Mathematics 133 -- Intensive Calculus

Problem Set 8

November 11, 2000

• A) For each of the following functions,
  • find all critical points,
  • determine the intervals on which f(x) is increasing or decreasing,
  • identify each critical point as a local maximum, local minimum, or neither,
  • determine the intervals on which y = f(x) is concave up or down,
  • determine all inflection points
  
  
  1. f(x) = 2x³ - 3x² - 12x
  2. f(x) = x sqrt(x² + 1)
  3. f(x) = x - 2 sin(x) for 0 < x < 3Pi
  
  
• B) Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.
• C) Find the point on the line y = 4x+7 that is closest to the origin.
• D) A piece of wire 10 m long is cut into two pieces. The first is bent to form a square, and the second to form an equilateral triangle. What are the maximum and minimum total areas that can be enclosed.
• E) A cylindrical can without a top is made to contain V cubic inches of liquid. Find the dimensions that will minimize the cost of the can, assuming the metal costs a constant amount per square inch.
• F) A conical drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges. Find the maximum capacity of such a cup.

From the text:

• Section 5.5/1,17,19
• Section 3.1/1,3,5,7