Riemann Sum & Table Problems (Type 5)

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

Riemann Sum & Table Problems (Type 5)

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.

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 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 1 – Limits and Continuity

This is the first of a series of blog posts that I plan to write over the next few months, staying a little ahead of where you are so you can use anything you find useful in your planning. Look for this series every 2 – 4 weeks.

Unit 1 contains topics on Limits and Continuity. (CED – 2019 p. 36 – 50). These topics account for about 10 – 12% of questions on the AB exam and 4 – 7% of the BC questions.

Logically, limits come before continuity since limit is used to define continuity. Practically and historically, continuity comes first. Newton and Leibnitz did not have the concept of limit the way we use it today. It was in the early 1800’s that the epsilon-delta definition of limit was first given by Bolzano (whose work was overlooked) and then by Cauchy and Weierstrass. But their formulation did not use the word “limit”, rather the use was part of their definition of continuity. Only later was it pulled out as a separate concept and then returned to the definition of continuity as a previously defined term.

Students should have plenty of experience in their math courses before calculus with functions that are and are not continuous. They should know the names of the types of discontinuities – jump, removable, infinite, etc. As you go through this unit, you may want to quickly review these terms and concepts as they come up.

(As a general technique, rather than starting the year with a week or three of review – which the students need but will immediately forget again – be ready to review topics as they come up during the year as they are needed – you will have to do that anyway. See Getting Started #2)

Topics 1.1 – 1.9: Limits

Topic 1.1: Suggests an introduction to calculus to give students a hint of what’s coming. See Getting Started #3

Topic 21.: Proper notation and multiple-representations of limits.

There is an exclusion statement noting that the delta-epsilon definition of limit is not tested on the exams, but you may include it if you wish. The epsilon-delta definition is not tested probably because it is too difficult to write good questions. Specifically, (1) the relationship for a linear function is always  $\delta =\frac{\varepsilon }{{\left| m \right|}}$  where m is the slope and is too complicated to compute for other functions, and (2) for a multiple-choice question the smallest answer must be correct. (Why?)

Topic 1.3: One-sided limits.

Topic 1.4: Estimating limits numerically and from tables.

Topic 1.5: Algebraic properties of limits.

Topic 1.6: Simplifying expressions to find their limits. This can and should be done along with learning the other concepts and procedures in this unit.

Topic 1.7: Selecting the proper procedure for finding a limit. The first step is always to substitute the value into the limit. If this comes out to be number than that is the limit. If not, then some manipulation may be required. This can and should be done along with learning the other concepts and procedures in this unit.

Topic 1.8: The Squeeze Theorem is mainly used to determine $\underset{{x\to 0}}{\mathop{{\lim }}}\,\frac{{\sin \left( x \right)}}{x}=1$ which in turn is used in finding the derivative of the sin(x). (See Why Radians?) Most of the other examples seem made up just for exercises and tests. (See 2019 AB 6(d)). Thus, important, but not too important.

Topic 1.9: Connecting multiple-representations of limit. This can and should be done along with learning the other concepts and procedures in this unit. Dominance, Topic 15, may be included here as well (EK LIM-2.D.5)

Topics 1.10 – 1.16 Continuity

Topic 1.10: Here you can review the different types of discontinuities with examples and graphs.

Topic 1.11: The definition of continuity. The EK statement does not seem to use the three-hypotheses definition. However, for the limit to exist and for f(c) to exist, they must be real numbers (i.e. not infinite). This is tested often on the exams, so students should have practice with verifying that (all three parts of) the hypothesis are met and including this in their answers.

Topic 1.12: Continuity on an interval and which Elementary Functions are continuous for all real numbers.

Topic 1.13: Removable discontinuities and handing piecewise – defined functions

Topic 1.14: Vertical asymptotes and unbounded functions. Here be sure to explain the difference between limits “equal to infinity” and limits that do not exist (DNE). See Good Question 5: 1998 AB2/BC2.

Topic 1.15: Limits at infinity, or end behavior of a function. Horizontal asymptotes are the graphical manifestation of limits at infinity or negative infinity. Dominance is included here as well (EK LIM-2.D.5)

Topic 1.16: The Intermediate Value Theorem (IVT) is a major and important result of a function being continuous. This is perhaps the first Existence Theorem students encounter, so be sure to stop and explain what an existence theorem is.

The suggested number of 40 – 50 minute class periods is 22 – 23 for AB and 13 – 14 for BC. This includes time for testing etc. If time seems to be a problem you can probably combine topics 3 – 5, topics 6 -7, topics 11 – 12. Topics 6, 7, and 9 are used with all the limit work.

There are three other important limits that will be coming in later Units:

The definition of the derivative in Unit 2, topics 1 and 2

L’Hospital’s Rule in Unit 4, topic 7

The definition of the definite integral in Unit 6, topic 3.

