The Derivative II

(In this activity I am paraphrasing and expanding the suggestions of Alan Lipp in a posting “Derivatives of Trig Functions” August 29, 2012 to the Calculus Electronic Discussion Group.)

This activity parallels the one in my last post here using technology.

    1. Enter the function you are investigating as Y1 in your calculator. Later you will change this to other functions but will not have to change the following entries.  Start with Y1 = x2.
    2. Enter

      \displaystyle Y2=\frac{Y1(x+0.0001)-Y1(x)}{0.0001}.

      This will approximate the derivative. This expression is called the forward difference quotient.

    1. Graph in a square window.
    1. Guess the equation of the graph you see for Y2, enter you guess in Y3 and graph it. If your guess is correct what should you see?
    1. Deselect Y1 and produce a table for the Y2 and Y3 graph. Do the values of Y2 look like what you guessed for Y3? If not, adjust your guess for Y3. (Hint: because Y2 is an approximation, they will be close but not exact.)
  1. Another way to check your guess is to graph Y4 = Y2/Y3. If your guess is close, Y4 should be the line y = 1. If their guess is wrong, the graph of Y4 may give a clue as to the correct answer. If the guess the derivative of x2 is x, then Y4 = 2 hinting that the correct guess is 2x.

A comment: Calculators have a built in numerical derivative function usually called nDeriv or d/dx. You may use this in step 2 above. However, entering and using the expression for the approximate derivative as above, reinforces the concept and is more transparent for the student than using some strange new built-in function.

Now repeat the exercise above with other functions. Chose functions whose derivatives are easy to guess for example, y = x3, y = x4, y = x5, etc., and  y = sin(x), and y = cos(x).

Keep a list of the results, so you can check it later.


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