3 and page 4 of the reading.
√ Creating Graphs of Functions
To review this topic, focus on pages 4 and 5 of the reading.
√ Reading Graphs
To review this topic, focus on pages 6  8 of the reading. 
1.4 Combinations of Functions
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.4: Combinations of Functions”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.4: Combinations of Functions” (PDF)
Instructions: Read Section 1.4 on pages 1  11 for an introduction to combinations of functions, then work through practice problems 1  9. For solutions to the practice problems, see pages 18  20.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.4: Combinations of Functions”

1.4.1 Problems for Solutions
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.4: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.4: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  31 on pages 11  16. Once you have completed the problem set, check your answers here.
Note that there is an error in the answer to question 7 beneath Section 1.4 on page 4. You can find a correction to this answer here.
Completing this problem set should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.4: Problems for Solution”

1.4.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Multiline Definition of Functions
To review this topic, focus on page 1 of the reading.
√ Wind Chill Index Sample
To review this topic, focus on pages 1  3 of the reading.
√ Composition of Functions – Functions of Functions
To review this topic, focus on page 3 and page 4 of the reading.
√ Shifting and Stretching Graphs
To review this topic, focus on page 5 and page 6 of the reading.
√ Iteration of Functions
To review this topic, focus on page 6 and page 7 of the reading.
√ Absolute Value and Greatest Integer
To review this topic, focus on pages 7  9 of the reading.
√ Broken Graphs and Graphs with Holes
To review this topic, focus on page 10 and page 11 of the reading. 
1.5 Mathematical Language
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.5: Mathematical Language”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.5: Mathematical Language” (PDF)
Instructions: Read Section 1.5 on pages 1  5 for an introduction to mathematical language, then work through practice problems 1  4. For the solutions to the practice problems, see page 7 and page 8.
Reading this section and completing the practice problems should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.5: Mathematical Language”

1.5.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.5: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.5: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  25 on pages 5  7. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing this problem set should take approximately 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 1.5: Problems for Solution”

1.5.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Equivalent Statements
To review this topic, focus on page 1 of the reading.
√ The Logic of “And” and “Or”
To review this topic, focus on page 1 of the reading.
√ Negation of a Statement
To review this topic, focus on page 2 of the reading.
√ “IfThen” Statements
To review this topic, focus on page 2 and page 3 of the reading.
√ Contrapositive of “IfThen” Statements
To review this topic, focus on page 4 of the reading.
√ Converse of “IfThen” Statements
To review this topic, focus on page 4 and page 5 of the reading. 
Unit 1: Assessment
 Assessment: The Saylor Foundation’s “Problem Set 1”
Link: The Saylor Foundation’s “Problem Set 1” (HTML)
Instructions: You are now ready to complete “Problem Set 1.” If you have not already done so, create a free account on the Moodle website in order to access the quiz and then work on answering the 10 multiplechoice questions. When you have completed the quiz, click on “submit all and finish” to tabulate your score.
Completing this assessment should take approximately 30 minutes.
 Assessment: The Saylor Foundation’s “Problem Set 1”

Unit 2: Functions, Graphs, Limits, and Continuity
The concepts of continuity and the meaning of a limit 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.
Unit 2 Time Advisory show close
A student taking a calculus course during a winter term came up with the best analogy that I have ever heard for tying these concepts together: The weather was raining ice  the kind of weather in which no human being in his right mind would be driving a car. When he stepped out on the front porch to see whether the icerain 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 student’s 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, so his limiting value was not correct.
Unit 2 Learning Outcomes show close

2.1 Tangent Lines, Velocities, and Growth
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.1: Tangent Lines, Velocities, and Growth”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.1: Tangent Lines, Velocities, and Growth” (PDF)
Instructions: Read Section 2.1 on pages 1  7 for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1  4. For the solutions to these practice problems, see page 10 and page 11.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.1: Tangent Lines, Velocities, and Growth”

2.1.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.1: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.1: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  9 on pages 7  9. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing this problem set should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.1: Problems for Solution”

2.1.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ The Slope of a Tangent Line
To review this topic, focus on pages 1  3.
√ Average Velocity and Instantaneous Velocity
To review this topic by examining an example with a falling tomato, focus on pages 3  5.
√ Average Population Growth Rate and Instantaneous Population Growth Rate
To review this topic by examining an example of growing bacteria, focus on pages 5  7. 
2.2 The Limit of a Function
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.2: The Limit of a Function”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.2: The Limit of a Function” (PDF)
Instructions: Read Section 2.2 on pages 1  7 for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1  4. For solutions to these practice problems, see page 10.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.2: The Limit of a Function”

2.2.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.2: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.2: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  19 on pages 7  9. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing this problem set should take approximately 2 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.2: Problems for Solution”

2.2.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Informal Notion of a Limit
To review this topic, focus on pages 1  3 of the reading.
√ Algebra Method for Evaluating Limits
To review this topic, focus on pages 4  6 of the reading.
√ Table Method for Evaluating Limits
To review this topic, focus on pages 4  6 of the reading.
√ Graph Method for Evaluating Limits
To review this topic, focus on pages 4  6 of the reading.
√ OneSided Limits
To review this topic, focus on page 6 and page 7 of the reading. 
2.3 Properties of Limits
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.3: Properties of Limits”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.3: Properties of Limits” (PDF)
Instructions: Read Section 2.3 on pages 1  8 to learn about the properties of limits. Work through practice problems 1  6. For the solutions to these problems, see page 14.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.  Web Media: YouTube: RootMath’s “Solving Limits (Rationalization)”
Link: YouTube: RootMath’s “Solving Limits (Rationalization)” (YouTube)
Instructions: Watch this video on finding limits algebraically. Be warned that removing x  4 from the numerator and denominator in Step 4 of this video is only legal inside this limit. The function (x  4)/(x  4) is not defined at x = 4; however, when x is not 4, it simplifies to 1. Because the limit as x approaches 4 depends only on values of x different from 4, inside that limit (x  4)/(x  4) and 1 are interchangeable. Outside that limit, they are not! However, this kind of cancellation is a key technique for finding limits of algebraically complicated functions.
Watching this video and pausing to take notes should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to RootMath.  Web Media: Khan Academy’s “Calculating Slope of Tangent Line Using Derivative Definition”
Link: Khan Academy’s “Calculating Slope of Tangent Line Using Derivative Definition” (YouTube)
Instructions: Watch this video on limits as the slopes of tangent lines.
An earlier Khan Academy video (not used in this course) defined the limit that gives the slope of the tangent line to a curve as y = f(x) at a point x = a and called it the derivative of f(x) at a. Your text will introduce this term in Unit 3.
Watching this video and taking notes should take approximately 30 minutes.
Terms of Use: This video is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 United States License. It is attributed to the Khan Academy.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.3: Properties of Limits”

