Riemann Sum & Table Problems (Type 5)

Posted on March 25, 2022 by Lin McMullin

AP Questions Type 5: Riemann Sum & Table Problems

Information given in tables may be used to test a variety of ideas in calculus including analysis of functions, accumulation, theory and theorems, and position-velocity-acceleration, among others. Numbers and working with numbers are part of the Rule of Four and table problems are one way they are tested. This question often includes an equation in a latter part of the problem that refers to the same situation.

What students should be able to do:

Find the average rate of change over an interval
Approximate the derivative using a difference quotient. Use the two values closest to the number at which you are approximating. This amounts to finding the slope or rate of change. Show the quotient even if you can do the arithmetic in your head and even if the denominator is 1.
Use a left-, right-, or midpoint- Riemann sums or a trapezoidal approximation to approximate the value of a definite integral using values in the table (typically with uneven subintervals). The Trapezoidal Rule, per se, is not required; it is expected that students will add the areas of a small number of trapezoids without reference to a formula.
Average value, average rate of change, Rolle’s theorem, the Mean Value Theorem, and the Intermediate Value Theorem. (See 2007 AB 3 – four simple parts that could be multiple-choice questions; the mean on this question was 0.96 out of a possible nine points.)
These questions are usually presented in context and answers should be in that context. The context may be something growing (changing over time) or linear motion.
Use the table to find a value based on the Mean Value Theorem (2018 AB 4(b)) or Intermediate Value Theorem. Also, 2018 AB 4 (d) asked a related question based on a function given by an equation.
Unit analysis.

Dos and Don’ts

Do: Remember that you do not know what happens between the values in the table unless additional information is given. For example, do not assume that the largest number in the table is the maximum value of the function, or that the function is decreasing between two values just because a value is less than the preceding value.

Do: Show what you are doing even if you can do it in your head. If you’re finding a slope, show the quotient even if the denominator is 1.

Do Not do arithmetic: A long expression consisting entirely of numbers such as you get when doing a Riemann sum, does not need to be simplified in any way. If you simplify a correct answer incorrectly, you will lose credit.

Do Not leave expression such as R(3) – pull its numerical value from the table.

Do Not: Find a regression equation and then use that to answer parts of the question. While regression is perfectly good mathematics, regression equations are not one of the four things students may do with their calculator. Regression gives only an approximation of our function. The exam is testing whether students can work with numbers.

This question typically covers topics from Unit 6 of the CED but may include topics from Units 2, 3, and 4 as well.

Free-response examples:

2007 AB 3 (4 simple parts on various theorems, yet the mean score was 0.96 out of 9),
2017 AB 1/BC 1, and AB 6,
2016 AB 1/BC 1
2018 AB 4
2021 AB 1/ BC 1
2022 AB4/BC4 – average rate of change, IVT, Rieman sum, Related Rate (part (d) good question)

Multiple-choice questions from non-secure exams:

2012 AB 8, 86, 91
2012 BC 8, 81, 86 (81 and 86 are the same on both the AB and BC exams)

Revised March 12, 2021, March 25, 2022

Foreshadowing the MVT

Posted on October 15, 2021 by Lin McMullin

The Mean Value Theorem (MVT) is proved by writing the equation of a function giving the (directed) length of a segment from the given function to the line between the endpoints as you can see here. Since the function and the line intersect at the endpoints of the interval this function satisfies the hypotheses of Rolle’s theorem and so the MVT follows directly. This means that the derivative of the distance function is zero at the points guaranteed by the MVT. Therefore, these values must also be the location of the local extreme values (maximums and minimums) of the distance function on the open interval. *

Here is an exploration in three similar examples that use this idea to foreshadow the MVT. You, of course, can use your own favorite function. Any differentiable function may be used, in which case a CAS calculator may be helpful. Answers are at the end.

First example:

Consider the function $\displaystyle f\left( x \right)=x+2\sin \left( {\pi x} \right)$ defined on the closed interval [–1,3]

Write the equation of the line thru the endpoints of the function.
Write an expression for h(x) the vertical distance between f(x) and the line found in part 1.
Find the x-coordinates of the local extreme values of h(x) on the open interval (–1,3).
Find the slope of f(x) at the values found in part 3.
Compare your answer to part 4 with the slope of the line. Is this a coincidence?

