Table & Riemann Sum Questions (Type 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 one way this is tested.

 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. 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, 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. However, 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.

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.


Next Posts:

Tuesday Match 21: Differential Equations (Type 6)

Friday March 24: Others (Type 7: related rates, implicit differentiation, etc.)

Tuesday March 28: for BC Parametric Equation (Type 8)


 

 

 

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4 thoughts on “Table & Riemann Sum Questions (Type 5)

  1. Never mind. I got the idea from your blog. Desmos works great. Also, I checked mr. Goldstein’s webpage and I got the equation. Thanks

    Sonal Patel

    Begin forwarded message:

    > From: Sonal Patel > Date: March 20, 2017 at 8:25:12 AM EDT > To: Teaching Calculus > Subject: Re: [New post] Table & Riemann Sum Questions (Type 5) > > Lin, > Thank you so much for giving valuable advice for these many years. > I had been using winplot file for series for so many years to show how adding power gets better curve. But now that file is not working due to “incompatible version”. > Can you please give me the directions how to make equation in the winplot. > I lost my written notes for that. I was so dependent on my old files. > Thank you in advance. > > > Sonal Patel > > >>

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  2. Lin, Thank you so much for giving valuable advice for these many years. I had been using winplot file for series for so many years to show how adding power gets better curve. But now that file is not working due to “incompatible version”. Can you please give me the directions how to make equation in the winplot. I lost my written notes for that. I was so dependent on my old files. Thank you in advance.

    Sonal Patel

    >

    Like

    • Sonal
      The Winplot website is no longer up it seems. You can try downloading it from http://www.winportal.com/winplot Rich Parris past away a few years ago and I guess there was no one to maintain it. I still use it often. I have had trouble with it today too. I don’t know what’s up with that, since it ran fine in Windows 10 recently. The format for series is under F1: sum(general term in terms of x and n, n , 0, A) On the A slider: setL = 0 and setR = 100 so it will increase by 1. So for example the cosine series is sum((-1)^(n)*x^(2n)/(2n)!,n,0, A)
      Desmos will also do this nicely. Here is a link for sin(x) https://www.desmos.com/calculator/jcooohh9im and for ln(x) centered at x = 2 https://www.desmos.com/calculator/qjfbkwwhq2 You should be able to figure out how to do others by studying the equation on the left of the screen.
      Hope this Helps/

      Like

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