(1) (25points)

(a)  State the limit definition of .

(b) State two of the three main interpretations of a derivative  .  Be specific.

(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.)

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

(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  .

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

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

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

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

(f) Suppose .  Estimate .

(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?