Calculus I Test #2 October  22, 2001 Name____________________ R.  Hammack Score ______

(1)   Suppose   f(x) = .     Use the limit definition of the derivative to find  f '(x).

(2) Find the derivatives of the following functions. You may use any applicable rule.

(a)    f(x) = 4+ 3+

(b)   f(x) =

(c)  y = x tan(x)

(d) [ 25 + cos( ) ] =

(e) [ ] =

(3)
Suppose f(x) equals the number of dollars it costs to erect an x-foot-high transmitting tower.
(a) What are the units of f '(x)?

(b) Suppose that f '(100) = 105.    Explain, in ordinary English, what this means.

(4) This problem concerns the function f that is graphed below

(a)    Sketch the graph of f '(x). (Use the same coordinate axis)

(b)    Suppose g(x) = sin(f(x)). Find g '(4).

(c)   Suppose h(x) = 4 + + f(x).   Find h'(2).

(5) Sketch the graph of a function  f  whose derivative has the following properties:
f(0) = 2,     f '(0) = 0,    f '(3) = 0,   and  f '(x) ≤ 0 for all values of x.

(6) Consider the function  f(x) = x +

(a) Find the slope of the tangent line to the graph of f at the point where x = 4.

(b) Find the equation of the tangent line to the graph of f at the point where x = 4.

(7) Find all values of x for which the slope of the tangent to the graph of   y = sin x   at the point   x   is

(8)  Find the slope of the tangent to the graph of    at the point (π/4,  1).

(9) Suppose a 10-foot-long ladder is sliding down a wall in such a way that the base of the ladder moves away from the wall at a constant rate of 2 feet per second.  How fast is the top of the ladder moving down the wall when it is 6 feet above the floor?