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

(1) (25points)

(a)  State the limit definition of [Graphics:Images/T2BF02sol_gr_1.gif].

[Graphics:Images/T2BF02sol_gr_2.gif]= [Graphics:Images/T2BF02sol_gr_3.gif]

(b) State two of the three main interpretations of a derivative  [Graphics:Images/T2BF02sol_gr_4.gif].  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 [Graphics:Images/T2BF02sol_gr_5.gif].

[Graphics:Images/T2BF02sol_gr_6.gif]= [Graphics:Images/T2BF02sol_gr_7.gif] [Graphics:Images/T2BF02sol_gr_8.gif] [Graphics:Images/T2BF02sol_gr_9.gif][Graphics:Images/T2BF02sol_gr_10.gif]


(d) Use the derivative rules to find the derivative of  [Graphics:Images/T2BF02sol_gr_17.gif] without using a limit. (Answer should agree with Part c.)

  [Graphics:Images/T2BF02sol_gr_18.gif]   Using the general power rule (version of chain rule), we get:

(e) Find the equation of the tangent line to  [Graphics:Images/T2BF02sol_gr_22.gif] at the point [Graphics:Images/T2BF02sol_gr_23.gif].

The tangent passes through point  [Graphics:Images/T2BF02sol_gr_24.gif]and its slope is f '(8)[Graphics:Images/T2BF02sol_gr_25.gif]
By the point-slope formula, the equation of the tangent line is



(2) (20 points) The problems on this page concern the function  [Graphics:Images/T2BF02sol_gr_29.gif] that is graphed below.


(a)  Using the same coordinate axis, sketch the graph of  [Graphics:Images/T2BF02sol_gr_31.gif]. (Drawn above in red)

(b) For which value(s) of x is [Graphics:Images/T2BF02sol_gr_32.gif] increasing most rapidly?  x = 0

(c)  For which value(s) of x is [Graphics:Images/T2BF02sol_gr_33.gif] greatest? x = 0

(d) For which value(s) of x is [Graphics:Images/T2BF02sol_gr_34.gif] decreasing most rapidly? x = 2

(e)  For which value(s) of x is [Graphics:Images/T2BF02sol_gr_35.gif] smallest?  x = 2

(f) Suppose [Graphics:Images/T2BF02sol_gr_36.gif].  Estimate [Graphics:Images/T2BF02sol_gr_37.gif].
g '(x) = f '( f(x) ) f '(x)
g '(0) = f '( f(0) ) f '(0) =  f '( 2 ) 2 = (-2)(2) = -4

(g) Suppose [Graphics:Images/T2BF02sol_gr_38.gif].  Estimate [Graphics:Images/T2BF02sol_gr_39.gif].

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

(a)       [Graphics:Images/T2BF02sol_gr_42.gif]           [Graphics:Images/T2BF02sol_gr_43.gif][Graphics:Images/T2BF02sol_gr_44.gif]

(b)     [Graphics:Images/T2BF02sol_gr_45.gif]          [Graphics:Images/T2BF02sol_gr_46.gif]

(c)     [Graphics:Images/T2BF02sol_gr_47.gif][Graphics:Images/T2BF02sol_gr_48.gif]

(d)      [Graphics:Images/T2BF02sol_gr_49.gif][Graphics:Images/T2BF02sol_gr_50.gif][Graphics:Images/T2BF02sol_gr_51.gif][Graphics:Images/T2BF02sol_gr_52.gif]

(e)      [Graphics:Images/T2BF02sol_gr_53.gif]  [Graphics:Images/T2BF02sol_gr_54.gif]

(f)      [Graphics:Images/T2BF02sol_gr_55.gif][Graphics:Images/T2BF02sol_gr_56.gif]

(g)     [Graphics:Images/T2BF02sol_gr_57.gif]  [Graphics:Images/T2BF02sol_gr_58.gif][Graphics:Images/T2BF02sol_gr_59.gif]

(5) (10 points)  Find [Graphics:Images/T2BF02sol_gr_62.gif]by implicit differentiation:    [Graphics:Images/T2BF02sol_gr_63.gif]


(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 [Graphics:Images/T2BF02sol_gr_70.gif]

We seek [Graphics:Images/T2BF02sol_gr_71.gif]?

By Pythagorean Theorem, [Graphics:Images/T2BF02sol_gr_72.gif]





Now, to find z, we use the Pythagorean theorem again.  [Graphics:Images/T2BF02sol_gr_78.gif], so [Graphics:Images/T2BF02sol_gr_79.gif]

Thus [Graphics:Images/T2BF02sol_gr_80.gif]miles per hour