This is an introduction to the use of a calculator in Mathematics 180
and 181.  It is designed to be calculator independent.  You will need
some form of graphing calculator.  Most any graphing calculator will do.
Make sure that you become sufficiently familiar with your graphing
calculator so that you are able to do these exercises.

A graphing calculator displays a rectangular portion of the graph of a function
in a graphing window.  This is the viewing rectangle.  In general you may set
this rectangle.  Most graphing calculators have a "default" window which is
set upon powerup.   For example, to graph the function
                        f(x) = x^2 + 4
over the windows [-2,2] by [-2,2] you would set xmin=-2, xmax=2,
ymin=-2 and ymax=2.  You would generally enter the function by the key
strokes   x^2+4  or you might use the square key or even the power key
to indicate x^2.  If you do this you will see a BLANK window.  The viewing
rectangle doesn't cover any points which are occupied by the function's graph!
Now try the windows: [-5,5] by [-5,5], [-10,10] by [-10,10] and
[-100,1000] by [-40,40].  Notice that the viewing rectangle need not be
centered on the origin.  Some indication of the DOMAIN and RANGE will help
choose the appropriate viewing rectangle.

Problem: Determine an appropriate viewing rectangle for
f(x) = sqrt(16-4*x^2) and use it to graph the function.
Solution: 16-4*x^2 >= 0 must be the case for f(x) to be real and well
defined.  Hence 4*x^2 <= 16 or x^2 <= 4 or |x| <= 2.
When x = 2 or -2 then f(2)=0, f(-2)=0 but
0<=sqrt(16-4*x^2) <=sqrt(16) <=4 so the appropriate viewing window is
[-2,2] by [0,4].  We will take the window slightly larger so that the
coordinate axis are included.  Hence we would use [-3,3] by [-1,5].

Problem: Graph the function y=x^3-64*x
Solution: Assuming you start with a window [-5,5] by [-5,5] what do you see?
Most calculators have a zoom feature that permits you to zoom in or out.
In this case you must zoom out. Doing so will suggest the best window to
use in order to see this graph.  Good results can be achieved by a window
[-15,15] by [-200,200].  Notice that if you did not zoom out you would not
have gotten a clear understanding of what the graph looked like.

Problem: Graph the function f(x)=sin(60*x) in viewing window which gives you
a good understanding of the function.
Solution: Since the RANGE of sin is always bounded by [-1,1] we can work on
finding the xmin and xmax while keeping the y window set at [-1.5,1.5].
First try [-15,15] by [-1.5,1.5].  Next compare this with x intervals of
[-10,10], [-15,15].  What is going on?  What is the true picture?
We must first notice that the sin(60*x) is a periodic function of period
2*Pi/60 = Pi/30 approx equal to 0.105.  This suggests that we should
use a window in x which covers only a few oscillations of the function in the
viewing window.  We can now guess why the other windows gave such
misleading results.  The function oscillates so rapidly between -1 and +1 that
when the calculator plots say 30 points in the window and connects them
then inaccurate pictures will result.

Problem: Graph the function f(x)= sin(x) + 1/200*cos(100*x)
Solution:   No single window will give an accurate graphical description of
this function.  The second term is so much smaller than the first that
we really need at least two different windows to see the behavior.
For example: [-5,5] by [-2,2] AND [-0.1,0.1] by [-0.1,0.1]

Problem: Graph the function y = 1/(x-1)
Solution:  Try various x and y window widths.  An extraneous line may exist
in the window near the pint x = 1.  This line is actually not present.
The graph is made up of two separate pieces.

Problem: Graph the function y = (x)^(1/3)
Solution: The final graph should show symmetry with respect to the origin.
If it only has a portion along the positive x axis it is because
of the way your calculator computes x^(1/3) as e^((1/3)*ln(x)) which
gives only non-negative results.

Problem: Graph the circle x^2+y^2=1 and make the result look like
a circle.
Solution: We must solve for the function y before we can
proceed, y = sqrt(1-x^2) and y = -1*sqrt(1-x^2).  We can graph
both pieces together.  We have to make sure that the y window
has the same width as the x window or else the result will not
look like a circle, but will actually look like an ellipse.
The upshot is that it is hard to tell from a graph if a curve is a
circle or an ellipse.  It is much easier to determine that
by looking at the equation.

HW Problem I: Graph the function y = x^3 - c*x
Sketch various different shapes the graph can take on depending on the
various vales of c.

HW Problem II: Using the graphics mode of your calculator, find the
solution to sin(x)=x-1 correct to two decimal places.
Hint: Graph the functions y=sin(x) and y=x-1 and find the place where
the graphs intersect by zooming it.  You may also be able to move the cursor on
your calculator to locate the point of intersection.