Extreme Average

A recent post on the AP Calculus bulletin board observed that the maximum value of the average value of a function on an interval occurred at the point where the graph of the average value and the function intersect. I am not sure if this concept is important in and of itself, but it does make an interesting exercise.

For a function f(x), we may treat its average value as a function, A(x), defined for all x ∈ [a, b], interval [a, x] as

\displaystyle A\left( x \right)=\left\{ {\begin{array}{*{20}{c}} {\tfrac{1}{{x-a}}\int_{a}^{x}{{f\left( t \right)dt}}} & {x\ne a} \\ {f\left( a \right)} & {x=a} \end{array}} \right.

Graphically, the segment drawn at y = A(x) is such that the regions between the line and the function above and below the segment have equal areas. See figure 1 in which the red curve is the function, and the blue curve is the average value function. The two shaded regions have the same area.

Figure 1: The shaded regions have the same area.

Regardless of the starting value, the function and its average value start at the same value. If the function is increasing the average value is less than the function and increasing. When the function starts to decrease, the average value will continue to increase for a while. When the two graphs nest intersect, the process starts over, and the average value will now start to decrease. Therefore, the intersection value is when the average value function change from increasing to decreasing and this is its (local) maximum value. See Figure 2.

Figure 2:The maximum value of A(x) is at the intersection of the two graphs

This continues until the graphs intersect again after the function starts to increase: a (local) minimum value of the average value function. The process continues with the extreme values of the average value function (blue graph) occurring at its intersections with the function. Figure 3

Figure 3: A(x) has its extreme values where it intersects the function.

This can be proved by finding the extreme values of the average value function by considering its derivative. Begin by finding its derivative using the product rule (or quotient rule) and the FTC.

\displaystyle {A}'\left( x \right)=\tfrac{1}{{x-a}}f\left( x \right)+\left( {-\tfrac{1}{{{{{\left( {x-a} \right)}}^{2}}}}} \right)\int_{a}^{x}{{f\left( t \right)dt}}

\displaystyle {A}'\left( x \right)=\tfrac{1}{{x-a}}f\left( x \right)-\tfrac{1}{{x-a}}\left( {\tfrac{1}{{x-a}}\int_{a}^{x}{{f\left( t \right)dt}}} \right)

\displaystyle {A}'\left( x \right)=\tfrac{1}{{x-a}}\left( {f\left( x \right)-A\left( x \right)} \right)

The critical points of a(x) occur when its derivative is equal to zero (or undefined). This is when f\left( x \right)=A\left( x \right) (or when x = a, the endpoint). This is where the graphs intersect.

How to use this in your class

This is not a concept that is likely to be tested on the AP Calculus Exams. Nevertheless, it is an easy enough idea to explore when teaching the average value of a function and at the same time reviewing some earlier concepts such as product (or quotient) rule, the FTC (differentiating an integral), and some non-ordinary simplification.

You could have your students use their own favorite function and show that the extreme values of its average value occur where the average value intersects the function. This is good practice in equation solving on a calculator since the points do not occur at “nice” numbers. Here’s an example.

If \displaystyle f\left( x \right)=\sin \left( x \right), then its average value on the interval \displaystyle [0,\infty ) is

\displaystyle  A\left( x \right)=\tfrac{1}{x}\int_{0}^{x}{{\sin \left( t \right)dt}}=\frac{{-\cos \left( x \right)+1}}{x}.

The intersections of f(x) and A(x) can be found by solving

\displaystyle\sin \left( x \right)=\frac{{-\cos \left( x \right)+1}}{x}

The extreme values of \displaystyle \frac{{-\cos \left( x \right)+1}}{x} may also be found using a calculator.

The points are the same. the first is approximately (2.331, 0.725) and the second is (6.283, 0) or (2π, 0). This second is reasonable since at 2π the sine function has completed one period and its average value zero. (See figure 3 again.).

Other questions you could ask (for my function anyway) are what is the absolute maximum and how can you be sure? Why are all the minimums zero?