2.3.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.3: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.3: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  21 on pages 9  14. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing this problem set should take approximately 3 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.3: Problems for Solution”

2.3.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Main Limit Theorem
To review this topic, focus on page 1 of the reading.
√ Limits by Substitution
To review this topic, focus on page 2 of the reading.
√ Limits of Combined or Composed Functions
To review this topic, focus on pages 2  4 of the reading.
√ Tangent Lines as Limits
To review this topic, focus on page 4 and page 5 of the reading.
√ Comparing the Limits of Functions
To review this topic, focus on page 5 and page 6 of the reading.
√ Showing that a Limit Does Not Exist
To review this topic, focus on pages 6  8 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.” If you have not already done so, create a free account on the Moodle website in order to access the quiz and then work on answering the 10 multiplechoice questions. When you have completed the quiz, click on “submit all and finish” to tabulate your score.
Completing this assessment should take approximately 1 hour.
 Assessment: The Saylor Foundation’s “Problem Set 2”

2.4 Continuous Functions
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.4: Continuous Functions”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.4: Continuous Functions” (PDF)
Instructions: Read Section 2.4 on pages 1  11 for an introduction to what we mean when we say a function is continuous. Work through practice problems 1 and 2. For solutions to these problems, see page 16.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.4: Continuous Functions”

2.4.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.4: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.4: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  23 on pages 12  15. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing this problem set should take approximately 3 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.4: Problems for Solution”

2.4.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Definition and Meaning of Continuous
To review this topic, focus on page 1 of the reading.
√ Graphic Meaning of Continuity
To review this topic, focus on pages 1  4 of the reading.
√ The Importance of Continuity
To review this topic, focus on page 5 of the reading.
√ Combinations of Continuous Functions
To review this topic, focus on page 5 and page 6 of the reading.
√ Which Functions Are Continuous?
To review this topic, focus on pages 6  8 of the reading.
√ Intermediate Value Property
To review this topic, focus on page 8 and page 9 of the reading.
√ Bisection Algorithm for Approximating Roots
To review this topic, focus on pages 9  11 of the reading. 
2.5 Definition of a Limit
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.5: Definition of a Limit”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.5: Definition of a Limit” (PDF)
Instructions: Read Section 2.5 on pages 1  11 to learn how a limit is defined. Work through practice problems 1  6. For solutions to these problems, see pages 14  16.
Reading this section and completing the practice problems should take approximately 1 hour and 15 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.  Web Media: Khan Academy’s “Epsilon Delta Limit Definition 1”
Link: Khan Academy’s “Epsilon Delta Limit Definition 1” (YouTube)
Instructions: Watch this video for the epsilondelta definition of a limit.
Watching this video and taking notes should take approximately 30 minutes.
Terms of Use: This video is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 United States License. It is attributed to the Khan Academy.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.5: Definition of a Limit”

2.5.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.5: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.5: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  23 on pages 12  14. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 2.5: Problems for Solution”

2.5.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Intuitive Approach to Defining a Limit
To review this topic, focus on pages 1  7 of the reading.
√ The Formal Definition of a Limit
To review this topic, focus on pages 7  10 of the reading.
√ Two Limit Theorems
To review this topic, focus on page 10 and page 11 of the reading. 
Unit 2: Assessment
 Assessment: The Saylor Foundation’s “Problem Set 3”
Link: The Saylor Foundation’s “Problem Set 3” (HTML)
Instructions: You are now ready to complete “Problem Set 3.” If you have not already done so, create a free account on the Moodle website in order to access the quiz and then work on answering the 10 multiplechoice questions. When you have completed the quiz, click on “submit all and finish” to tabulate your score.
Completing this assessment should take approximately 1 hour.
 Assessment: The Saylor Foundation’s “Problem Set 3”

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, so what was previously an abstract concept can now be something we can see. Nothing makes calculus more tangible than to recognize that the first derivative of an automobile’s position is its velocity and the second derivative of that position is its acceleration. We are at the very point that started Isaac Newton on his quest to master this mathematics, what 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 Fis the force on any object, Mis its mass, and A is the second derivative of its position. Thus, he could connect all the variables of a moving object mathematically, including its acceleration, velocity, and position, and he could explain what really makes motion happen.
Unit 3 Time Advisory show close
Unit 3 Learning Outcomes show close

3.1 Introduction to Derivatives
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.1: Introduction to Derivatives”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.1: Introduction to Derivatives” (PDF)
Instructions: Read Section 3.1 on pages 1  5 to lay the groundwork for introducing the concept of a derivative. Work through practice problems 1  5. For solutions to these problems, see page 8 and page 9.
Reading this section and completing the practice problems should take approximately 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.1: Introduction to Derivatives”

3.1.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.1: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.1: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  17 on pages 5  7. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 1 hour and 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.1: Problems for Solution”

