IntroCal / Mr. Hansen

Name: _______________________________________

3/7/2007

 

 

Quest (70 pts.) on FTC, Definite Integrals, and Optimization

 

Instructions: For all problems, show your work for full credit. There are certain permitted calculator operations (solving an equation, finding a derivative at a point, finding an intersection of curves, finding a definite integral, graphing, and function evaluation at a point) for which you need not show work unless specifically told to. Each problem is worth 10 points, except for #4, which has two parts and is worth double.

 

  1. The letters “FTC” mean Federal Trade Commission in official Washington. To us, they mean ____________________________________________________ .
  2. State the FTC, including the conditions (i.e., the “if” part).








  3. Use the formula  to find the mean value of the function

    f (x) = cos x – x2 + 3x on the interval [–2, 1.5]. Show your FTC algebra, and check your answer with MATH 9.















    4.
  4. Useful formulas:

    An aluminum beverage can is to be manufactured so as to have a volume of 350 ml (i.e., 350 cubic centimeters). The radius r must satisfy 2 ≤ r ≤ 12, where r is measured in centimeters, since obviously a can cannot be manufactured if it is too thin, and if it is too fat it ceases to be a beverage can and starts to become something like a sardine can. Find the dimensions (r and h) that produce
    (a) the minimum surface area
    (b) the maximum surface area































  1. We learned a certain rule regarding odd functions. State the rule, giving an illustrated example. (Make a sketch and write a sentence or two to show that you understand the rule.)









  2. Compute the area between the line y = –4 and the curve y = 9 – x2, below the x-axis. You may omit the FTC algebra to save time, but you must show the integral(s) you are using to get your answer.