The message on the AP Calculus discussion boards that inspired this post was started by Neema Salimi an AP Calculus teacher from Georgia. He made the original observation. You can read his original post and proof, and comments by others here.

The Rule of Four

Not much has been heard of the Rule of Four lately. The Rule of Four suggests that mathematical concepts should be looked at graphically, numerically, analytically, and verbally. It has not gone away. The Rule of Four has a new name: multiple representations. (In the latest Course and Exam Description, you will find it in Mathematical Practices (p. 14), specifically practices 2.B, 2.C, 2.D, 2.E, 3.E, 3.F, 4.A, and 4.C)

I have used the Rule of Four in this post. The post started with a verbal discussion of the concept and how the result can be seen graphically. That was followed by analytic proof. At the end is a numerical example.  

Other posts on the average value of a function:

Finding the average value of a function on an interval is Topic 8.1 in the Course and Exam Description (p. 149)

Average Value of a Function – or How do you average an infinite number of numbers?

Most Triangles Are Obtuse! An obvious observation, but here’s how to figure the exact proportion of obtuse to acute triangles.

Half-full or Half-empty Visualizing the average value of a function

What’s a Mean Old Average Anyway? Be sure to distinguish between the average rate of change, the average value of a function, and the mean value theorem.

…but what does it look like?

It will soon be time to teach about finding the volumes of solid figures using integration techniques. Here is a list of links to posts that will help your students what these figures look like and how they are generated.

Visualizing Solid Figures 1 Here are ideas for making physical models of solid figures. These make good projects for students.

A Little Calculus is an iPad app that does an excellent job in helping students visualize many of the concepts of the calculus. Volumes with regular cross section, disk method, washer method, cylindrical shells are all illustrated.

The first illustrations show square cross sections on a semicircular base. The base is in the lower part and the solid in the upper. By using the plus and minus button (lower right) you can increase or decrease the number of sections in real time and see the figures change. The upper figure may be rotated by moving your finger on the screen.

The illustration below shows a washer situation.

The following older posts show how to use Winplot to generate and explore solid figures. Unfortunately, Winplot seems to have gone out of favor. I’m not sure why; it is one of the best. I still use it and like it. You may download Winplot here for free (PC only).

Visualizing Solid Figures 2 This post demonstrates how to use Winplot to generate solids with regular cross sections and solids of rotation.

Visualizing Solid Figures 3 The washer method is illustrated using Winplot. These post all relate to finding volumes by washers: Subtract the Hole from the Whole and Does Simplifying Make Things Simpler?

Visualizing Solid Figures 4 Using Winplot to see the method cylindrical shells. Note that this method is not tested on either the AB or BC Calculus exams, so you do not have to teach it. Many teachers present this topic after the exams are given. As a footnote you may also find Why You Never Need Cylindrical Shells interesting. (However, this is not the reason it is not tested on the AP Calculus exams.)

Visualizing Solid Figures 5 An exercise demonstrating how “half” can mean different things and shows that how the figures are generated makes a difference.

Unit 7 – Differential Equations

Unit 7 is an introduction to the initial ideas and easy techniques related to differential equations . (CED – 2019 p. 129 – 142 ). These topics account for about 6 – 12% of questions on the AB exam and 6 – 9% of the BC questions.

Topics 7.1 – 7.9

Topic 7.1 Modeling Situations with Differential Equations Relating a functions and its derivatives.

Topic 7.2 Verifying Solutions for Differential Equations A proposed solution of a differential equation can be checked by substituting the function and its derivative(s) into the original differential equation. There may be an infinite number of general solutions (solutions with one or more constants).

Topic 7.3 Sketching Slope Fields Slope fields are a graphical representation of a differential equation and provide information about the behavior of the solutions.

Topic 7.4 Reasoning Using Slope Fields 

Topic 7.5 Approximating Solutions Using Euler’s method (BC ONLY) A numerical approach to approximating solutions of a differential equation.

Topic 7.6 Finding General Solutions Using Separation of Variable Since this unit is only an introduction to differential equations, the method of separation of variable is the only solution method tested on the AB and BC exams.

