A very typical calculus problem is given the equation of a function, to find information about it (extreme values, concavity, increasing, decreasing, etc., etc.). This is usually done by computing and analyzing the first derivative and the second derivative. All the textbooks show how to do this with copious examples and exercises. I have nothing to add to that. One of the “tools” of this approach is to draw a number line and mark the information about the function and the derivative on it.

A very typical AP Calculus exam problem is given the graph of the derivative of a function, but not the equation of either the derivative or the function, to find all the same information about the function. For some reason, students find this difficult even though the two-dimensional graph of the derivative gives all the same information as the number line graph and, in fact, a lot more.

Looking at the graph of the derivative in the x,y-plane it is easy to determine the important information. Here is a summary relating the features of the graph of the derivative with the graph of the function.

 Feature the function ${y}'$> 0 is increasing ${y}'$ < 0 is decreasing ${y}'$ changes – to + has a local minimum ${y}'$changes + to – has a local maximum ${y}'$ increasing is concave up ${y}'$ decreasing is concave down ${y}'$ extreme value has a point of inflection

Here’s a typical graph of a derivative with the first derivative features marked. Here is the same graph with the second derivative features marked. The AP Calculus Exams also ask students to “Justify Your Answer.” The table above, with the columns switched does that. The justifications must be related to the given derivative, so a typical justification might read, “The function has a relative maximum at x-2 because its derivative changes from positive to negative at x = -2.”

 Conclusion Justification y is increasing ${y}'$> 0 y is decreasing ${y}'$< 0 y has a local minimum ${y}'$changes  – to + y has a local maximum ${y}'$changes + to – y is concave up ${y}'$increasing y is concave down ${y}'$decreasing y has a point of inflection ${y}'$extreme values

For notes on asymptotes see Asymptotes and the Derivative and Other Asymptotes.

Originally posted on October 26, 2012, and my single most viewed post over the years.

# 2019 CED Unit 5 Analytical Applications of Differentiation

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.

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

### 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 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

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

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

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

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

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

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation  Consider teaching Unit 5 before Unit 4

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

# An Exploration in Differential Equations

This is an exploration based on the AP Calculus question 2018 AB 6. I originally posed it for teachers last summer. This will make, I hope, a good review of many of the concepts and techniques students have learned during the year. The exploration, which will take an hour or more, includes these topics:

• Finding the general solution of the differential equation by separating the variables
• Checking the solution by substitution
• Using a graphing utility to explore the solutions for all values of the constant of integration, C
• Finding the solutions’ horizontal and vertical asymptotes
• Finding several particular solutions
• Finding the domains of the particular solutions
• Finding the extreme value of all solutions in terms of C
• Finding the second derivative (implicit differentiation)
• Considering concavity
• Investigating a special case or two

I also hope that in working through this exploration students will learn not so much about this particular function, but how to use the tools of algebra, calculus, and technology to fully investigate any function and to find all its foibles. The exploration is here in a PDF file. Here are the solutions.

The College Board is pleased to offer a new live online event for new and experienced AP Calculus teachers on March 5th at 7:00 PM Eastern.

I will be the presenter.

The topic will be AP Calculus: How to Review for the Exam:  In this two-hour online workshop, we will investigate techniques and hints for helping students to prepare for the AP Calculus exams. Additionally, we’ll discuss the 10 type questions that appear on the AP Calculus exams, and what students need know and to be able to do for each. Finally, we’ll examine resources for exam review.

Registration for this event is \$30/members and \$35/non-members. You can register for the event by following this link: http://eventreg.collegeboard.org/d/xbqbjz

# Teaching Concavity

As you’ve probably noticed, different authors use different definitions based on how they plan to present topics later in their texts. So, the same concept seems to have different definitions. A definition in one book is a theorem in another. Students should be aware of this; looking at different definitions and related theorems about the same idea helps their mathematical education.

Here is a thought on exploring how concepts may be defined and learning about concavity are the same time.

Start by drawing the 4 shapes of (non-linear) graphs (See Concepts Related to Graphs and The Shapes of a Graph.

• Increasing, concave up
• Increasing concave down
• Decreasing concave up
• Decreasing concave down

Look at the sine or cosine graphs which show all four and the tangent graph that shows only two. Look at other functions as well and identify which parts (intervals) exhibit each shape.

Next, challenge the students, in groups, individually, in class, or for homework to find analytic ways to say what concavity is or to identify it from equations. If they need hints suggest they look at derivatives (first and second) and tangent lines. Don’t limit them – explain that there are other ways. They should try to find several ways.

Hopefully, they will come up with some of these. (I have purposely listed these in a rough preliminary form. That’s how math is done – get an idea and then develop and formalize it)

A function is concave up (down) when:

1. The slope (derivative) is increasing (decreasing)
2. The second derivative is positive (negative).
3. The tangent line lies below (above) the graph
4. All (any, every) segments joining points in the interval lie above (below) the graph.
5. Others ???

Finally, clean these up. Help the students:

• State them as “if, then” or “if, and only if” statements giving the hypotheses for each.
• Consider how one can be used to imply the others – in either direction.
• Determine which implies the inclusion of endpoints (1 and maybe 3).
• Discuss which the class thinks would make the best definition, and which should become theorems.
• Taking whichever definition your book uses, show how the others can be proved.

The point here is the thinking, forming ideas, doing the mathematics; the understanding of concavity will follow.

# Summer Fun

Every Spring I have a lot of fun proofreading Audrey Weeks’ new Calculus in Motion illustrations for the most recent AP Calculus Exam questions. These illustrations run on Geometers’ Sketchpad. In addition to the exam questions Calculus in Motion (and its companion Algebra in Motion) include separate animations illustrating most of the concepts in calculus and algebra. This is a great resource for your classes.