3.1.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Slopes of Tangent Lines
To review this topic, focus on page 1 and page 2 of the reading.
√ Tangents to y = x^{2 }
To review this topic, focus on pages 2  5 of the reading. 
3.2 The Definition of a Derivative
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.2: The Definition of a Derivative”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.2: The Definition of a Derivative” (PDF)
Instructions: Read Section 3.2 on pages 1  10 to understand the definition of a derivative. Work through practice problems 1  8. For solutions to these problems, see page 14 and page 15.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.2: The Definition of a Derivative”

3.2.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.2: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.2: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  37 on pages 11  14. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 3 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.2: Problems for Solution”

3.2.2 Reading Review
√ Formal Definition of a Derivative
To review this topic, focus on page 1 and page 2 of the reading.
√ Calculations Using the Definition
To review this topic, focus on pages 2  6 of the reading.
√ Tangent Line Formula
To review this topic, focus on page 4 of the reading.
√ sin and cos Examples
To review this topic, focus on page 4 and page 5 of the reading.
√ Interpretations of the Derivative
To review this topic, focus on pages 6  8 of the reading.
√ A Useful Formula: D(x^{n})
To review this topic, focus on pages 8  10 of the reading.
√ Important Definitions, Formulas, and Results for the Derivative, Tangent Line Equation, and Interpretations of f′(x)
To review this topic, focus on page 10 of the reading. 
3.3 Derivatives, Properties and Formulas
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.3: Derivatives, Properties and Formulas”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.3: Derivatives, Properties and Formulas” (PDF)
Instructions: Read Section 3.3 on pages 1  9 to understand the properties of derivatives. Work through practice problems 1  11. For solutions to these problems, see page 16 and page 17.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.  Web Media: Khan Academy’s “Applying the Product Rule for Derivatives”
Link: Khan Academy’s “Applying the Product Rule for Derivatives” (YouTube)
Instructions: Watch this video on the product rule for differentiation.
Watching this video and taking notes should take less than 15 minutes.
Terms of Use: This video is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 United States License. It is attributed to the Khan Academy.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.3: Derivatives, Properties and Formulas”

3.3.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.3: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.3: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  55 on pages 10  15. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 3 hours and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.3: Problems for Solution”

3.3.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Which Functions Have Derivatives?
To review this topic, focus on pages 1  3 of the reading.
√ Derivatives of Elementary Combination of Functions
To review this topic, focus on pages 3  6 of the reading.
√ Using the Differentiation Rules
To review this topic, focus on page 7 and page 8 of the reading.
√ Evaluative a Derivative at a Point
To review this topic, focus on page 9 of the reading.
√ Important Results for Differentiability and Continuity
To review this topic, focus on page 9 of the reading. 
3.4 Derivative Patterns
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.4: More Differentiation Problems”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.4: More Differentiation Problems” (PDF)
Instructions: Read Section 3.4 on pages 1  9 to learn about patterns of derivatives. Work through practice problems 1  8. For solutions to these problems, see pages 12  14.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.4: More Differentiation Problems”

3.4.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.4: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.4: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  47 on pages 9  14. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 3 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.4: Problems for Solution”

3.4.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ A Power Rule for Functions: D(f ^{n}(x))
To review this topic, focus on pages 1 and 2 of the reading.
√ Derivatives of Trigonometric and Exponential Functions
To review this topic, focus on pages 3  6 of the reading.
√ Higher Derivatives – Derivatives of Derivatives
To review this topic, focus on page 6 and page 7 of the reading.
√ Bent and Twisted Functions
To review this topic, focus on page 7 and page 8 of the reading.
√ Important Results for Power Rule of Functions and Derivatives of Trigonometric and Exponential Functions
To review this topic, focus on page 9 of the reading. 
3.5 The Chain Rule
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.5: The Chain Rule”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.5: The Chain Rule” (PDF)
Instructions: Read Section 3.5 on pages 1  7 to learn about the Chain Rule. Work through practice problems 1  8. For solutions to these problems, see page 11 and page 12.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.  Web Media: Khan Academy’s “Chain Rule Introduction”
Link: Khan Academy’s “Chain Rule Introduction” (YouTube)
Instructions: Watch this video for an introduction to the chain rule for differentiation.
Watching this video and taking notes should take approximately 15 minutes.
Terms of Use: This video is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 United States License. It is attributed to the Khan Academy.  Web Media: Khan Academy’s “Chain Rule Definition and Example”
Link: Khan Academy’s “Chain Rule Definition and Example” (YouTube)
Instructions: Watch this video for a definition and example of the chain rule for differentiation.
Watching this video and taking notes should take approximately 15 minutes.
Terms of Use: This video is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 United States License. It is attributed to the Khan Academy.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.5: The Chain Rule”

3.5.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.5: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.5: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  83 on pages 7  11. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 5 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.5: Problems for Solution”

3.5.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Chain Rule for Differentiating a Composition of Functions
To review this topic, focus on page 1 of the reading.
√ The Chain Rule Using Leibnitz Notation Form
To review this topic, focus on page 2 of the reading.
√ The Chain Rule Composition Form
To review this topic, focus on pages 2  5 of the reading.
√ The Chain Rule and Tables of Derivatives
To review this topic, focus on page 5 and page 6 of the reading.
√ The Power Rule for Functions
To review this topic, focus on page 7 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.” If you have not already done so, create a free account on the Moodle website in order to access the quiz and then work on answering the 10 multiplechoice questions. When you have completed the quiz, click on “submit all and finish” to tabulate your score.
Completing this assessment should take approximately 1 hour.
 Assessment: The Saylor Foundation’s “Problem Set 4”

3.6 Some Applications of the Chain Rule
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.6: Some Applications of the Chain Rule”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.6: Some Applications of the Chain Rule” (PDF)
Instructions: Read Section 3.6 on pages 1  8 to learn how to apply the Chain Rule. Work through practice problems 1  8. For solutions to these problems, see page 13 and page 14.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.6: Some Applications of the Chain Rule”