Posts on Continuity

CONTINUITY To help understand limits it is a good idea to look at functions that are not continuous. Historically and practically, continuity should come before limits. On the other hand, the definition of continuity requires knowing about limits. So, I list continuity first. The modern definition of limit was part of Weierstrass’ definition of continuity.

Continuity (8-13-2012)

Continuity (8-21-2013) The definition of continuity.

Continuous Fun (10-13-2015) A fuller discussion of continuity and its definition

Right Answer – Wrong Question (9-4-2013) Is a function continuous even if it has a vertical asymptote?

Asymptotes (8-15-2012) The graphical manifestation of certain limits

Fun with Continuity (8-17-2012) the Diriclet function

Far Out! (10-31-2012) When the graph and dominance “disagree” From the Good Question series

Posts on Limits

Why Limits? (8-1-2012)

Deltas and Epsilons (8-3-2012) Why this topic is not tested on the AP Calculus Exams.

Finding Limits (8-4-2012) How to…

Limit of Composite Functions

Dominance (8-8-2012) See limits at infinity

Determining the Indeterminate (12-6-2015) Investigating an indeterminate form from a differential equation. From the Good Question series.

Locally Linear L’Hôpital (5-31-2013) Demonstrating L’Hôpital’s Rule (a/k/a L’Hospital’s Rule)

L’Hôpital’s Rules the Graph (6-5-2013)

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

Continuity

Karl Weierstrass (1815 – 1897) was the mathematician who (finally) formalized the definition of continuity. Included in that definition was the epsilon-delta definition of limit. This definition has been pulled out, so to speak, and now is usually presented on its own. So, which came first – continuity or limit? The ideas and situations that required continuity could only be formalized with the concept of limit. So, looking at functions that are or are not continuous helps us understand what limits are and why we first need them.

In the ideal world, students would have plenty of work with continuous and not continuous functions before starting the calculus. The vocabulary and notation, if not the formal definitions, would be used as early as possible. Then when students got to calculus, they would know the ideas and be ready to formalize the ideas.

The Intermediate Value Theorem (IVT) is an important property of continuous functions.

Using the definition of continuity to show that a function is or is not continuous at a point is a common question of the AP exams, as is the IVT.

Continuity The definition of continuity.

Continuity Should continuity come before limits?

From One Side or the Other One-sided limits and one-sided differentiability

How to Tell Your Asymptote from a Hole in the Graph  From the technology series. Showing holes and asymptotes on a graphing calculator.

Fun with Continuity Defined everywhere and continuous nowhere. Continuous only at a single point.

Theorems The Intermediate Value Theorem (IVT) and suggestions on teaching theorems.

Intermediate Weather  Using the IVT

Right Answer – Wrong Question Continuity or continuity “on its domain”?

Revised from a post of August 22, 2017

Intermediate Weather

I found a curious fact in a textbook today that relates to the Intermediate Value Theorem [1]. It claimed that if you draw a circle of any size on a map, there will be two diametrically opposite points on the circle at which the temperature will be the same!

This might make an interesting example when you are talking about the IVT.

The proof goes like this:

Assume the temperature varies continuously between every pair of points. Draw a circle of any radius on your map. Put a polar coordinate system on the map with the pole at the center of the circle and let $theta$ be the angle formed by a line through the pole and the polar axis in the usual way. Let $\theta$ be the temperature at any point on the circle on the line that makes an angle of $\theta$ with the polar axis. Then consider the function, f that gives the difference in temperature between two diametrically opposite points at angles of $\theta$ and $\theta +\pi$:

$f(\theta )=T(\theta )-T(\theta +\pi )$

Case I:If $f(0)=T(0)-T(\pi )=0$ then $T(0)=T(\pi )$ and we have our two points.

Case II: If $f(0)=T(0)-T(\pi )>0$, then$~T(0)>T(\pi )$, indicating that as the angle increases from 0 to $\pi$, the temperature has a net decrease. Then, on the other half of the circle from $\pi$ to $2\pi$ the temperature must increase $f(\pi )=T(\pi )-T(2\pi )=T(\pi )-T(0)<0$. (Since $T(0)=T(\pi )$). Therefore, by the Intermediate Value Theorem there is some value $\theta =c$, between 0 and $\pi$ where $f(c)=0$ and $T(c)=T(c+\pi )$

Case III is like Case II with $f(0)<0$.

Without so many equations, this says that if you keep track of the temperature difference at the ends of the diameter on the way around the first half of the circle and find a net decrease in the temperature difference, then on your way around the second half of the circle (returning to the starting point) you must see a net increase. Somewhere between the decrease and the increase you must have a point where the difference is zero – the temperatures at the ends will be the same.

Now, I wasn’t really convinced. Yes, I believe proofs, but still…. So, I looked at a weather map [2]

Consider the circle drawn on the map. From Iowa to Canada the temperature decreases from 79 to 66. Meanwhile over in California the temperature increases from 73 near San Francisco to 82 in Los Angeles. With a little visual interpolation, the temperatures at the ends of the diameter seem to be about equal. Try it with your own weather map.