Second example: slightly more difficult than the first.

Consider the function $\displaystyle f\left( x \right)=1+x+2\cos (x)$ defined on the closed interval $\displaystyle [\tfrac{\pi }{2},\tfrac{{9\pi }}{2}]$ .

Write the equation of the line thru the endpoints of the function.
Write an expression for h(x) the vertical distance between f(x) and the line found in part 1.
Find the x-coordinates of the local extreme values of h(x) on the open interval $\displaystyle \left( {\tfrac{\pi }{2},\tfrac{{9\pi }}{2}} \right)$ .
Find the slope of f(x) at the values found in part 3.
Compare your answer to part 4 with the slope of the line. Is this a coincidence?

Third example: In case you think I cooked the numbers. You may want to use a CAS for this one.

Consider the function $\displaystyle f(x)={{x}^{3}}$ defined on the closed interval $\displaystyle [-4,5]$ .

Write the equation of the line thru the endpoints of the function.
Write an expression for h(x) the vertical distance between f(x) and the line found in part 1.
Find the x-coordinates of the local extreme values of h(x) on the open interval $\displaystyle \left( {-4,5} \right)$ .
Find the slope of f(x) at the values found in part 3.
Compare your answer to part 4 with the slope of the line. Is this a coincidence?

Answers

First example:

y = x
$\displaystyle h(x)=f(x)-y(x)=x-\left( {x+2\sin (\pi x} \right)=\left( {2\sin (\pi x} \right)$
$\displaystyle {h}'\left( x \right)=2\pi \cos \left( {\pi x} \right)=0$ when x = –1/2, ½, 3/2 and 5/2
$\displaystyle {f}'\left( x \right)=1+2\pi \cos \left( {\pi x} \right)=1$ , the slope = 1 at all four points
They are the same. Not a coincidence.

Second example:

The endpoints are $\displaystyle \left( {\tfrac{\pi }{2},1+\tfrac{\pi }{2}} \right)$ and $\displaystyle \left( {\tfrac{9\pi }{2},1+\tfrac{{9\pi }}{2}} \right)$ ; the line is $\displaystyle y=x+1$
$\displaystyle h(x)=f(x)-y(x)=\left( {1+x+2\cos (x)} \right)-(x+1)=2\cos (x)$
$\displaystyle {h}'\left( x \right)=-2\sin (x)=0$ when $\displaystyle x=\pi ,2\pi ,3\pi ,\text{ and }4\pi$
$\displaystyle {f}'\left( x \right)=1-2\sin \left( x \right)=1$ ; at the points above the slope is 1.
They are the same. Not a coincidence.

Third example:

The endpoints are (-4, -64) and (5, 125), the line is $\displaystyle y=125+21\left( {x-5} \right)=21x+20$ .
$\displaystyle h\left( x \right)={{x}^{3}}-21x-20$
$\displaystyle {h}'\left( x \right)=3{{x}^{2}}-21=0$ when $\displaystyle x=\sqrt{7},-\sqrt{7}$
$\displaystyle {f}'\left( {\pm \sqrt{7}} \right)=3{{\left( {\pm \sqrt{7}} \right)}^{2}}=21$
They are the same. Not a coincidence.

See this post for links to other posts discussing the full development of the MVT

* It is possible that the derivative is zero and the point is not an extreme value. This is like the situation with a point of inflection when the first derivative is zero but does not change sign.

This post was originally published on October 19, 2018.

Unit 5 – Analytical Applications of Differentiation

Posted on October 12, 2021 by Lin McMullin

Unit 5 covers the application of derivatives to the analysis of functions and graphs. Reasoning and justification of results are also important themes in this unit. (CED – 2019 p. 92 – 107). These topics account for about 15 – 18% of questions on the AB exam and 8 – 11% of the BC questions.

You may want to consider teaching Unit 4 after Unit 5. Notes on Unit 4 are here.

