Math 165

Spring 2002

Exam 2

 

  1. (36pt) Compute the derivatives of the following functions:

a)       b)      

c)        d)

e)

  1. (15 pt) Consider the function.

a)      Compute the derivative of this function.

b)      Find all the critical numbers of the function.

c)      Find the maximum and minimum values of the function on the interval [0,4].

 

  1. (15 pt) A rocket lifts off and is tracked by a radar station 5 miles from the launch pad. How fast is the rocket rising at the moment that it is 12 miles high if the radar station detects that the rocket is receding from the station at 2000 miles per hour at that instant?

 

  1. (15 pt) An asteroid enters the earth’s atmosphere and begins to burn up. Assume that the asteroid is always spherical and that the rate at which it “vaporizes” is proportional to its surface area. Show that the radius of the asteroid decreases at a constant rate.

 

  1. (15 pt) Suppose that I wish to measure a perfect cube and use the measurement to compute the volume. Suppose that I want the relative error of the volume to be less than .03 (that is less than a 3% error). How well do I have to measure the length of the side (how big can my relative error be in measuring the side)?
  2. (6 pt) Use implicit differentiation to verify that  You may use the fact that  and that

 

  1. (8 pt) Recall that  

a)      Use a linearization to estimate .

b)      Is your estimate too large or too small and why? (Hint: Consider the picture of  below.