Math 165

Fall 2002

Exam 2

 

  1. (40 pt) Find derivatives for the following functions:

a)                          b)                  

c)                        d)

e)

 

  1. (12 pt) Find the tangent line to the curve defined by the equation  at the point

 

  1. (18 pt) Consider the function

a)      Find the derivative of the function.

b)      Find all of the critical numbers of the function.

c)      Find the maximum and minimum values of the function on [2,3].

 

4.      (10 pt) I wish to paint a large hemispherical dome of radius 50 feet. If my coat of paint is to be 0.1 inches thick, use differentials to estimate the amount of paint that I will need (in either cubic feet or cubic inches).

 

  1. (8 pt) Let  be a continuous function and let

a)      Show that  is a critical number for  if and only if is a critical number for

b)      Explain why  is a local maximum if and only if  is a local maximum.

 

  1. (12 pt) A ball 12 feet to the right of a 30-foot tall lamppost drops at a constant rate of 10 feet per second. How fast is the shadow of the ball moving along the ground at the instant that the ball is 20 feet above the ground?

 

  1. (10 pt) A regular pyramid (with square base of length 20 feet and height 30 feet) with base side down is filled with water. The depth of the water is decreasing at a rate of 1 ft/min when the depth of the water is 3 feet. How fast is the water leaking out?