To understand the derivative function of sinx and
the derivative function of e^{x}.
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.2'' 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) = sinx where x is in
. Our goal is
to guess the function g, which is the derivative function of
sinx.
You are going to use a 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 sinx.
Estimate the slope of sinx 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! :-(
).
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/sine.html
by using the ``Edit'' menu. Do NOT exit the 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. Notice that
Mathematica uses "Pi" for
.
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 trigonometric function best describes your graph?
This function is the derivative of sinx.
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/sine2.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 tan(x), I would enter ``Tan''
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 in PART A for the function f(x) = e^{x}
so that you may fill in the table below.
Question B1.
Table: Slope for f(x) = e^{x}
a
-3
-2
-1
0
1
2
3
4
5
5.5
6
g(a) = slope
of f at a
Step 2
Repeat Step 2 in PART A for this table of data using the commands at
http://www.uta.edu/math/Calculus/exp.html.
This time 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 B2.
Which exponential function best describes your graph?
This function is the derivative of e^{x}.
To check your answer to question B2, 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/exp2.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 e^{x2}, I would enter ``Exp[x^2]'' for
``???''.) The first line
computes your guess function at the same values of x as those
appearing in the above table and at some other values too. 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.
Question B3.
How are the function e^{x} and the function you found for
the derivative of e^{x} related?
If c is a constant real number, how are the function
ce^{x} and the function obtained by differentiating
ce^{x} related?
Note
Labs I.3 & I.4 will repeat this lab for the functions 2 ^{x},
3 ^{x}, b ^{x} and lnx.
END LAB I.2
Save any work you wish to save to a disk.
Exit the browser and Mathematica.