Math 132, Calculus 2, Exam 1 Practice Problems
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Tom Linton, Spring 2002
The following questions contain many of the important concepts for our first exam. This practice sheet is by no means a complete list of the topics you should know, it is simply a collection of problems to help prepare you for the exam. Most likely, none of these questions will appear on the exam, but if you can do all of these problems without much trouble, you should be well prepared for the test.
  1. Cars arrive at a certain toll booth at the rate r(t) shown in the plot below (the units of r(t) are cars per hour). The toll booth operator can process cars at the booth at the rate of 80 cars per hour.
    1. When do cars start to have to wait to get through the toll booth? How can you tell?
    2. How many cars are waiting in line at the toll booth at noon?
    3. What time is the line longest, and how many cars are in line at that time?
    4. What is the total number of cars that arrive between 10 AM and 2 PM?
    5. What is the average rate that cars arrive at this toll booth from 9 AM to Noon?
    6. When does the line at the toll booth disappear?
  1. The first few seconds worth of velocities (in meters / sec) for Carl Lewis in the 100 meter finals at the 1987 World Championships in Rome are given in the table below.
Carl Lewis Velocities, 1987 Rome
Time (sec) 0.0 0.5 1.0 1.5 2.0 2.5 3.0
Velocity ( m /sec) 0.0 3.61 6.11 7.82 9.01 9.82 10.38
    1. Give lower and upper estimates of how far Carl ran during the first 3 seconds of this race.?
    2. Now suppose instead that Carl's velocity, t seconds into the race, was given by v(t) = 11.63*( 1 - e-0.7446t) meters per second. Use your calculator to estimate how far Carl ran from t = 0 to 10 seconds.
    3. Again using your calculator and the formula for velocity in part (b), estimate how far Carl ran in 9.94 seconds.
    4. Give a decent estimate of how long it took Carl to finish the 100 meter race.
  1. If I wanted to estimate  with an accuracy of 0.02, how many rectangles would I need? Would a left sum be an over estimate or an under estimate in this case?
  2. Suppose that water is leaking into a boat at the rate of r(t) gallons per minute, where t represents time in minutes, and t = 0 corresponds to 8 AM. Assume that at 8 AM, the boat had no water in it.
    1. In common terms, what does  represent?
    2. In common terms, what does r(60) represent?
  3. Let F(x) = , where f(t) is plotted below.
    1. Calculate F(x) for x = 0,2,4, and 8.
    2. What is F'(3)? How about F'(6)?
    3. Which is larger, F(4) or F(5)? How do you know?
    4. Where (for x from 0 to 8) does F(x) obtain its maximum value? What is that maximum value?
    5. Is F(x) concave up or concave down at x = 9?
    6. Can you find a value of x (other than x = 0), where F(x) = 0?
  1. State (in a precise manner) both versions of the Fundamental Theorem of Calculus, and give an example of the use of each version.
  2. Shown below is a graph of f(x), with certain areas marked. Suppose that F(x) is an antiderivative of f(x) and that F(0) = 3. Use the plot of f(x) to sketch a fairly accurate graph of F(x).
  1. Suppose that an egg is thrown straight up, at 60 feet per second, from a dorm window that is 50 feet above ground.
    1. Find formulas for the acceleration, velocity, and height of this egg as a function of time,  measured in seconds.
    2. How high does the egg go? At what time is the egg at its maximal height?
    3. When does the egg hit the ground? How fast is going at that time?
    4. When the egg returns to its original height (50 feet), how fast is it traveling?
  2. Use the Fundamental Theorem of Calculus to evaluate the following integrals.