
Objective

To understand the derivative function of the exponential
function b^{x} 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.

Duedate

Your instructor should tell you if any of this
lab is to be turned in for grading and what the duedate 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)= 2^{x}.
Our goal is to guess the function g, which is the derivative
function of 2^{x}.
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 xycoordinate 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 2^{x}.
Estimate the slope of 2^{x} 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 (doubleclick 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 xcoordinates
without their corresponding ycoordinates. Edit the commands by
entering the corresponding ycoordinates (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 2^{x}, say a multiple/fraction of it.
This function is the derivative of 2^{x}.
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(100^{x}), 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 ycoordinates 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)= 3^{x}.
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 2^{x} (you'll need to edit
the ycoordinates). 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 3^{x}, say a multiple/fraction of it.
This function is the derivative of 3^{x}.
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 2^{x}?
Question C2.
What was your answer for the derivative of 3^{x}?
Question C3.
What would you guess is the derivative of 4^{x}?
(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 5^{x}?
Question C5.
What would you guess is the derivative of b^{x} for any
positive real number b?
Question C6.
What is the derivative of e^{x}?
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
b^{x} 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 lnx.