Topic 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables An initial condition (i.e. a point on the particular solution) allows you to evaluate the constant in the general solution and find the one solution that contains the initial condition. Also, if \displaystyle \frac{{dy}}{{dx}}=f\left( x \right) has the initial condition\displaystyle \left( {a,F(a))} \right), then the solution is\displaystyle F\left( x \right)=F\left( a \right)+\int_{a}^{x}{{f\left( x \right)dx}}. Solution may also be subject to domain restrictions

Topic 7.8 Exponential Models with Differential Equations Applications include linear motion and exponential growth and decay. The growth and decay model is \displaystyle \frac{{dy}}{{dt}}=kt with the initial condition \displaystyle \left( {0,y\left( 0 \right)} \right) has the solution \displaystyle y=y\left( 0 \right){{e}^{{kt}}}

Topic 7.9 Logistic Models with Differential Equations (BC ONLY) The model of logistic growth, \displaystyle \frac{{dy}}{{dx}}=ky\left( {a-y} \right), can be solved by separating the variables and using partial fraction decomposition. This has never been tested (probably because solving requires a large amount of complicated algebra). Students are expected to know how to interpret the properties of the solution directly from the differential equation (asymptotes, carrying capacity, point where changing the fastest, etc.) and discuss what they mean in context without actually solving the equation.


The suggested time for Unit 7 is  8 – 9 classes for AB and 9 – 10 for BC of 40 – 50-minute class periods, this includes time for testing etc.

Previous posts on these topics for both AB and BC include:

Differential Equations  A summary of the terms and techniques of differential equation and the method of separation of variables

Domain of a Differential Equation – On domain restrictions.

Accumulation and Differential Equations 

Slope Fields

An Exploration in Differential Equations An exploration illustrating many of the ideas of differential equations. The exploration is here in PDF form and the solution is here. The ideas include: 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, 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, and investigating a special case or two. 

Posts on BC Only Topics

Euler’s Method

Euler’s Method for Making Money

The Logistic Equation 

Logistic Growth – Real and Simulated

Adapting 2021 AB 6

Adapting 2021 BC 5

Posts on Accumulation

One of the main uses of the definite integral is summed up (pun intended) in the idea of accumulation. When you integrate a rate of change you get the (net) amount of change. This important idea is often treated very lightly, if at all, in textbooks.

Here are a series of past posts that use, explain, and illustrate that concept.

Accumulation – Need an Amount? The Fundamental Theorem of Calculus says that the integral of a rate of change (a derivative) is the net amount of change. This post shows how that works in practice.

AP Accumulation Questions and Good Question 7 – 2009 AB 3 the “Mighty Cable Company” show how accumulation is tested on the AP Calculus exams. The “Mighty Cable Company” question is a particularly good and difficult example.

The next two posts show how to use the concept of accumulation to analyze a function and its graph without reference to the derivative. The graphical idea of a Riemann sum rectangle moving across the interval of integration makes the features of function much more intuitive than the common approach. You will not find these ideas in textbooks. Nevertheless, a lesson on this idea may help your students.

Graphing with Accumulation 1 explains how to analyze the derivative to determine when a function is increasing or decreasing and finding the locations of extreme values. By thinking of the individual Rieman sum rectangles moving across the interval the features of the function are easy to see and easier to remember. Once understood, this method will help students with their graph analysis work.

Graphing with Accumulation 2 continues the idea of using accumulation to determine information about the concavity of a function.

Unit 8 – Applications of Integration

I haven’t missed Unit 7! This unit seems to fit more logically after the opening unit on integration (Unit 6). The Course and Exam Description (CED) places Unit 7 Differential Equations before Unit 8 probably because the previous unit ended with techniques of antidifferentiation. My guess is that many teachers will teach Unit 8: Applications of Integration immediately after Unit 6 and before Unit 7: Differential Equations. The order is up to you. Unit 7 will post next Tuesday.

Unit 8 includes some standard problems solvable by integration (CED – 2019 p. 143 – 161). These topics account for about 10 – 15% of questions on the AB exam and 6 – 9% of the BC questions.

