Math 165

Spring 2002

Final Exam

 

  1. (20 pt) Find the following limits:

a)           b)       c)

 

d)  where  is a function such that  for all

  1. (20 pt) Find the derivatives of the following functions:

a)        b)   c)

d)

  1. (6 pt) Use the definition of the derivative to find the derivative of the function
  2. (20 pt) Evaluate the following integrals:

a)            b)    c)     d)

 

 

5.       (8 pt) A rock is thrown into a pond resulting in ripples that flow outward in the shape of a circle. If the ripples are going out at a rate of 2 ft/sec, then how fast is the area of the circle changing after 5 seconds?

  1. (8 pt) A circular piece of paper with radius  has a circular sector removed from it. The remaining piece is then folded as to make a cone shaped paper cup. What is the maximum volume of this cup?

 


                                                         

 

 

 

 

 

 


  1. (5 pt) Use the definition of the definite integral to evaluate  Check your answer.
  2. (5 pt) Suppose that I use Newton’s method to approximate the positive square root of a positive number. Explain why all my approximations (besides my first guess) are overestimates if I start with a positive initial approximation. What happens if I start with a negative initial approximation? (Hint: a picture might help you make your argument).
  3. (6 pt) Find the tangent line to the curve  at the point
  4. (12 pt) Sketch the graph of the function