3.6.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.6: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.6: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  49 on pages 9  11. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 2 hours and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.6: Problems for Solution”

3.6.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Derivatives of Logarithms
To review this topic, focus on page 1 and page 2 of the reading.
√ Derivative of a^{x}
To review this topic, focus on page 2 and page 3 of the reading.
√ Applied Problems
To review this topic, focus on pages 3  5 of the reading.
√ Parametric Equations
To review this topic, focus on page 5 and page 6 of the reading.
√ Speed
To review this topic, focus on page 8 of the reading. 
3.7 Related Rates
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.7: Related Rates: An Application of Derivatives”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.7: Related Rates: An Application of Derivatives” (PDF)
Instructions: Read Section 3.7 on pages 1  7 to learn to connect derivatives to the concept of the rate at which things change. Work through practice problems 1  3. For solutions to these problems, see page 12 and page 13.
Reading this section and completing the practice problems should take approximately 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.7: Related Rates: An Application of Derivatives”

3.7.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.7: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.7: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  21 on pages 8  12. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.7: Problems for Solution”

3.7.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ The Derivative as a Rate of Change
To review this topic, focus on pages 1  7 of the reading. 
3.8 Newton’s Method for Finding Roots
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.8: Newton’s Method for Finding Roots”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.8: Newton’s Method for Finding Roots” (PDF)
Instructions: Read Section 3.8 on pages 1  8. Work through practice problems 1  6. For solutions to these problems, see page 10 and page 11.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.  Web Media: YouTube: MIT: Christine Breiner’s “Using Newton’s Method”
Link: YouTube: MIT: Christine Breiner’s “Using Newton’s Method” (YouTube)
Instructions: Watch this video on Newton’s method.
Watching this video and taking notes should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License. It is attributed to Christine Breiner.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.8: Newton’s Method for Finding Roots”

3.8.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.8: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.8: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  21 on page 8 and page 9. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.8: Problems for Solution”

3.8.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Newton’s Method Using the Tangent Line
To review this topic, focus on pages 1  3 of the reading.
√ The Algorithm for Newton’s Method
To review this topic, focus on pages 3  5 of the reading.
√ Iteration
To review this topic, focus on page 5 of the reading.
√ What Can Go Wrong with Newton’s Method?
To review this topic, focus on page 5 and page 6 of the reading.
√ Chaotic Behavior and Newton’s Method
To review this topic, focus on pages 6  8 of the reading. 
3.9 Linear Approximation and Differentials
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.9: Linear Approximation and Differentials”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.9: Linear Approximation and Differentials” (PDF)
Instructions: Read Section 3.9 on pages 1  10 to learn how linear approximation and differentials are connected. Work through practice problems 1  10. For the solutions to these problems, see page 14 and page 15.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.  Web Media: YouTube: RootMath’s “Linear Approximation”
Link: YouTube: RootMath’s “Linear Approximation” (YouTube)
Instructions: Watch this video on linear approximation and differentials.
Watching this video and taking notes should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to RootMath.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.9: Linear Approximation and Differentials”

3.9.1 Problems for Solution
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.9: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.9: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  19 on pages 10  13. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 2 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.9: Problems for Solution”

3.9.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Linear Approximation and Its Process
To review this topic, focus on pages 1  4 of the reading.
√ Applications of Linear Approximation to Measurement Error
To review this topic, focus on pages 4  6 of the reading.
√ Relative Error and Percentage Error
To review this topic, focus on page 6 and page 7 of the reading.
√ The Differential of a Function
To review this topic, focus on page 7 and page 8 of the reading.
√ The Linear Approximation Error
To review this topic, focus on pages 8  10 of the reading. 
3.10 Implicit and Logarithmic Differentiation
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.10: Implicit and Logarithmic Differentiation”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.10: Implicit and Logarithmic Differentiation” (PDF)
Instructions: Read Section 3.10 on pages 1  5 to learn about implicit and logarithmic differentiation. Work through practice problems 1  6. For solutions to these problems, see page 8 and page 9.
Reading this section and completing the practice problems should take approximately 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.  Web Media: Khan Academy’s “Implicit Differentiation”
Link: Khan Academy’s “Implicit Differentiation” (YouTube)
Instructions: Watch this video on implicit differentiation.
Watching this video and taking notes should take approximately 30 minutes.
Terms of Use: This video is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 United States License. It is attributed to the Khan Academy.  Web Media: Khan Academy’s “Derivative of x^(x^x)”
Link: Khan Academy’s “Derivative of x^(x^x)” (YouTube)
Instructions: Watch this video on logarithmic differentiation.
While aimed at implicit differentiation, this video gives an example of using logarithms to simplify a complicated function in order to differentiate it.
Watching this video and taking notes should take approximately 30 minutes.
Terms of Use: This video is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 United States License. It is attributed to the Khan Academy.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.10: Implicit and Logarithmic Differentiation”

3.10.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.10: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.10: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  55 on pages 5  8. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 2 hours and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 3.10: Problems for Solution”

3.10.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Implicit Differentiation
To review this topic, focus on pages 1  3 of the reading.
√ Logarithmic Differentiation
To review this topic, focus on pages 3  5 of the reading. 
Unit 3: Assessment
 Assessment: The Saylor Foundation’s “Problem Set 5”
Link: The Saylor Foundation’s “Problem Set 5” (HTML)
Instructions: You are now ready to complete “Problem Set 5.” If you have not already done so, create a free account on the Moodle website in order to access the quiz and then work on answering the 10 multiplechoice questions. When you have completed the quiz, click on “submit all and finish” to tabulate your score.
Completing this assessment should take you approximately 1 hour.
 Assessment: The Saylor Foundation’s “Problem Set 5”

