**For updated information for the special 2020 exams click here.**

**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)

**Schedule of future posts for reviewing for the 2019 Exams**

**NOTE:** The type number I’ve assigned to each type DO NOT correspond to the ** 2019 CED** Unit numbers. Many AP Exam questions have parts from different Units. The CED Unit numbers will be referenced in each post.

**For updated information for the special 2020 exams click here.**

Tuesday February 25 – AP Exam Review 2020 https://wp.me/p2zQso-286

Friday, February 28 – Reviewing Resources 2020 https://wp.me/p2zQso-28f

Tuesday March 3, 2020: Rate and accumulation questions (Type 1) https://wp.me/p2zQso-265

Friday March 6, 2020: Linear motion problems (Type 2) https://wp.me/p2zQso-269

Tuesday March 10, 2020: Graph analysis problems (Type 3) https://wp.me/p2zQso-26e

Friday March 13, 2020: Area and volume problems (Type 4) https://wp.me/p2zQso-26m

Tuesday March 17, 2020: Table and Riemann sum questions (Type 5) https://wp.me/p2zQso-27K

Friday March 20, 2020: Differential equation questions (Type 6) https://wp.me/p2zQso-27U

Tuesday March 24, 2020: Other questions (Type 7) https://wp.me/p2zQso-28q

Friday March 27, 2020: Sequences and Series questions (Type 10) BC Topic https://wp.me/p2zQso-298