Name: ______Date: 02/15/13
AP Calculus ABMs. Wilson
Chapter 5 Problem Set – DUE Friday, March 1st
Answer all questions on looseleaf or graph paper. Make sure all answers are complete and show all work where necessary.
1.) An object is shot straight upward from sea level with an initial velocity of 400 ft/sec.
a.) Assuming gravity is the only force acting on the object, give an upper estimate for its velocity after 5 seconds have elapsed. Use g = 32 ft/sec2 for the gravitational constant.
b.) Find a lower estimate for the height attained after 5 seconds.
2.) The nose “cone” of a rocket is a paraboloid obtained by revolving the curve about the x-axis, where x is measured in feet. Estimate the volume V of the nose cone by partitioning [0,5] into five subintervals of equal length, slicing the cone with planes perpendicular to the x-axis at the subintervals left endpoints, constructing cylinders of height 1 based on cross sections at these points and finding the volume of these cylinders.
Repeat using cylinders based on cross sections at the midpoints of the subintervals.
3.) It can be shown by mathematical induction that . Use this fact to give a formal proof that
by following these steps:
a)Partition [0,1] into n subintervals of length 1/n. Show that the RRAM Riemann sum for the integral is .
b)Show that this sum can be written as.
c)Show that the sum can therefore be written as .
d)Show that
e)Explain why the equation in part (d) proves that .
4.) A dam released 1000 cubic meters of water at a rate of 10 cubic meters per minute, and then released another 1000 cubic meters at a rate of 20 cubic meters per minute. What was the average rate at which the water was released? Give reasons for your answer.
5.) Archimedes discovered that the area under a parabolic arch is always two-thirds the base times the height.
a.) Find the area under the parabolic arch
b.) Find the height of the arch.
c.) Show that the area is two-thirds the base times the height.
6.) The marginal cost of printing a poster when x posters have to be printed is dollars.
a.) Find c(100) – c(1), the cost of printing posters 2 to 100.
b.) Find c(400) – c(100), the cost of printing posters 101 to 400.
7.) Suppose that a company’s marginal revenue from the manufacture and sale of eggbeaters is where r is measured in thousands of dollars and x in thousands of units. How much money should the company expect from a production run of x = 3 thousand eggbeaters?
8.) The design of a new airplane requires a gasoline tank of constant cross section area in each wing. A scale drawing of a cross section is shown on page 313 in your textbook (#30). The tank must hold 5000 lb of gasoline, which has a density of 42 pounds per cubic foot. Estimate the length of the tank.