Unit 4: Derivatives and Graphs
A visual person should find this unit extremely helpful in understanding the concepts of calculus, as a major emphasis in this unit is to display those concepts graphically. That allows us to see what, so far, we could only imagine. Graphs help us to visualize ideas that are hard enough to conceptualize  like limits going to infinity but still having a finite meaning, or asymptotes  lines that approach each other but never quite get there.
Unit 4 Time Advisory show close
Graphs can also be used in a kind of reverse by displaying 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 Learning Outcomes show close

4.1 Finding Maximums and Minimums
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.1: Finding Maximums and Minimums”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.1: Finding Maximums and Minimums” (PDF)
Instructions: Read Section 4.1 on pages 1  9 to learn about maximums, minimums, and extreme values for functions. Work through practice problems 1  5. For solutions to these problems, see page 13 and page 14.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.1: Finding Maximums and Minimums”

4.1.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.1: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.1: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  43 on pages 9  13. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 3 hours and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.1: Problems for Solution”

4.1.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Methods for Finding Maximums and Minimums
To review this topic, focus on page 1 of the reading.
√ Terminology: Global Maximum, Local Maximum, Maximum Point, Global Minimum, Local Minimum, Global Extreme, and Local Extreme
To review this topic, focus on page 2 of the reading.
√ Finding Maximums and Minimums of a Function
To review this topic, focus on pages 3  5 of the reading.
√ Is f(a) a Maximum, Minimum, or Neither?
To review this topic, focus on page 5 of the reading.
√ Endpoint Extremes
To review this topic, focus on pages 5  7 of the reading.
√ Critical Numbers
To review this topic, focus on page 7 of the reading.
√ Which Functions Have Extremes?
To review this topic, focus on page 7 and page 8 of the reading.
√ Extreme Value Theorem
To review this topic, focus on page 8 and page 9 of the reading. 
4.2 The Mean Value Theorem and Its Consequences
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.2: The Mean Value Theorem and Its Consequences”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.2: The Mean Value Theorem and Its Consequences” (PDF)
Instructions: Read Section 4.2 on pages 1  6 to learn about the Mean Value Theorem and its consequences. Work through practice problems 1  3. For solutions to these problems, see pages 9 and 10.
Reading this section and completing the practice problems should take approximately 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.2: The Mean Value Theorem and Its Consequences”

4.2.1 Problems for Solution
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.2: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.2: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  35 on pages 6  9. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 2 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.2: Problems for Solution”

4.2.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Rolle’s Theorem
To review this topic, focus on page 1 and page 2 of the reading.
√ The Mean Value Theorem
To review this topic, focus on pages 2  4 of the reading.
√ Consequences of the Mean Value Theorem
To review this topic, focus on pages 4  6 of the reading. 
4.3 The First Derivative and the Shape of a Function f(x)
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.3: The First Derivative and the Shape of f”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.3: The First Derivative and the Shape of f” (PDF)
Instructions: Read Section 4.3 on pages 1  8 to learn how the first derivative is used to determine the shape of functions. Work through practice problems 1  9. For the solution to these problems, see pages 10  12.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.  Web Media: YouTube: RootMath’s “First Derivative Test, Example 2, Part 1” and “First Derivative Test, Example 2, Part 2”
Link: YouTube: RootMath’s “First Derivative Test, Example 2, Part 1” (YouTube) and “First Derivative Test, Example 2, Part 2” (YouTube)
Instructions: Watch both parts of this video on the first derivative test.
Watching these videos and taking notes should take approximately 30 minutes.
Terms of Use: These videos are licensed under a Creative Commons Attribution 3.0 Unported License. They are attributed to RootMath.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.3: The First Derivative and the Shape of f”

4.3.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.3: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.3: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  29 on pages 8  10. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 1 hour and 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.3: Problems for Solution”

4.3.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Definitions of the Functionf
To review this topic, focus on page 1 of the reading.
√ First Shape Theorem
To review this topic, focus on pages 2  4 of the reading.
√ Second Shape Theorem
To review this topic, focus on pages 4  7 of the reading.
√ Using the Derivative to Test for Extremes
To review this topic, focus on page 7 and page 8 of the reading. 
4.4 The Second Derivative and the Shape of a Function f(x)
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.4: Second Derivative and the Shape of f”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.4: Second Derivative and the Shape of f” (PDF)
Instructions: Read Section 4.4 on pages 1  6 to learn how the second derivative is used to determine the shape of functions. Work through practice problems 1  9. For solutions to these problems, see page 8 and page 9.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.  Web Media: Khan Academy’s “Concavity, Concave upwards and Concave downwards Intervals”
Link: Khan Academy’s “Concavity, Concave upwards and Concave downwards Intervals” (YouTube)
Instructions: Watch this video on the second derivative test. This video describes a way to identify critical points as minima or maxima other than the first derivative test, using the second derivative.
Watching this video and taking notes should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 United States License. It is attributed to the Khan Academy.  Web Media: YouTube: RootMath’s “Concavity and the Second Derivative”
Link: YouTube: RootMath’s “Concavity and the Second Derivative” (YouTube)
Instructions: Watch this video, which works through an example of the second derivative test.
Watching this video and taking notes should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to RootMath.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.4: Second Derivative and the Shape of f”

4.4.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.4: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.4: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  17 on pages 6  8. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.4: Problems for Solution”

4.4.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Concavity
To review this topic, focus on page 1 and page 2 of the reading.
√ The Second Derivative Condition for Concavity
To review this topic, focus on page 2 and page 3 of the reading.
√ Feeling the Second Derivative: Acceleration Applications
To review this topic, focus on page 3 and page 4 of the reading.
√ The Second Derivative and Extreme Values
To review this topic, focus on page 4 and page 5 of the reading.
√ Inflection Points
To review this topic, focus on page 5 and page 6 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.” If you have not already done so, create a free account on the Moodle website in order to access the quiz and then work on answering the 10 multiplechoice questions. When you have completed the quiz, click on “submit all and finish” to tabulate your score.
Completing this assessment should take you approximately 1 hour.
 Assessment: The Saylor Foundation’s “Problem Set 6”