Make your own circle and space the temperatures evenly on both sides. The diameter with the same temperatures will be ¼ of the way around. Try again with unevenly spaced temperatures; you will still find a place.

This is similar to the mountain climbing problem: If you climb a mountain during certain hours one day and climb back down during the same hours the next day, then there will be a place that you pass at the same time on both days.

_______________________________

References:

[1] Calculus by Rogowski and Cannon, Second edition, Section 2.8 exercise 26

[2] Air Sports New Weather for September 1, 2014, 14:20 EDT

Revised September 3, 2014 12:50 to fix some problems with the equations appearing properly.

Proof

When math books present a theorem, they almost always immediately present its proof. I tend to skip the proofs. I assume they are correct. I want to get on with the ideas in the text. Later I may come back and read through them. Is this a good thing to advise students to do? I don’t know.

There are reasons to read proofs. One reason is to help understand why a theorem is true, by seeing the reasoning that leads to the result. Another is to check the reasoning yourself. A third is to learn how to do proofs.

Learning to write original proofs is not usually one of the goals of a beginning calculus course. That comes later in a course with “analysis” in its title. There are many theorems that involve some one-off that rarely will be used again. I’m thinking of a proof like that of the sum of the limits is equal to the limit of the sums, where you add and subtract the same expression and this more complicated form allows you to group and factor the terms of the numerator and arrive at the result. Another example is in the Mean Value Theorem where you consider a new function that gives the vertical distance between a function and its secant line. These always bring the question, “How did you know to do that?”

If a student can accept things like that, then the proof is usually easy enough to follow. But I would never spend a lot of time making every student fight his or her way through each and every proof.

On this other hand, I would never just present a theorem and not give some explanation as to why it is true (and why it is important enough to mention). Unfortunately, I have seen teachers write the Fundamental Theorem of Calculus on the board and proceed to show how to use it to evaluate definite integrals, with no hint of why this important theorem is true. Sure kids can memorize it and use it, but it seems to me they should also have a hint as to why it is true.

Some theorems are easy to understand if explained in ways other than giving a proof. For an example of this, see my post of October 1, 2012 on the Mean Value Theorem. Almost every book will bail out on the Intermediate Value Theorem by claiming (quite rightly) that, “the proof is beyond the scope of this book,” or they give the proof in an appendix. But a simple drawing will convince you that it is true.

So my feeling is that you do not need to labor over a proof for every theorem, BUT, big BUT, you should provide a good explanation of why it is true.

This is important for all students and especially for young women. Jo Boaler writes

“As I interviewed more and more boys and girls, I noticed that the desire to know why was something that separated the girls from the boys. The girls were able to accept the method that were shown them and practice them, but they wanted to know why they worked, where they came from, and how they connected with other methods…. When they could not get access to the depth of understanding they wanted, the girls started to turn away from the subject…. Classes in which students discuss concepts, giving them access to a deep and connected understanding of math, are good for boys and girls. Boys may be willing to work in isolation on abstract rules, but such approaches do not give many students, girls or boys, access to the understanding they need. In addition, high-level work in mathematics, science and engineering is not about isolated, abstract rule following, but about collaboration and connection making.”

[Jo Boaler, What’s Math Got to Do with It? Helping Children Learn to Love Their Most Hated Subject – And Why It’s Important for America, © 2008 Penguin Group, New York. From Chapter 6]

Theorems

Theorems are statements that summarize the results that are true in mathematics. Theorems are statements that have been proved true; but the emphasis in AP Calculus is not on proof. Rather, it is on what the theorems mean and how to use them.

Theorems have two parts: the “if …” clause called the hypothesis and the “then …” clause called the conclusion. Students need to know both parts. In many theorems the conclusion is some sort of formula. The students need to know this, but also need to know when they can use it (the hypothesis tells them that).

An early important theorem is the Intermediate Value Theorem (IVT). Take some time with this theorem. “Play” with it. The hypothesis requires that the function be continuous on a closed interval. Use graphs (sketches, no equation needed) to show cases where the conclusion is both true and false when the function is not continuous. Can the function take on values not between f(a) and f(b)? Can you find a case where the hypothesis is met, but the conclusion is false? (Let’s hope not!)

Consider the theorem ($p\to q$), its converse ($q\to p$), its inverse ($\sim p\to \sim q$) and its contrapositive ($\sim q\to \sim p$) by looking at graphs of each case. (For the IVT the converse and inverse are false. The contrapositive of any true theorem is also true.)

Finally, for this and for all the important theorems that you use this year, express them in words, “play” with them by making change to the hypothesis, and look at graphs. Don’t just state the theorem and expect students to understand it, remember it and use it correctly.

The next post will be about definitions, which are similar to theorems in lots of ways.