Calculus I Test #2 November 1, 2002 Name____________________ R.  Hammack Score ______

(1) (25points)

(a)  State the limit definition of .

=

(b) State two of the three main interpretations of a derivative  .  Be specific.
1.  f '(x) is the slope of the tangent to y = f(x) at the point (x, f(x)).
2.  f '(x) is the rate of change of the quantity f(x) at x.
3. f '(x) is the velocity at time x of an object moving on a straight line and whose distance from a fixed point at time x is f(x).

(c) Use the limit definition from Part a to find the derivative of .

=

(d) Use the derivative rules to find the derivative of   without using a limit. (Answer should agree with Part c.)

Using the general power rule (version of chain rule), we get:

(e) Find the equation of the tangent line to   at the point .

The tangent passes through point  and its slope is f '(8)
By the point-slope formula, the equation of the tangent line is

(2) (20 points) The problems on this page concern the function   that is graphed below.

(a)  Using the same coordinate axis, sketch the graph of  . (Drawn above in red)

(b) For which value(s) of x is increasing most rapidly?  x = 0

(c)  For which value(s) of x is greatest? x = 0

(d) For which value(s) of x is decreasing most rapidly? x = 2

(e)  For which value(s) of x is smallest?  x = 2

(f) Suppose .  Estimate .
g '(x) = f '( f(x) ) f '(x)
g '(0) = f '( f(0) ) f '(0) =  f '( 2 ) 2 = (-2)(2) = -4

(g) Suppose .  Estimate .

(3) (35 points)  Find the derivatives of the following functions.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(5) (10 points)  Find by implicit differentiation:

(6) (10 points)  A rocket, rising vertically, is tracked by a radar station that is on the ground 5 miles from the launchpad.  How fast is the rocket rising when it is 4 miles high and its distance from the radar station is increasing at a rate of 2000 miles per hour?

Let  z be the distance between the radar station and the rocket.
Let h be the height of the rocket

We know

We seek ?

By Pythagorean Theorem,

Now, to find z, we use the Pythagorean theorem again.  , so

Thus miles per hour