4.5 Applied Maximum and Minimum Problems
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.5: Applied Maximum and Minimum Problems”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.5: Applied Maximum and Minimum Problems” (PDF)
Instructions: Read Section 4.5 on pages 1  6 to learn how to apply previously learned principles to maximum and minimum problems. Work through practice problems 1  3. For solutions to these problems, see page 15 and page 16. There is no reading review for this section; instead, make sure to study the problems carefully to become familiar with applied maximum and minimum problems.
Reading this section and completing the practice problems should take approximately 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.  Web Media: Khan Academy’s “Minimizing Sum of Squares”
Link: Khan Academy’s “Minimizing Sum of Squares” (YouTube)
Instructions: Watch this video on optimization.
Watching this video and taking notes should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 United States License. It is attributed to the Khan Academy.  Web Media: YouTube: MIT: Christine Breiner’s “Minimum Triangle Area”
Link: YouTube: MIT: Christine Breiner’s “Minimum Triangle Area” (YouTube)
Instructions: Watch this video on optimization.
Watching this video and taking notes should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 Unported License. It is attributed to Christine Breiner.  Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.5: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.5: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  33 on pages 6  15. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 3 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.5: Applied Maximum and Minimum Problems”

4.6 Infinite Limits and Asymptotes
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.6: Infinite Limits and Asymptotes”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.6: Infinite Limits and Asymptotes” (PDF)
Instructions: Read Section 4.6 on pages 1  10 to learn how to use and apply infinite limits to asymptotes. Work through practice problems 1  8. For solutions to these problems, see page 13 and page 14.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.6: Infinite Limits and Asymptotes”

4.6.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.6: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.6: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  59 on pages 10  12. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 3 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.6: Problems for Solution”

4.6.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Limits as x Approaches Infinity
To review this topic, focus on pages 1  4 of the reading.
√ Using Calculators to Find Limits as x Goes to Infinity
To review this topic, focus on page 5 of the reading.
√ The Limit Is Infinite
To review this topic, focus on page 5 and page 6 of the reading.
√ Horizontal Asymptotes
To review this topic, focus on page 6 and page 7 of the reading.
√ Vertical Asymptotes
To review this topic, focus on page 7 and page 8 of the reading.
√ Other Asymptotes as x Approaches Infinity
To review this topic, focus on page 8 and page 9 of the reading.
√ Definition of lim _{x >∞}_{ }(x) = k
To review this topic, focus on page 9 and page 10 of the reading. 
4.7 L’Hopital’s Rule
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.7: L’Hopital’s Rule”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.7: L’Hopital’s Rule” (PDF)
Instructions: Read Section 4.7 on pages 1  6 to learn how to use and apply L’Hopital’s Rule. Work through practice problems 1  3. For solutions to these problems, see page 7 and page 8.
Reading this section and completing the practice problems should take approximately 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.  Web Media: Khan Academy’s “Introduction to L’Hopital’s Rule”
Link: Khan Academy’s “Introduction to L’Hopital’s Rule” (YouTube)
Instructions: Watch this video for an introduction to L’Hopital’s Rule.
Watching this video and taking notes should take approximately 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons AttributionNonCommercialShareAlike 3.0 United States License. It is attributed to the Khan Academy.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.7: L’Hopital’s Rule”

4.7.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.7: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.7: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  29 on page 6 and page 7. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 4.7: Problems for Solution”

4.7.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ A Linear Example
To review this topic, focus on page 1 of the reading.
√ 0/0 Form of L’Hopital’s Rule
To review this topic, focus on page 2 of the reading.
√ Strong Version of L’Hopital’s Rule
To review this topic, focus on page 2 and page 3 of the reading.
√ Which Function Grows Faster?
To review this topic, focus on page 4 of the reading.
√ Other Indeterminate Forms
To review this topic, focus on pages 4  6 of the reading. 
Unit 4: Assessment
 Assessment: The Saylor Foundation’s “Problem Set 7”
Link: The Saylor Foundation’s “Problem Set 7” (HTML)
Instructions: You are now ready to complete “Problem Set 7.” If you have not already done so, create a free account on the Moodle website in order to access the quiz and then work on answering the 10 multiplechoice questions. When you have completed the quiz, click on “submit all and finish” to tabulate your score.
Completing this assessment should take you approximately 1 hour.
 Assessment: The Saylor Foundation’s “Problem Set 7”

Unit 5: The Integral
While previous units dealt with differential calculus, this unit starts the study of integral calculus. As you may recall, differential calculus began with the development of the intuition behind the notion of a tangent line. Integral calculus begins with understanding the intuition behind the notion of an area.In fact, we will be able to extend the notion of the area and apply these more general areas to a variety of problems. This will allow us to unify differential and integral calculus through the Fundamental Theorem of Calculus. Historically, this theorem marked the beginning of modern mathematics and is extremely important in all applications.
Unit 5 Time Advisory show close
Unit 5 Learning Outcomes show close

5.1 Introduction to Integration
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.1: Introduction to Integration”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.1: Introduction to Integration” (PDF)
Instructions: Read Section 5.1 on pages 1  7 to learn about area. Work through practice problems 1  9. For solutions to these problems, see page 10.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.1: Introduction to Integration”

5.1.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.1: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.1: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  15 on page 8 and page 9. Once you have completed the problem set, check your answers for the oddnumbered questions against here.
Completing the problem set should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.1: Problems for Solution”

