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

(1) (25points)

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

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


(c) Use the limit definition from Part a to find the derivative of [Graphics:Images/T2BF02sol_gr_5.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.)


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


(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].

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

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

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

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

(f) Suppose [Graphics:Images/T2BF02sol_gr_36.gif].  Estimate [Graphics:Images/T2BF02sol_gr_37.gif].

(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]

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

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

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

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

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

(g)     [Graphics:Images/T2BF02sol_gr_57.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?