AP Type Questions 5
Tables may be used to test a variety of ideas in calculus including analysis of functions, accumulation, position-velocity-acceleration, theory and theorems among others. Numbers and working with numbers is part of the Rule of Four and table problems are how this is tested.
What students should be able to do
- Find the average rate of change over an interval or 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. 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 several 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, don’t assume that the largest number in the table is the maximum value of the function.
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 entire of numbers such as you get when doing a Riemann sum, does not need to be simplified in any way. If you simplify correct answer incorrectly, you will lose credit.
Do Not: Use a calculator to 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.
Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.