Before we look at the table and Riemann sum problem take a look at this: A Google Employee has calculated 
On to Riemann sums and 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.
- Use Riemann sums (left, right, midpoint), 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.
- 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 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.
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.
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
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 review postings:
- Tuesday February 27 – AP Exam Review
- Friday, March 2 – Resources for reviewing
- Tuesday March 6 – Type 1 questions – Rate and accumulation questions
- Friday March 9 – Type 2 questions – Linear motion problems
- Tuesday March 13 – Type 3 questions – Graph analysis problems
- Friday March 16 – Type 4 questions – Area and volume problems
- Tuesday Match 19 Type 5 questions – Table and Riemann sum questions (this post)
- Friday March 22 Type 6 questions – Differential equation questions
- Tuesday March 26 – Type 7 questions – miscellaneous
- Friday March 29 Type 8 questions – Parametric and vector questions (BC topic)
- Tuesday April 2 Type 9 questions – Polar equations
- Friday April 5 Type 10 questions – Sequences and Series
Teaching Calculus 