Name: Problem Possible Score Number Points Total 140

Transcription

1 Name: Math Xa Final Examination Part II Thursday, January 20, 2005 Please circle your section: Thomas Judson Angela Vierling-Claassen David Harvey Ben Weinkove Jennie Schiffman (CA) Connie Zong (CA) Margaret Barusch (CA) Evan Hepler-Smith (CA) MWF MWF MWF MWF Problem Possible Score Number Points Total 140 Directions Please Read Carefully! You have three hours to take this exam. Make sure to use correct mathematical notation. Any answer in decimal form must be accurate to three decimal places, unless otherwise specified. Pace yourself by keeping track of how many problems you have left to go and how much time remains. You do not have to answer the problems in any particular order, so move to another problem if you find you are stuck or spending too much time on a single problem. To receive full credit on a problem, you will need to justify your answers carefully unsubstantiated answers will receive little or no credit (except if the directions for that question specifically say no justification is necessary, such as in a True/False section). Please be sure to write neatly illegible answers will receive little or no credit. If more space is needed, use the back of the previous page to continue your work. Be sure to make a note of this on the problem page so that the grader knows where to find your answers. You may use a calculator on this part of the exam, but no other aids are allowed. Good Luck!!! 1

2 1. (10 points) Flight PI314 takes off from Logan airport in Boston at 1pm and lands in Miami, Florida at 7pm the same day. The following are all functions of time. Which of these functions are invertible? You should take the initial time t = 0 to be the moment when the plane lifts off the ground in Boston, and take the final time t = 6 (measured in hours) to be the moment when the plane touches the ground in Miami. Assume that nothing goes wrong and that the engine runs continuously during the entire flight. You must state whether each function is invertible or not and give a brief explanation in either case. (a) The height of the airplane above sea level at time t. (b) The total distance travelled at time t. (c) The speed of the airplane at time t. (d) The total number of mini pretzels consumed by the passengers at time t. (Mini pretzels are handed out 30 minutes after take-off.) (e) The amount of fuel in the airplane at time t. 2

3 2. (10 points) An island community that has been living without electric power has decided to have electricity brought to the island. The electric lines must be brought from the nearest town to the island. The island is 5 miles from the nearest point on the shore. The town is located on the shore and is 13 miles from the point on the shore that is nearest to the island. The community is considering how to run the cables. It costs 50% more to run the cable under water than it does to run it under the ground. How far should the cable be run along the shore so that the community will have to pay the smallest amount possible? Be sure to carefully justify your answer. 3

4 3. (10 points) Roger was speeding in his car along a straight desert highway towards Las Vegas. He s going at constant speed (above the limit) when he sees a police car in his rear view mirror pulling him over. He immediately starts to slow down to a stop, and the police car stops behind him. He argues with the police officer for a little while, but she s taking no excuses and gives him a ticket. Then he sets off again, this time carefully driving at exactly 5 miles an hour below the speed limit. Here is a graph of the distance travelled by Roger as a function of time. The units of time are minutes and the units of distance are miles. Take t = 0 to be 1pm. Distance miles Time min (a) When did Roger see the police car? (b) For how long was he stopped by the side of the road? (c) How fast was he driving at 1:16 p.m? (d) How many miles per hour above the speed limit was he driving before he got pulled over? 4

5 4. (10 points) Suppose that after you stuff a turkey its temperature is 50 F and you then put it into a 325 F oven. After an hour the meat thermometer indicates that the temperature of the turkey is 93 F and after two hours it indicates 129 F. Use linear approximation to predict the temperature after two hours and fifteen minutes. 5

6 5. (8 points) The following is a graph of the derivative of f(x) (a) On what interval(s) is the original function f(x) increasing? (b) On what interval(s) is the original function concave up? (c) For what values of x is f (x) = 0? (d) Suppose that f(3) = 1. Is f(4) positive, negative, or zero? Briefly explain why. 6