5.1.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Area
To review this topic, focus on pages 1  4 of the reading.
√ Applications of Area like Distance and Total Accumulation
To review this topic, focus on pages 5  7 of the reading. 
5.2 Sigma Notation and Riemann Sums
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.2: Sigma Notation and Riemann Sums”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.2: Sigma Notation and Riemann Sums” (PDF)
Instructions: Read Section 5.2 on pages 1  10 to learn about area. Work through practice problems 1  9. For solutions to these problems, see page 15 and page 16.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.2: Sigma Notation and Riemann Sums”

5.2.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.2: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.2: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  61 on pages 10  15. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 3 hours and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.2: Problems for Solution”

5.2.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Sigma Notation
To review this topic, focus on page 1 and page 2 of the reading.
√ Sums of Areas of Rectangles
To review this topic, focus on page 3 and page 4 of the reading.
√ Area under a Curve – Riemann Sums
To review this topic, focus on pages 5  8 of the reading.
√ Two Special Riemann Sums – Lower and Upper Sums
To review this topic, focus on page 9 and page 10 of the reading. 
5.3 The Definite Integral
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.3: The Definite Integral”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.3: The Definite Integral” (PDF)
Instructions: Read Section 5.3 on pages 1  6 to learn about the definite integral and its applications. Work through practice problems 1  6. For solutions to these problems, see page 11.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.3: The Definite Integral”

5.3.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.3: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.3: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  29 on pages 6  10. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 2 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.3: Problems for Solution”

5.3.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ The Definition of the Definite Integral
To review this topic, focus on pages 1  3 of the reading.
√ Definite Integrals of Negative Functions
To review this topic, focus on pages 3  5 of the reading.
√ Units for the Definite Integral
To review this topic, focus on page 5 and page 6 of the reading. 
5.4 Properties of the Definite Integral
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.4: Properties of the Definite Integral”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.4: Properties of the Definite Integral” (PDF)
Instructions: Read Section 5.4 on pages 1  8 to learn about properties of definite integrals and how functions can be defined using definite integrals. Work through practice problems 1  5. For solutions to these problems, see page 11.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.4: Properties of the Definite Integral”

5.4.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.4: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.4: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  51 on pages 9  11. Once you have completed the problem set, check your answers for the oddnumbered questions against here.
Completing the problem set should take approximately 2 hours and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.4: Problems for Solution”

5.4.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Properties of the Definite Integral
To review this topic, focus on page 1 and page 2 of the reading.
√ Properties of Definite Integrals of Combinations of Functions
To review this topic, focus on pages 3  5 of the reading.
√ Functions Defined by Integrals
To review this topic, focus on page 5 and page 6 of the reading.
√ Which Functions Are Integrable?
To review this topic, focus on page 6 and page 7 of the reading.
√ A Nonintegrable Function
To review this topic, focus on page 8 of the reading. 
5.5 Areas, Integrals, and Antiderivatives
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.5: Areas, Integrals, and Antiderivatives”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.5: Areas, Integrals, and Antiderivatives” (PDF)
Instructions: Read Section 5.5 on pages 1  6 to learn about the relationship among areas, integrals, and antiderivatives. Work through practice problems 1  5. For solutions to these problems, see page 10 and page 11.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.5: Areas, Integrals, and Antiderivatives”

5.5.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.5: Problems for Solution
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.5: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  25 on pages 7  9. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 2 hours and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.5: Problems for Solution

5.5.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Area Functions as an Antiderivative
To review this topic, focus on page 1 and page 2 of the reading.
√ Using Antiderivatives to Evaluate Definite Integrals
To review this topic, focus on pages 2  4 of the reading.
√ Integrals, Antiderivatives, and Applications
To review this topic, focus on pages 4  6 of the reading. 
5.6 The Fundamental Theorem of Calculus
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.6: The Fundamental Theorem of Calculus”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.6: The Fundamental Theorem of Calculus” (PDF)
Instructions: Read Section 5.6 on pages 1  9 to see the connection between derivatives and integrals. Work through practice problems 1  5. For solutions to these problems, see page 14 and page 15.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.6: The Fundamental Theorem of Calculus”

5.6.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.6: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.6: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  67 on pages 10  13. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 4 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.6: Problems for Solution”

5.6.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Antiderivatives
To review this topic, focus on pages 1  3 of the reading.
√ Evaluating Definite Integrals
To review this topic, focus on page 4 and page 5 of the reading.
√ Steps for Calculus Application Problems
To review this topic, focus on pages 6  8 of the reading.
√ Leibnitz’s Rule for Differentiating Integrals
To review this topic, focus on page 9 of the reading. Assessment: The Saylor Foundation’s “Problem Set 8”
Link: The Saylor Foundation’s “Problem Set 8” (HTML)
Instructions: You are now ready to complete “Problem Set 8.” If you have not already done so, create a free account on the Moodle website in order to access the quiz and then work on answering the 10 multiplechoice questions. When you have completed the quiz, click on “submit all and finish” to tabulate your score.
Completing this assessment should take you approximately 1 hour.
 Assessment: The Saylor Foundation’s “Problem Set 8”

5.7 Finding Antiderivatives
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.7: Finding Antiderivatives”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.7: Finding Antiderivatives” (PDF)
Instructions: Read Section 5.7 on pages 1  9 to see how one can (sometimes) find an antiderivative. In particular, we will discuss the change of variable technique. Change of variable, also called substitution or usubstitution (for the most commonlyused variable), is a powerful technique that you will use time and again in integration. It allows you to simplify a complicated function to show how basic rules of integration apply to the function. Work through practice problems 1  4. For solutions to these problems, see page 12 and page 13.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.  Web Media: YouTube: RootMath’s “Integration: USubstitution  Ex. 5” and “Integration: USubstitution  Ex. 6”
Link: YouTube: RootMath’s “Integration: USubstitution  Ex. 5” (YouTube) and “Integration: USubstitution  Ex. 6” (YouTube)
Instructions: Watch these videos on change of variable, also called substitution or usubstitution.
Watching these videos and taking notes should take approximately 30 minutes.
Terms of Use: These resources are licensed under a Creative Commons Attribution 3.0 Unported License. They are attributed to RootMath.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.7: Finding Antiderivatives”