Topics 8.1 – 8.3 Average Value and Accumulation

Topic 8.1 Finding the Average Value of a Function on an Interval Be sure to distinguish between average value of a function on an interval, average rate of change on an interval and the mean value

Topic 8.2 Connecting Position, Velocity, and Acceleration of Functions using Integrals Distinguish between displacement (= integral of velocity) and total distance traveled (= integral of speed)

Topic 8. 3 Using Accumulation Functions and Definite Integrals in Applied Contexts The integral of a rate of change equals the net amount of change. A really big idea and one that is tested on all the exams. So, if you are asked for an amount, look around for a rate to integrate.

Topics 8.4 – 8.6 Area

Topic 8.4 Finding the Area Between Curves Expressed as Functions of x

Topic 8.5 Finding the Area Between Curves Expressed as Functions of y

Topic 8.6 Finding the Area Between Curves That Intersect at More Than Two Points Use two or more integrals or integrate the absolute value of the difference of the two functions. The latter is especially useful when do the computation of a graphing calculator.

Topics 8.7 – 8.12 Volume

Topic 8.7 Volumes with Cross Sections: Squares and Rectangles

Topic 8.8 Volumes with Cross Sections: Triangles and Semicircles

Topic 8.9 Volume with Disk Method: Revolving around the x– or y-Axis Volumes of revolution are volumes with circular cross sections, so this continues the previous two topics.

Topic 8.10 Volume with Disk Method: Revolving Around Other Axes

Topic 8.11 Volume with Washer Method: Revolving Around the x– or y-Axis See Subtract the Hole from the Whole for an easier way to remember how to do these problems.

Topic 8.12 Volume with Washer Method: Revolving Around Other Axes. See Subtract the Hole from the Whole for an easier way to remember how to do these problems.

Topic 8.13  Arc Length BC Only

Topic 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled  BC ONLY


The suggested time for Unit 8 is  19 – 20 classes for AB and 13 – 14 for BC of 40 – 50-minute class periods, this includes time for testing etc.

Previous posts on these topics for both AB and BC include:

Average Value and Accumulation

Average Value of a Function and 

Most Triangles Are Obtuse!

Half-full or Half-empty

Accumulation: Need an Amount?

AP Accumulation Questions

Good Question 7 – 2009 AB 3 Accumulation, explain the meaning of an integral in context, unit analysis

Good Question 8 – or Not Unit analysis

Graphing with Accumulation 1 Seeing increasing and decreasing through integration

Graphing with Accumulation 2 Seeing concavity through integration

Adapting AB 1 / BC 1


Area Between Curves

Under is a Long Way Down  Avoiding “negative area.”

Improper Integrals and Proper Areas  BC Topic

Math vs. the “Real World”  Improper integrals  BC Topic

Adapting 2021 AB 3 / BC 3


Volumes of Solids with Regular Cross-sections

Volumes of Revolution

Why You Never Need Cylindrical Shells

Visualizing Solid Figures 1

Visualizing Solid Figures 2

Visualizing Solid Figures 3

Visualizing Solid Figures 4

Visualizing Solid Figures 5

Painting a Point

Subtract the Hole from the Whole and Does Simplifying Make Things Simpler?

Adapting 2021 AB 3 / BC 3

Other Applications of Integrals

Density Functions have been tested in the past, but are not specifically listed on the CED then or now.

Who’d a Thunk It? Some integration problems suitable for graphing calculator solution

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

A Calculus Journey

I think that the path leading up to and including the Fundamental Theorem of Calculus (FTC) is one of the most beautiful walks in mathematics. I have written several posts about it. You will soon be ready to travel that path with your students. (I always try to post on topics shortly before most teachers will get to them, so that you have some time to consider them and work the ideas you like into your lessons.)

Here is an annotated list of some of the posts to guide you on your journey.

Working Towards Riemann Sums gives the preliminary definitions you will need to define and discuss Riemann sums.

