Math 132, Calculus 2, Exam 1 Practice Problems,
 Tom Linton, Fall 2003 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 60 cars per hour.

1. When do cars start having to wait to get through the toll booth? How can you tell?
2. How many cars are waiting in line at noon?
3. When 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. Write down a definite integral, with integrand r(t), that represents the answer to part (d).
   
6. What is the average rate that cars arrive at this toll booth from 10 AM to 1 PM?
7. When does the line at the toll booth disappear?
   
2. 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.

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. Like 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.
   
   
3. Consider
   
1. Give a left hand sum with 3 rectangles that approximates this definite integral. Sketch a graph showing the integrand and your rectangles.
   
2. Would a right sum with 50 rectangles be an over estimate or an under estimate in this case?
3. If I wanted to estimate this definite integral with an error less than 0.02, how many rectangles would I need?
   
4. 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?
   
5. Let F(t) be the antiderivative of f(t) with F(0) = -2, where f(t) is plotted below.

1. Calculate F(t) for t = 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 t from 0 to 8) does F(t) obtain its maximum value? What is that maximum value?
5. Is F(t) concave up or concave down at x = 9? How do you know?
   
6. Can you find a value of t (other than t = 2), where F(t) = 0?
   
6. Use the Fundamental Theorem of Calculus to evaluate the following integrals.
1. 
2. 
3. 
4. Hint: you'll need to guess an antiderivative. F(x) = e^2x isn't quite right, it needs to be divided by some constant.
   
5. 
   
   
   
   
7. 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). Pay careful attention to where F(x) switches concavity, and has local maximums (hilltops) and local minimums (valley bottoms).
   

8. If F(x) is an antiderivative of f(x) = 2^sin(x) - 1, and F(3) = 5, use fnInt and the Fundamental Theorem of Calculus to estimate F(0) and F(6).