**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)
- 2023 AB 1/ BC 1 – Right Riemann sum, slope equals zero (may be done by max/min analysis, MVT, or Rolle’s Theorem), Equation extension – average value, meaning of derivative.

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, June 6, 2023

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Hi Lin â

Great posting on Riemann sums. This info will be of great help to fellow instructors as they prepare students for the AP Exam.

As you know, during the first week of class I introduce my AB and BC students to limits, derivatives, and definite integrals using numeric, graphical, and verbal methods. The analytic methods follow as âshortcutsâ to find exact answers, once they know that derivatives are limits of average rates and a definite integral equals the area of a region under an x-y graph, which represents the product of x and y if y varies with x.

For a day or two students find integrals approximately by counting squares under an accurate graph. When they ask, âWill I always have to count squares?â I tell them, âNo. You can divide the region into trapezoids, and calculate an approximate area. After doing an example once by hand with 4 trapezoids, they download a program to calculate the sum of areas for n trapezoids. They find that the areas approach a limit as n increases, thus reinforcing the concept of limit, and introducing the idea that a definite integral is a limit of a sum. As a result, the students later move easily to Riemann sums, the formal definition of definite integral, and applications, even if it is not obvious that the integrand they are accumulating is a rate of change.

Interestingly, the trapezoidal rule becomes a first resort in evaluating integrals, rather than a last resort when you canât find an antiderivative. This information was to be the topic of my presentation at the July APAC in Austin before COVID canceled it.

Iâve attached the Exploration (and its solutions) that Iâve used for many years to introduce students to the trapezoidal rule in Week 1. You are welcome to share this with all on your Teaching Calculus listserv. You may publish my email address, foerster@idweorld.net, so that I can sent instructors the other 6 Explorations, and the power point presentation supporting them.

Thanks for all you do to enlighten our students.

Paul ====================âââââââ

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