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).

f '(x) =  = = =

=   =   =

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

(a)    f(x) = 4+ 3+
f '(x) = 40 + 9

(b)   f(x) =

f ' (x) =

(c)  y = x tan(x)
dy/dx = (1)tan x + x = tan x + x

(d) [ 25 + cos( ) ] = 0 + [  cos( ) ] = -sin ( ) 4= -4sin ( )

(e) [ ] = [ ] = 1/2(2x) =

(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)?
dy/dx = dollars per foot

(b) Suppose that f '(100) = 105.    Explain, in ordinary English, what this means.
It will cost \$105 to build the 101st foot

(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).
By the chain rule,  g '(x) = cos( f(x) ) f '(x),
so g '(4) = cos(f(4)) f '(4) = cos(1)(0) = 0

(c)   Suppose h(x) = 4 + + f(x).   Find h'(2).
h '(x) = 2x + 2x f(x) + f '(x),
so h '(2) = 2(2) + 2(2)(f(2)) + f '(2) =  2(2) + 2(2)(1.5) + (-1) = 4 + 6 - 4 = 6

(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.
f '(x) = 1 +, thus the slope we seek is f '(4) = 1 += 1+1/4 = 5/4

(b) Find the equation of the tangent line to the graph of f at the point where x = 4.
The slope is 5/4 (from part a), and the line passes through the point (4, f(4)) = (4, 6).
Using the point-slope formula:
y - 6 = 5/4(x - 4)
y - 6 = 5/4 x - 5
y = 5/4 x + 1

(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

Slope = dy/dx = cos x, so we are looking for all values of x for which cos x = 1/2.
Looking at the unit circle, we see that these values of x are x = π/3 + k2π  and x = -π/3 + k2π , where k is an integer.

(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?

Let x be the distance from the wall to the base of the ladder. Let y be the height of the top of the ladder.
We know = 2, and we want to find .
By the Pythagorean Theorem, . Differentiating both sides with respect to t:

2x

,   so

Now use the Pythagorean Theorem again to find that x = 8 when y = 6.
Then =