Make sketches of the calculator screen and hand those in when
called for. Be sure to indicate the axis scales on any sketch.

1.) Investigate the family of curves given by
f(x) = x - c*arcsin(x)
a.) What happens to the number of maxima and minima as c changes?
b.) Graph several members of the family to illustrate your results, and
hand in the sketches.


2.) Let f(x) = sinh(x)-(x-1)*cosh(x)
a.) Find the local maximum and minimum values of f(x).
b.) Estimate the x coordinate of the inflection point correct to two decimal
places by graphical means.
c.) Sketch the graph of f(x) and hand it in.

3.) Graph both f(x)/g(x) and f'(x)/g'(x) near x=0 and show in the following
examples that both ratios have the same limit as x goes to zero.
a.) f(x) = E^x-1, g(x) = x^4 - x
b.) f(x) = 2*x*ln(x), g(x) = sec(x) - 1
c.) Calculate the exact limiting value of f(x)/g(x) in each case at x=0.
Hand in the sketches in parts a and b, as well as the value obtained in c.

4.) Let f(x) be given by |x|^x when x is not equal to 0
and 1 when x = 0.
a.) Investigate graphically if f(x) is continuous at x=0.
b.) Investigate graphically if f(x) is differentiable at x=0.
c.) Use calculus to show each of the above results.
Hint:  take logs. Hand in your sketches, and your work.