5.7.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.7: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.7: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  69 on pages 10  12. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 4 hours.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.7: Problems for Solution”

5.7.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Indefinite Integrals and Antiderivatives
To review this topic, focus on page 1 of the reading.
√ Properties of Antiderivatives (Indefinite Integrals)
To review this topic, focus on page 2 and page 3 of the reading.
√ Antiderivatives of More Complicated Functions
To review this topic, focus on page 3 and page 4 of the reading.
√ Getting the Constant Right
To review this topic, focus on page 4 and page 5 of the reading.
√ Making Patterns More Obvious  Changing Variables
To review this topic, focus on pages 5  8 of the reading.
√ Changing the Variables and Definite Integrals
To review this topic, focus on page 8 and page 9 of the reading.
√ Special Transformations  Antiderivatives of sin^{2}(x) and cos^{2}(x)
To review this topic, focus on page 9 of the reading. 
5.8 First Application of Definite Integral
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.8: First Application of Definite Integrals”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.8: First Application of Definite Integrals” (PDF)
Instructions: Read Section 5.8 on pages 1  8 to see how some applied problems can be reformulated as integration problems. Work through practice problems 1  4. For solutions to these problems, see page 10 and page 11.
Reading this section and completing the practice problems should take approximately 1 hour.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.8: First Application of Definite Integrals”

5.8.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.8: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.8: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  41 on pages 8  10. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 1 hour and 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.8: Problems for Solution”

5.8.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Area between Graphs of Two Functions
To review this topic, focus on pages 1  4 of the reading.
√ Average (Mean) Value of a Function
To review this topic, focus on pages 4  6 of the reading.
√ A Definite Integral Application  Work
To review this topic, focus on pages 6  8 of the reading. 
5.9 Using Tables to Find Antiderivatives
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.9: Using Tables to Find Antiderivatives”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.9: Using Tables to Find Antiderivatives”(PDF)
Instructions: Read Section 5.9 on pages 1  3 to learn how to use tables to find antiderivatives. See the following “Calculus Reference Facts” for the table of integrals mentioned in the reading. Work through practice problems 1  5. For solutions to these problems, see page 6.
Reading this section and completing the practice problems should take approximately 45 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.9: Using Tables to Find Antiderivatives”

5.9.1 Problems for Solution
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.9: Problems for Solution”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.9: Problems for Solution” (PDF)
Instructions: Work through the oddnumbered problems 1  55 on page 4 and page 5. Once you have completed the problem set, check your answers for the oddnumbered questions here.
Completing the problem set should take approximately 2 hours and 30 minutes.
Terms of Use: This resource is licensed under a Creative Commons Attribution 3.0 Unported License. It is attributed to Dale Hoffman, and the original version can be found here.
 Assessment: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 5.9: Problems for Solution”

5.9.2 Reading Review
Before moving on to the next assigned reading, you should be comfortable with each of the topics listed in the reading review below:
√ Table of Integrals
To review this topic, focus on pages 1  3 of the reading.
√ Using Recursive Formulas
To review this topic, focus on page 3 of the reading. 
Unit 5: Assessment
 Assessment: The Saylor Foundation’s “Problem Set 9”
Link: The Saylor Foundation’s “Problem Set 9” (HTML)
Instructions: You are now ready to complete “Problem Set 9.” If you have not already done so, create a free account on the Moodle website in order to access the quiz and then work on answering the 10 multiplechoice questions. When you have completed the quiz, click on “submit all and finish” to tabulate your score.
Completing this assessment should take you approximately 1 hour.
 Assessment: The Saylor Foundation’s “Problem Set 9”

Unit 6: 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 the following: “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.
Unit 6 Time Advisory show close
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: This resource is licensed under a Creative Commons AttributionShareAlike 3.0 Unported License. The original Wikipedia version can be found here.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 6.1: Calculus Reference”
Link: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 6.1: Calculus Reference” (PDF)
Instructions: There are neither readings nor problems associated with this section. Rather, it consists of two pages of formulas that can be useful to you in your further explorations of calculus, including the final exam. You should be able to quickly print out these two pages or save them where they can be easily located as a quick reference when needed.
 Reading: Washington State Board for Community and Technical Colleges: Dale Hoffman’s Contemporary Calculus: “Section 6.1: Calculus Reference”

Final Exam
 Final Exam: The Saylor Foundation’s “MA005 Final Exam”
Link: The Saylor Foundation’s “MA005 Final Exam” (HTML)
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.
Note: While taking the final exam, you are welcome to make use of this Formula Sheet (PDF) and this Online Calculator.  Optional Final Exam: The Saylor Foundation’s NCCRS Creditrecommended “MA005 Final Exam”
Link: The Saylor Foundation’s NCCRS Creditrecommended “MA005 Final Exam” (HTML)
Instructions: The above linked exam has been specially created as part of our National College Credit Recommendation Service (NCCRS) review program. Successfully passing this exam will make students eligible to receive a transcript with 4 hours of recommended college credit.
Note: While taking the final exam, you are welcome to make use of this Formula Sheet (PDF) and this Online Calculator.
Please note that because this exam has the possibility to be a creditbearing exam, it must be administered in a proctored environment, and is therefore password protected. Further information about Saylor’s NCCRS program and the options and requirements for proctoring, can be found here. Make sure to read this page carefully before attempting this exam.
If you choose to take this exam, you may want to first take the regular, certificatebearing MA005 Final Exam as a practice test, which you can find above.
 Final Exam: The Saylor Foundation’s “MA005 Final Exam”