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.

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.

Completing this unit should take you approximately 29 hours.
- Upon successful completion of this unit, you will be able to:
  
  state whether a given point on a graph is a global/local maximum/minimum;
  
  find critical points and extreme values (max/min) of functions by using derivatives;
  
  determine the values of a function guaranteed to exist by Rolle's Theorem and by the Mean Value Theorem;
  
  use the graph of $f(x)$ to sketch the shape of the graph of $f(x)$ ;
  
  use the values of $f'(x)$ to sketch the graph of $f(x)$ and state whether $f(x)$ is increasing or decreasing at a point;
  
  use the values of $f''(x)$ to determine the concavity of the graph of $f(x)$ ;
  
  use the graph of $f(x)$ to determine if $f''(x)$ is positive, negative, or zero;
  
  solve maximum and minimum problems by using derivatives;
  
  restate in words the meanings of the solutions to applied problems, attaching the appropriate units to an answer;
  
  determine asymptotes of a function by using limits; and
  
  determine the values of indeterminate form limits by using derivatives and L'Hopital's Rule.
- 4.1: Finding Maximums and Minimums
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.1: Finding Maximums and Minimums" URL
    Read this section 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 pages 13-14.
  - 4.1.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.1: Practice Problems" URL
    Work through the odd-numbered problems 1-43 on pages 9-13. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 4.1.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Methods for Finding Maximums and Minimums; page 1.
    Terminology: Global Maximum, Local Maximum, Maximum Point, Global Minimum, Local Minimum, Global Extreme, and Local Extreme; page 2.
    Finding Maximums and Minimums of a Function; pages 3-5.
    Is $f(a)$ a Maximum, Minimum, or Neither?; page 5.
    Endpoint Extremes; pages 5-7.
    Critical Numbers; page 7.
    Which Functions Have Extremes?; pages 7-8.
    Extreme Value Theorem; pages 8-9.
- 4.2: The Mean Value Theorem and Its Consequences
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.2: The Mean Value Theorem and Its Consequences" URL
    Read this section 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.
  - 4.2.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.2: Practice Problems" URL
    Work through the odd-numbered problems 1-35 on pages 6-9. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 4.2.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Rolle's Theorem; pages 1-2.
    The Mean Value Theorem; pages 2-4.
    Consequences of the Mean Value Theorem; pages 4-6.
- 4.3: The First Derivative and the Shape of a Function f(x)
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.3: The First Derivative and the Shape of f" URL
    Read this section 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.
  - RootMath: "First Derivative Test, Example 2" Page
    Watch both parts of this video on the first derivative test.
  - 4.3.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.3: Practice Problems" URL
    Work through the odd-numbered problems 1-29 on pages 8-10. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 4.3.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Definitions of the Function; page 1.
    First Shape Theorem; pages 2-4.
    Second Shape Theorem; pages 4-7.
    Using the Derivative to Test for Extremes; pages 7-8.
- 4.4: The Second Derivative and the Shape of a Function f(x)
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.4: Second Derivative and the Shape of f" URL
    Read this section 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 pages 8-9.
  - Khan Academy: "Concavity, Concave upwards and Concave downwards Intervals" Page
    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.
  - RootMath: "Concavity and the Second Derivative" Page
    Watch this video, which works through an example of the second derivative test.
  - 4.4.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.4: Practice Problems" URL
    Work through the odd-numbered problems 1-17 on pages 6-8. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 4.4.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Concavity; pages 1-2.
    The Second Derivative Condition for Concavity; pages 2-3.
    Feeling the Second Derivative: Acceleration Applications; pages 3-4.
    The Second Derivative and Extreme Values; pages 4-5.
    Inflection Points; pages 5-6.
- Problem Set 6
  - Problem Set 6 Quiz
    Take this assessment to see how well you understood these concepts.
    
    This assessment does not count towards your grade. It is just for practice!
    
    You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
    
    You can take this assessment as many times as you want, whenever you want.
- 4.5: Applied Maximum and Minimum Problems
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.5: Applied Maximum and Minimum Problems" URL
    Read this section 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 pages 15-16. There is no review for this section; instead, make sure to study the problems carefully to become familiar with applied maximum and minimum problems.
  - Khan Academy: "Minimizing Sum of Squares" Page
    Watch this video on optimization.
  - Massachusetts Institute of Technology: Christine Breiner's "Minimum Triangle Area" Page
    Watch this video on optimization.
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.5: Practice Problems" URL
    Work through the odd-numbered problems 1-33 on pages 6-15. Once you have completed the problem set, check your answers for the odd-numbered questions here.
- 4.6: Infinite Limits and Asymptotes
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.6: Infinite Limits and Asymptotes" URL
    Read this section 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 pages 13-14.
  - 4.6.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.6: Practice Problems" URL
    Work through the odd-numbered problems 1-59 on pages 10-12. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 4.6.2: Review
    Before moving on, you should be comfortable with each of these topics:
    Limits as $x$ Approaches Infinity; pages 1-4.
    Using Calculators to Find Limits as $x$ Goes to Infinity; page 5.
    The Limit Is Infinite; pages 5-6.
    Horizontal Asymptotes; pages 6-7.
    Vertical Asymptotes; pages 7-8.
    Other Asymptotes as $x$ Approaches Infinity; pages 8-9.
    Definition of $\lim_{x\rightarrow \infty}{x}=k$ ; pages 9-10.
- 4.7: L'Hopital's Rule
  - Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.7: L'Hopital's Rule" URL
    Read this section 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 8.
  - Khan Academy: "Introduction to L'Hopital's Rule" Page
    Watch this video for an introduction to L'Hopital's Rule.
  - 4.7.1: Practice Problems
    Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 4.7: Practice Problems" URL
    Work through the odd-numbered problems 1-29 on pages 6-7. Once you have completed the problem set, check your answers for the odd-numbered questions here.
  - 4.7.2: Review
    Before moving on, you should be comfortable with each of these topics found in the resources linked to above:
    A Linear Example; page 1.
    0/0 Form of L'Hopital's Rule; page 2.
    Strong Version of L'Hopital's Rule; pages 2-3.
    Which Function Grows Faster?; page 4.
    Other Indeterminate Forms; pages 4-6.
- Problem Set 7
  - Problem Set 7 Quiz
    Take this assessment to see how well you understood these concepts.
    
    This assessment does not count towards your grade. It is just for practice!
    
    You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
    
    You can take this assessment as many times as you want, whenever you want.

Unit 4: Derivatives and Graphs