This year, I really got into 2018 AB 6, the differential equation question. I wrote an exploration (or as the kids would say “worksheet”) on a function very similar to the differential equation in that question. The exploration, which is rather long, includes these topics:

• Finding the general solution of the differential equation by separating the variables
• Checking the solution by substitution
• Using a graphing utility to explore the solutions for all values of the constant of integration, C
• Finding the solutions’ horizontal and vertical asymptotes
• Finding several particular solutions
• Finding the domains of the particular solutions
• Finding the extreme value of all solutions in terms of C
• Finding the second derivative (implicit differentiation)
• Considering concavity
• Investigating a special case or two

I also hope that in working through this exploration students will learn not so much about this particular function, but how to use the tools of algebra, calculus, and technology to fully investigate any function and to find all its foibles.

Students will need to have studied calculus through differential equations before they start the exploration. I will repost it next January for them.

The exploration is here for you to try. Try it before you look at the solutions. It will give you something to do over the summer – well not all summer, only an hour or so.

There will be only occasional, very occasional, posts over the Summer. More regular posting will begin again in August. Enjoy the Explorations, and, more important, enjoy the Summer!

# Graphing with Accumulation 2

Accumulation 4: Graphing Ideas in Accumulation – Concavity

In the last post we saw how thinking about Riemann sum rectangles, RΣR, moving across the graph of the derivative made it easy to see when the function whose derivative was given increased and decreased and had its local extreme values. Today we will consider concavity.

Suppose a derivative is constant, its graph a horizontal line. In this case each successive RΣR is exactly the same size and adds exactly the same amount to the accumulated function. The function’s graph increases (or decreases) by exactly the same amount – it is linear.

For derivatives that are not constant, the change in the resulting function is not constant and the function’s graph bends up or down. This bending of the function is referred to as its concavity. If it bends up, the function increases faster and its graph is concave up; if it bends down, it is increasing slower (or decreasing faster) and concave down. The graph above shows pairs of RΣR in different intervals as they move along the graph of a derivative. Consider the dark blue rectangle to be the previous position of the red rectangle.

As the RΣR moves from a to b each red rectangle is larger than the dark blue one. Each move adds more to the accumulated sum than the previous one. The graph of the function increases more with each move – it is concave up.

As the RΣR moves from b to d (there are two pairs drawn) each red RΣR has a smaller value than the previous one. (Remember when they are below the x-axis the longer (red) RΣR has a smaller value.) In the interval [b, d] less is added to the accumulated sum (or more is subtracted) with each move to the right. Therefore, the graph of the function bends down – the function is concave down.

In the last section, from d to f the red rectangle now has a larger value than the dark blue one. (Again, remember that when the function has negative values, the shorter rectangle, has the larger value.) The graph of the function again bends up – is concave up.

Putting these ideas together with those in the last post we can see how the moving RΣR idea can distinguish the four shapes of the graph of the accumulating function:

• On [a, b] the function’s graph is increasing and concave up; the RΣR are positive and getting more positive (longer).
• On [b, c] the function’s graph is increasing and concave down; the RΣR are positive and getting less positive (shorter).
• On [c, d] the function’s graph is decreasing and concave down; the RΣR are negative and getting more negative (longer).
• On [d, e] the function’s graph is decreasing and concave up; the RΣR are negative and getting less negative (shorter).
• At the extreme values of the derivative, the concavity of the function changes from up to down or down to up. These are called points of inflection.

Questions in which students are asked about the properties of a function given the graph, but not the equation, of the derivative are very common. Many students (including me) find this approach easier and more intuitive than working strictly with derivative ideas.

A very typical calculus problem is given the equation of a function, to find information about it (extreme values, concavity, increasing, decreasing, etc., etc.). This is usually done by computing and analyzing the first derivative and the second derivative. All the textbooks show how to do this with copious examples and exercises. I have nothing to add to that. One of the “tools” of this approach is to draw a number line and mark the information about the function and the derivative on it.

A very typical AP Calculus exam problem is given the graph of the derivative of a function, but not the equation of either the derivative or the function, to find all the same information about the function. For some reason, student find this difficult even though the two-dimensional graph of the derivative gives all the same information as the number line graph and, in fact, a lot more.

Looking at the graph of the derivative in the x,y-plane it is easy to very determine the important information. Here is a summary relating the features of the graph of the derivative with the graph of the function.

 Feature the function ${y}'$> 0 is increasing ${y}'$ < 0 is decreasing ${y}'$ changes  – to + has a local minimum ${y}'$changes + to – has a local maximum ${y}'$ increasing is concave up ${y}'$ decreasing is concave down ${y}'$ extreme value has a point of inflection

Here’s a typical graph of a derivative with the first derivative features marked. Here is the same graph with the second derivative features marked. The AP Calculus Exams also ask students to “Justify Your Answer.” The table above, with the columns switched does that. The justifications must be related to the given derivative, so a typical justification might read, “The function has a relative maximum at x-2 because its derivative changes from positive to negative at x = -2.”

 Conclusion Justification y is increasing ${y}'$> 0 y is decreasing ${y}'$< 0 y has a local minimum ${y}'$changes  – to + y has a local maximum ${y}'$changes + to – y is concave up ${y}'$increasing y is concave down ${y}'$decreasing y has a point of inflection ${y}'$extreme values

For notes on vertical asymptotes see

For notes on horizontal asymptotes see Other Asymptotes