Please respond to the following in complete sentences. Construct your responses so that they communicate a clear sense of "how this works" and why it works the way it does. When asked to graph something, we expect you to use Graphing Calculator. Also, assume that all angles are measured in radians unless you are told otherwise.

1. Graph
sin(*x*), -10 < *x* < 10.

a) What
does *x* stand for? What does "-10 < *x* < 10" mean given what *x* stands for?

b)
What does "sin(*x*)" mean for some specific value of *x*?

c) Click on any point on the graph. The point's coordinates are shown at the top left. What are they? What story do these coordinates tell?

d) Why
does the graph of sin(*x*) appear as it
does?

2. Examine
the graphs of cos(*x*), cos(2*x*), cos(3*x*),
cos(4*x*), cos(5*x*), etc.

Explain the relationship
between the graph of cos(*x*) and the
graph of cos(*nx*).

3. Examine
the graphs of sin(*x*), sin(*x*/2), sin(*x*/3),
sin(*x*/4), sin(*x*/5), etc.

Explain the
relationship between the graph of sin(*x*)
and the graph of sin(*x*/*n*). Is this relationship different from the one you
came up with in Problem 2?

4.
Juan said, "The function sin( ) has a period of 2π, which means that it goes through a full cycle whenever whatever is in the parentheses varies by 2π. So, sin(5+3*x*) goes through a full cycle whenever 5+3*x* varies by 2π."

a) What does Juan mean? Say this in your own words?

b) How does his insight explain the behaviors of sin( ) and cos( ) in numbers 2 and 3?

c) How
does Juan?s insight explain the behavior of sin(*x*^{2})? sin(1/*x*)?

5. Explain the "microscopic" behavior of .[1]^{,} [2]

6. Use
a slider[3]
to explore the effect that *a* and *b* have on the behavior of these functions
(0<a<10, 0<*b*<10).
Explain both the parameter's effect and your reasoning while exploring it.

a) *g*(*x*) = a
sin(b*x*) See footnote [4]

b) *h*(*x*) =
cos(*g*(*x*)) [*g*(*x*)
defined as above; see footnote 4]

7.
Attend to these function's behavior both as |*x*| becomes very small and as |*x*| becomes very large.

a)
Describe what, in general, is happening as |*x*| gets very small.Why does this happen?

b)
Describe what, in general, is happening as |*x*| gets very large. Why does this happen?

c)
What does this pattern suggest about the value of *x*sin(1/*x*)
as |*x*| gets large? What does this suggest about the value of sin(*x*)/*x* as *x* gets very small?

[1] Use the vertical bar key "|" to begin an absolute value. Press "|" again to close the absolute value.

[2] Use the *zooming* feature of GC whereby you hold down the Shift key as you draw a box around part of the graph. The boxed region will *zoom out* to fill the graphing pane.

[3] For example, type "a=slider(0,10)". Then adjust the value of *a* by moving the slider button.

[4] In this question you will need to tell GC that "g(x)" and "f(x)" are *user-defined
functions* and not, for example, "g times x". To define *g*(*x*) as given in 6a, type "g ctrl-9 x = a sin(bx)". You graph a defined function by entering a new line (ctrl-m in Windows, command-return on Mac) and then typing "y = *g*(*x*)", remembering to type ctrl-9 to get "( )".