U.T. ARLINGTON CALCULUS I       LAB I.3           (Last revision June 2004.)
Objective
To understand the derivative function of the exponential function bx for any positive real number b. Also to find all the functions f such that f '(x) = r f(x) for all real numbers x where r is your favorite real number.
Procedure
Answer the questions below on a sheet of paper unless otherwise instructed. Write your answers as completely and neatly as possible. Put your name at the top right of the paper and put the title ``Calculus I   Lab I.3'' at the top center.
Due-date
Your instructor should tell you if any of this lab is to be turned in for grading and what the due-date is.  If not, then ask.

Introduction
The derivative of a differentiable function f at x = a is the slope of the tangent line to the graph of f at the point ( a, f(a) ) and is given by
which is the limit of the slopes of secant lines passing through (a, f(a)). In this lab, call this limit g(a).

PART A

Step 1
Suppose f(x)= 2x. Our goal is to guess the function g, which is the derivative function of 2x.

You are going to use a web browser to access a certain site given below. For this to work properly, java needs to be "allowed" or "on" in the preferences. You will alternate between that site and this one, so you might want to call up a new browser window, so that you can jump between that site and this one and look at them both at once. I recommend that you copy and paste the site's address into the new browser window rather than clicking on it.

You should eventually (!) get a page with an xy-coordinate system and instructions. Once you get to that site, read the instructions there (they are below the coordinate system) and apply those instructions to the function 2x. Estimate the slope of 2x at the values of a (or, as close as possible to the values of a) given in the following table. Record your data and be careful to notice if the slope is negative (it is difficult to see the negative sign on the screen!   :-(     ).

Go to http://www.uta.edu/math/Calculus/secant.html.

Question A1.

Table: Slope for f(x) = 2 x
a
   -3   -2   -1   0   1   2   2.5   3   3.5   4   4.5
g(a) = slope of f at a
                     
Step 2
We will now get Mathematica to plot these points for us on a graph. To do this, highlight and copy the commands located at http://www.uta.edu/math/Calculus/2x.html by using the ``Edit'' menu. Do NOT exit your web browser. Call up Mathematica (double-click on its icon or use ``program'' menu under ``start'' menu) and paste (using the ``Edit'' menu) into the untitled window. The commands pasted have the above x-coordinates without their corresponding y-coordinates. Edit the commands by entering the corresponding y-coordinates (your estimates in your table) where the ``?'' are, and then enter your commands by pressing the ``Enter'' key at the South East corner of the keyboard. (If you have trouble with the ``Enter'' key, try "Shift" and "Enter" together.) You should get a graph of your table of data. Sketch the graph on your sheet of paper (do NOT print yet).

Question A2.

Which exponential function best describes your graph? Hint: think in terms of 2x, say a multiple/fraction of it.
This function is the derivative of 2x.

To check your answer to question A2, you should plot both the above points and your answer together on the same axes. To do this, copy the commands at http://www.uta.edu/math/Calculus/2x2.html. Now, in your Mathematica document, you should paste these commands as plain text -- to do this, use the Edit menu in the top left of your Mathematica document, slide the mouse down to ``paste as'' and select, to the right, ``plain text''.) Once you have pasted them as plain text, you should edit them accordingly. For example, where ``???'' appears, enter your guess function. (If I had guessed 17(100x), I would enter ``17(100^x)'' for ``???''.) The first line computes your guess function at the same values of x as those appearing in the above table. Where you see ``?'' enter the y-coordinates from your table (alternatively, replace those lines of code with the relevant edited lines from your previous graph by copying and pasting them). Before you enter your 3 commands, make sure there are blank lines between the 3 commands (if need be, create them by using the ``enter'' key in the MIDDLE of the keyboard). Compare how the blue points match up with your guess function's points.

PART B

Step 1
Repeat Step 1 of Part A for f(x)= 3x.

Question B1.

Table: Slope for f(x) = 3 x
a
   -3   -2   -1   0   1   2   2.5   3   3.5   4   4.5
g(a) = slope of f at a
                     

Question B2.

Use Step 2 above to plot these points using the same Mathematica commands that you used for 2x (you'll need to edit the y-coordinates). Print out your graph. (To print: highlight the bracket on the far right of the window level with your graph; then go to the ``file'' menu and select ``print selection''. Your graph should print on the printer in the pc lab. )

Question B3.

Which exponential function best describes your graph? Hint: think in terms of 3x, say a multiple/fraction of it.
This function is the derivative of 3x.

To check your answer to question B3, you should plot both the above points and your answer together on the same axes. To do this, copy the commands at http://www.uta.edu/math/Calculus/2x2.html. Now, in your Mathematica document, you should paste these commands as plain text -- to do this, use the Edit menu in the top left of your Mathematica document, slide the mouse down to ``paste as'' and select, to the right, ``plain text''.) Once you have pasted them as plain text, you should edit them accordingly. (For example, where ``???'' appears, enter your guess function in the manner described in Part A.) Before you enter your 3 commands, make sure there are blank lines between the 3 commands (if need be, create them by using the ``enter'' key in the MIDDLE of the keyboard). Compare how the blue points match up with your guess function's points.

PART C

Question C1. What was your answer for the derivative of 2x?

Question C2. What was your answer for the derivative of 3x?

Question C3. What would you guess is the derivative of 4x? (You might need to rewrite your answers to Questions 1 & 2 before doing this question!)

Question C4. What would you guess is the derivative of 5x?

Question C5. What would you guess is the derivative of bx for any positive real number b?

Question C6. What is the derivative of ex?

Question C7. Does your answer to Question C5 agree with your answer to Question C6? Why/why not?

Question C8. What is the slope of the tangent line to the graph of bx at the point (0, 1)? What if b = e?

Question C9. Based on your above answers, which functions f have the property that f '(x) = f(x) for all x? Which functions F have the property that F'(x) = r F(x) for all x, where r is your favorite real number?

Note

Lab I.4 will repeat this lab for the functions lnx and ln|x|.

END LAB I.3
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