Reasoning and writing justification of results are mentioned and stressed in the introduction to the topic (p. 93) and for most of the individual topics. See Learning Objective FUN-A.4 “Justify conclusions about the behavior of a function based on the behavior of its derivatives,” and likewise in FUN-1.C for the Extreme value theorem, and FUN-4.E for implicitly defined functions. Be sure to include writing justifications as you go through this topic. Use past free-response questions as exercises and also as guide as to what constitutes a good justification. Links in the margins of the CED are also helpful and give hints on writing justifications and what is required to earn credit. See the presentation Writing on the AP Calculus Exams and its handout

Topics 5.1

Topic 5.1 Using the Mean Value Theorem While not specifically named in the CED, Rolle’s Theorem is a lemma for the Mean Value Theorem (MVT). The MVT states that for a function that is continuous on the closed interval and differentiable over the corresponding open interval, there is at least one place in the open interval where the average rate of change equals the instantaneous rate of change (derivative). This is a very important existence theorem that is used to prove other important ideas in calculus. Students often confuse the average rate of change, the mean value, and the average value of a function – See What’s a Mean Old Average Anyway?

Topics 5.2 – 5.9

Topic 5.2 Extreme Value Theorem, Global Verses Local Extrema, and Critical Points An existence theorem for continuous functions on closed intervals

Topic 5.3 Determining Intervals on Which a Function is Increasing or Decreasing Using the first derivative to determine where a function is increasing and decreasing.

Topic 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema Using the first derivative to determine local extreme values of a function

Topic 5.5 Using the Candidates’ Test to Determine Absolute (Global) Extrema The Candidates’ test can be used to find all extreme values of a function on a closed interval

Topic 5.6 Determining Concavity of Functions on Their Domains FUN-4.A.4 defines (at least for AP Calculus) When a function is concave up and down based on the behavior of the first derivative. (Some textbooks may use different equivalent definitions.) Points of inflection are also included under this topic.

Topic 5.7 Using the Second Derivative Test to Determine Extrema Using the Second Derivative Test to determine if a critical point is a maximum or minimum point. If a continuous function has only one critical point on an interval then it is the absolute (global) maximum or minimum for the function on that interval.

Topic 5.8 Sketching Graphs of Functions and Their Derivatives First and second derivatives give graphical and numerical information about a function and can be used to locate important points on the graph of the function.

Topic 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative First and second derivatives give graphical and numerical information about a function and can be used to locate important points on the graph of the function.

Topics 5.10 – 5.11

Optimization is an important application of derivatives. Optimization problems as presented in most textbooks, begin with writing the model or equation that describes the situation to be optimized. This proves difficult for students, and is not “calculus” per se. Therefore, writing the equation has not been asked on AP exams in recent years (since 1983). Questions give the expression to be optimized and students do the “calculus” to find the maximum or minimum values. To save time, my suggestion is to not spend too much time writing the equations; rather concentrate on finding the extreme values.

Topic 5.10 Introduction to Optimization Problems

Topic 5.11 Solving Optimization Problems

Topics 5.12

Topic 5.12 Exploring Behaviors of Implicit Relations Critical points of implicitly defined relations can be found using the technique of implicit differentiation. This is an AB and BC topic. For BC students the techniques are applied later to parametric and vector functions.

Timing

Topic 5.1 is important and may take more than one day. Topics 5.2 – 5.9 flow together and for graphing they are used together; after presenting topics 5.2 – 5.7 spend the time in topics 5.8 and 5.9 spiraling and connecting the previous topics. Topics 5.10 and 5.11 – see note above and spend minimum time here. Topic 5.12 may take 2 days.

The suggested time for Unit 5 is 15 – 16 classes for AB and 10 – 11 for BC of 40 – 50-minute class periods, this includes time for testing etc.

Finally, were I still teaching, I would teach this unit before Unit 4. The linear motion topic (in Unit 4) are a special case of the graphing ideas in Unit 5, so it seems reasonable to teach this unit first. See Motion Problems: Same thing, Different Context

This is a re-post and update of the third in a series of posts from last year. It contains links to posts on this blog about the differentiation of composite, implicit, and inverse functions for your reference in planning. Other updated post on the 2019 CED will come throughout the year, hopefully, a few weeks before you get to the topic.