Riemann Sums defines the several Riemann sums often used in the calculus left-side sums, right-side sums, midpoint sums and the trapezoidal sums. “The Area Under a Curve” in the iPad app A Little Calculus is a great visual display of these and shows what happens as you use more subintervals.

The Definition of the Definite Integral gives the definition of the definite integral as the limit of any Riemann sum. As with any definition, there is nothing to prove or argue about here. The thing to remember is that the limit of the Riemann sum and the definite integral are the same thing. Behind any definite integral is a Riemann sum. The advantage of the definition’s integral notation is that it shows the interval involved which the Riemann sum does not. (Any Riemann sum may be represented by many definite integrals. See Good Question 11 – Riemann Reversed.)

Foreshadowing the FTC is an example of how a definite integral may be evaluated. It is long and has a lot of notation, so you may not want to use this.

The Fundamental Theorem of Calculus is where the path leads. This post develops the FTC based on the other “big” idea of the calculus: the Mean Value Theorem. (I think the form here is preferable to the usual book notation that uses F(x) and its derivate f (x).)

Y the FTC? Tries to answer the question of what’s so important about the FTC. Example 1: The verbal interpretation of the FTC (the integral of a rate of change is the net amount of change over the interval.) will soon be used in many practical applications. While example 2 shows how the FTC allows one to evaluate a definite integral and, therefore the Riemann sum it represents, by evaluating a function whose derivative is the integrand (its antiderivative).

More About the FTC presents examples leading up to the other form of the FTC: the derivative of the integral is the integrand).

At this point you may go in the direction of learning how to find antiderivatives or working on applications. (See Integration itinerary.)

Bon Voyage!     

The Old Pump

A tank is being filled with water using a pump that is old and slows down as it runs. The table below gives the rate at which the pump pumps at ten-minute intervals. If the tank initially has 570 gallons of water in it, approximately how much water is in the tank after 90 minutes?

Elapsed time (minutes)   0   10  20   30   40   50   60   70   80   90
Rate (gallons / minute)   42   40   38   35   35   32   28   20   19   10

And so, integration begins.

Ask your students to do this problem alone. When they are ready (after a few minutes) collect their opinions.  They will not all be the same (we hope, because there is more than one reasonable way to approximate the amount). Ask exactly how they got their answers and what assumptions they made. Be sure they always include units (gallons).  Here are some points to make in your discussion – points that we hope the kids will make and you can just “underline.”

    1. Answers between 3140 and 3460 gallons are reasonable. Other answers in that range are acceptable. They will not use terms like “left-sum”, “right sum” and “trapezoidal rule” because they do not know them yet, but their explanations should amount to the same thing. An answer of 3300 gallons may be popular; it is the average of the other two, but students may not have gotten it by averaging 3140 and 3460.
    2. Ask if they think their estimate is too large or too small and why they think that.
    3. Ask what they need to know to give a better approximation – more and shorter time intervals.
    4. Assumptions: If they added 570 + 42(10) + 40(10) + … +19(10) they are assuming that the pump ran at each rate for the full ten minutes and then suddenly dropped to the next. Others will assume the rate dropped immediately and ran at the slower rate for the 10 minutes. Some students will assume the rate dropped evenly over each 10-minute interval and use the average of the rates at the ends of each interval (570 + 41(10) + 39(10) + … 14.5(10) = 3300).
    5. What is the 570 gallons in the problem for? Well, of course to foreshadow the idea of an initial condition. Hopefully, someone will forget to include it and you can point it out.
    6. With luck someone will begin by graphing the data. If no one does, you should suggest it; (as always) to help them see what they are doing graphically. They are figuring the “areas” of rectangles whose height is the rate in gallons/minute and whose width is the time in minutes. Thus the “area” is not really an area but a volume (gal/min)(min) = gallons). In addition to unit analysis, graphing is important since you will soon be finding the area between the graph of a function and the x-axis in just this same manner.

Follow up: Flying to Integrationland

Be sure to check the “Thoughts on ‘The Old Pump'” in the comments section below.

Revised from a post of November 30, 2012.