Previous posts on these topics include:

Then There Is This – Existence Theorems

What’s a Mean Old Average Anyway

Did He, or Didn’t He? History: how to find extreme values without calculus

Mean Value Theorem

Foreshadowing the MVT

Fermat’s Penultimate Theorem

Rolle’s theorem

The Mean Value Theorem I

The Mean Value Theorem II

Graphing

Concepts Related to Graphs

The Shapes of a Graph

Joining the Pieces of a Graph

Extreme Values

Extremes without Calculus

Concavity

Reading the Derivative’s Graph

Other Asymptotes

Real “Real-life” Graph Reading

Far Out! An exploration

Open or closed Should intervals of increasing, decreasing, or concavity be open or closed?

Others

Lin McMullin’s Theorem and More Gold The Golden Ratio in polynomials

Soda Cans Optimization video

Optimization – Reflections

Curves with Extrema?

Good Question 10 – The Cone Problem

Implicit Differentiation of Parametric Equations BC Topic

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

Limits and Continuity – Unit 1 (8-11-2020)

Definition of t he Derivative – Unit 2 (8-25-2020)

Differentiation: Composite, Implicit, and Inverse Function – Unit 3 (9-8-2020)

Contextual Applications of the Derivative – Unit 4 (9-22-2002) Consider teaching Unit 5 before Unit 4

Analytical Applications of Differentiation – Unit 5 (9-29-2020) Consider teaching Unit 5 before Unit 4 THIS POST

LAST YEAR’S POSTS – These will be updated in coming weeks

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions

2019 CED Unit 10 Infinite Sequences and Series

Discovering the MVT

Posted on October 6, 2020 by Lin McMullin

Today’s Blog is an exploration that will lead up to the Mean Value Theorem (MVT) and, I hope, help your students better understand the MVT and why it is true.

While you may do this by hand, using a graphing calculator will make things way easier. This is good calculator practice and can be done on a graphing, non-CAS, calculator without writing anything down. Try it that way.

Here are the steps to follow. My solution with screen pictures is below.

Choose your favorite differentiable function. Call it f(x) and enter it in your calculator as Y1.
Choose two values, a and b, in the domain of your function. Save (store) these on your calculator as a and b.
Find the slope of the line (a, f(a)) and (b, f(b)). It would be best, but not necessary, that the line intersects the function only at (a, f(a)) and (b, f(b)) not between them, and not be horizontal. Store this in your calculator as m.
Write the equation of the line through (a, f(a)) and (b, f(b)) and enter it as Y2.
Write a function, h(x), that gives the vertical distance between f(x) and the line found in step 3. (Hint: upper curve minus the lower.) Enter this as Y3
Find the x-coordinate local extreme value of h(x). Store this number to c.
Find the slope of the tangent line to f(x) at the value found in step 6.
What do you notice? Compare your result and conclusion with the other in your class. Discuss.

My solution.

Step 1: I choose $f\left( x \right)=x+2\sin \left( x \right)$ and entered this in my calculator as Y1

Step 2: I choose a = 1 and b = 3 and stored them in my calculator.

Step 3: I calculated the slope in my calculator – see first figure.

Step 4: The equation of the line is $y=f\left( a \right)+m\left( {x-a} \right)$ . I entered this as Y2 in my calculator.

Step 5 $h\left( x \right)=Y1(x)-Y2(x)=\left( {x+2\sin (x)} \right)-\left( {f\left( a \right)+m\left( {x-a} \right)} \right)$

Step 6: ${h}'\left( x \right)=1+2\cos (x)-m$

Solve ${h}'\left( x \right)=0$ for the value between a and b on your calculator. See second figure.

Step 6 and 7: I stored this value to C in my calculator and computed ${f}'(c)$ on the home screen. See third figure.

Step 8: It is no coincidence that ${f}'\left( c \right)=m$ .

The Mean Value Theorem states that for a function that is continuous on the interval [a, b] and differentiable on the open interval (a, b) there exists a number c in (a, b) such that

$\displaystyle {f}'\left( c \right)=\frac{{f\left( b \right)-f\left( a \right)}}{{b-a}}$

Additional Exploration:

Can you show why ${f}'\left( c \right)=m$ . ? Hint: Look at the expression for ${h}'\left( c \right)$ in step 5; set it equal to zero. Why must the solution be the value that makes ${f}'\left( x \right)=m$ ?
What does this mean graphically?

Pick a different value for a and/or b so that the line between (a, f(a)) and (b, f(b)) intersects f(x) two (or more) times. The derivative will now have two (or more) zeros. Find them and calculate the slope at each one. What do you notice?

Students often confuse the Mean Value Theorem, the Average Rate of Change of a function on an interval, and the Average Value of a function on an interval. This is understandable because of the similarity in their names and the similarity of their results. Be sure to point this out as you teach them and help them learn the meanings of each.

Other posts related to the Mean Value Theorem

Foreshadowing the MVT Other examples using this technique

Existence Theorems

Fermat’s Penultimate Theorem A lemma for Rolle’s Theorem: Any function extreme value(s) on an open interval must occur where the derivative is zero or undefined.

Rolle’s Theorem A lemma for the MVT: On an interval if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b) and f(a) = f(b), there must exist a number c in the open interval (a, b) where f ‘(c) = 0.

Mean Value Theorem I Proof

Mean Value Theorem II Graphical Considerations

Darboux’s Theorem the Intermediate Value Theorem for derivatives.

Mean Tables

The Definite Integral and the FTC

Analytical Applications of Differentiation – Unit 5

Posted on September 29, 2020 by Lin McMullin

You may want to consider teaching Unit 4 after Unit 5. Notes on Unit 4 are here.

Topics 5.1

Topics 5.2 – 5.9

Topic 5.2 Extreme Value Theorem, Global Verses Local Extrema, and Critical Points An existence theorem for continuous functions on closed intervals

Topic 5.3 Determining Intervals on Which a Function is Increasing or Decreasing Using the first derivative to determine where a function is increasing and decreasing.

Topic 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema Using the first derivative to determine local extreme values of a function

Topic 5.5 Using the Candidates’ Test to Determine Absolute (Global) Extrema The Candidates’ test can be used to find all extreme values of a function on a closed interval

Topics 5.10 – 5.11

Optimization is important application of derivatives. Optimization problems as presented in most text books, begin with writing the model or equation that describes the situation to be optimized. This proves difficult for students, and is not “calculus” per se. Therefore, writing the equation has not be asked on AP exams in recent years (since 1983). Questions give the expression to be optimized and students do the “calculus” to find the maximum or minimum values. To save time, my suggestion is to not spend too much time writing the equations; rather concentrate on finding the extreme values.

Topic 5.10 Introduction to Optimization Problems

Topic 5.11 Solving Optimization Problems

Topics 5.12

Timing

The suggested time for Unit 5 is 15 – 16 classes for AB and 10 – 11 for BC of 40 – 50-minute class periods, this includes time for testing etc.

Previous posts on these topics include:

Then There Is This – Existence Theorems

What’s a Mean Old Average Anyway

Did He, or Didn’t He? History: how to find extreme values without calculus

Mean Value Theorem

Foreshadowing the MVT

Fermat’s Penultimate Theorem

Rolle’s theorem

The Mean Value Theorem I

The Mean Value Theorem II

Graphing

Concepts Related to Graphs

The Shapes of a Graph

Joining the Pieces of a Graph

Extreme Values

Extremes without Calculus

Concavity

Reading the Derivative’s Graph

Other Asymptotes

Real “Real-life” Graph Reading

Far Out! An exploration

Open or Closed Should intervals of increasing, decreasing, or concavity be open or closed?

Others

Lin McMullin’s Theorem and More Gold The Golden Ratio in polynomials

Soda Cans Optimization video

Optimization – Reflections

Curves with Extrema?

Good Question 10 – The Cone Problem

Implicit Differentiation of Parametric Equations BC Topic

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

Limits and Continuity – Unit 1 (8-11-2020)

Definition of t he Derivative – Unit 2 (8-25-2020)

Differentiation: Composite, Implicit, and Inverse Function – Unit 3 (9-8-2020)

Contextual Applications of the Derivative – Unit 4 (9-22-2002) Consider teaching Unit 5 before Unit 4

Analytical Applications of Differentiation – Unit 5 (9-29-2020) Consider teaching Unit 5 before Unit 4 THIS POST

LAST YEAR’S POSTS – These will be updated in coming weeks

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions

2019 CED Unit 10 Infinite Sequences and Series

Riemann Sum & Table Problems (Type 5)

Posted on March 17, 2020 by Lin McMullin

AP Questions Type 5: Riemann Sum & Table Problems

Tables may be used to test a variety of ideas in calculus including analysis of functions, accumulation, theory and theorems, and position-velocity-acceleration, among others. Numbers and working with numbers is part of the Rule of Four and table problems are one way this is tested. This question often includes an equation in a latter part of the problem that refers to the same situation.

What students should be able to do:

Find the average rate of change over an interval
Approximate the derivative using a difference quotient. Use the two values closest to the number at which you are approximating. This amounts to finding the slope or rate of change. Show the quotient even if you can do the arithmetic in your head and even if the denominator is 1.
Use a left-, right-, or midpoint- Riemann sums or a trapezoidal approximation to approximate the value of a definite integral using values in the table (typically with uneven subintervals). The Trapezoidal Rule, per se, is not required; it is expected that students will add the areas of a small number of trapezoids without reference to a formula.
Average value, average rate of change, Rolle’s theorem, the Mean Value Theorem and the Intermediate Value Theorem. (See 2007 AB 3 – four simple parts that could be multiple-choice questions; the mean on this question was 0.96 out of a possible 9.)
These questions are usually presented in some context and answers should be in that context. The context may be something growing (changing over time) or linear motion.
Use the table to find a value based on the Mean Value Theorem (2018 AB 4(b)) or Intermediate Value Theorem.
One of the parts of this question asks a related question based on a function given by an equation.
Unit analysis.

Do’s and Don’ts

Do: Remember that you do not know what happens between the values in the table unless some other information is given. For example, do not assume that the largest number in the table is the maximum value of the function, or that the function is decreasing between two values just because a value is less than the preceding value.

Do: Show what you are doing even if you can do it in your head. If you’re finding a slope, show the quotient even if the denominator is 1.

Do Not do arithmetic: A long expression consisting entirely of numbers such as you get when doing a Riemann sum, does not need to be simplified in any way. If you a simplify correct answer incorrectly, you will lose credit.

Do Not leave expression such as R(3) – pull its numerical value from the table.

This question typically covers topics from Unit 6 of the 2019 CED but may include topics from Units 2, 3, and 4 as well.

Free-response examples:

2007 AB 3 (4 simple parts on various theorems, yet the mean score was 0.96 out of 9),
2017 AB 1/BC 1, and AB 6,
2016 AB 1/BC 1
2018 AB 4

Multiple-choice questions from non-secure exams:

2012 AB 8, 86, 91
2012 BC 8, 81, 86 (81 and 86 are the same on both the AB and BC exams)

Revised March 12, 2021

2019 CED Unit 5 Analytical Applications of Differentiation

Posted on October 15, 2019 by Lin McMullin

Topics 5.1

Topics 5.2 – 5.9

Topic 5.2 Extreme Value Theorem, Global Verses Local Extrema, and Critical Points An existence theorem for continuous functions on closed intervals

Topic 5.3 Determining Intervals on Which a Function is Increasing or Decreasing Using the first derivative to determine where a function is increasing and decreasing.

Topic 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema Using the first derivative to determine local extreme values of a function

Topic 5.5 Using the Candidates’ Test to Determine Absolute (Global) Extrema The Candidates’ test can be used to find all extreme values of a function on a closed interval

Topic 5.7 Using the Second Derivative Test to Determine Extrema Using the Second Derivative Test to determine if a critical point is a maximum or minimum point. If a continuous function has only one critical point on an interval, then it is the absolute (global) maximum or minimum for the function on that interval.

Topic 5.8 Sketching Graphs of Functions and Their Derivatives. First and second derivatives give graphical and numerical information about a function and can be used to locate important points on the graph of the function.

Topic 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative. First and second derivatives give graphical and numerical information about a function and can be used to locate important points on the graph of the function.