7 6. (10 points) Use the definition of the derivative to show that the derivative of f(x) = x 2 +x 1 is f (x) = 2x x 2. Do not just use the power rule you must calculate a difference quotient and take a limit. 7

8 7. (10 points) A box is to be built out of a rectangle of a 10 inch by 15 inch rectangle of cardboard by cutting a square out of each corner and folding the cardboard into a box as shown below. Note that the pieces removed from the corners must be square and they must all be the same size. Is it possible to maximize and/or minimize the volume of the box? If so, what size square must be removed from each corner? 8

9 8. (15 points) Storage company A will come to your house or apartment and pick up your things, take them to a storage facility, and then deliver them after the time of storage is complete. They charge a $300 fee that includes both the initial pickup and the final delivery and charge $400 for the first six months of storage. You can have them deliver sooner than 6 months, but it will still cost you $400 for storage, plus the fee for pickup and delivery. After 6 months, they charge $83 for each additional month. Storage company B charges a fee of $180 which includes both the initial pickup and the final delivery. Storage company B charges $104 per month for the first 6 months of storage, and $70 per month after that. (a) How much do 10 months of storage (including pickup and delivery) with Company B cost? (b) Write a piece-wise linear function A(t) that gives the cost of storage with Company A as a function of time t, in months, that your possessions will be in storage. 9

10 (c) Write a piece-wise linear function B(t) that gives the cost of storage with Company B as a function of time t, in months, that your possessions will be in storage. (d) Graph A(t) and B(t) on the axes below. Label each graph, and be sure to carefully label intercepts and any places where the two graphs intersect (e) For what values of t is Company A a better choice for you financially than Company B? 10

11 9. (10 points) Let f and g be the functions whose graphs are given below. Use the graphs to evaluate the limits and expressions. If the limit does not exist or the expression cannot be evaluated, say why. f x g x x x (a) lim f(x) + g(x) x 1 (b) (f + g)( 1) (c) lim f(x) + g(x) x 0 (d) lim x 0 f(x)g(x) (e) lim x 3 g(x) f(x) 11

12 10. (12 points) Census: World population growth slowing. 1 WASHINGTON (AP) The world s population growth is slowing because women are having fewer children and more people are dying from AIDS, especially in Africa, according to a Census Bureau report released Monday. The report forecasts there will be nearly 9.1 billion people by 2050, a nearly 50% increase from the 6.2 billion in However, the growth rate is slowing significantly. The global population grew 1.2% from 2001 to 2002, or about 74 million people, but growth will slow to 0.42% by That s far below the peak growth of 2.2% between 1963 and Suppose that P(t) is the world s population, where t is measured in years after 2000 (so P(0) is the world population in 2000 and P(1) is the population in 2001). Answer the following questions, assuming that the prediction stated in the article is correct. (a) Estimate the average rate of change of P(t) from t = 2 to t = 50. (b) Is P (2) positive or negative? Why? 1 Posted on USAToday.com on 3/22/

13 (c) Put the following in order from least to greatest: P (2), 0, P (50), P ( 37). (d) From the information given in the article, can you maximize P(t) on the interval [ 37,50]? If you can, estimate that maximum and explain how you know that is the maximum. If you cannot, explain why you cannot. 13

14 11. (10 points) Assume that f is a continuous function on the closed interval [ 2,4] with f( 2) = 3 and f(4) = 1. Also, assume that f and f exist and are continuous on ( 2,4). Use the information in the table below to sketch a possible graph of f. x 2 x < < x < < x < < x 4 f (x) f (x) f x x

15 12. (15 points) In the television series CSI the forensic scientists estimate the time of death using the rule of thumb that a body cools abut 2 F during the first hour after death and about 1 F for each additional hour. Assuming an air temperature of 68 F and a living body temperature of 98.6 F, the temperature T(t) of a body at a time t hours since death is more accurately modeled by T(t) = e kt. (a) For what value of k will the body cool by 2 F in the first hour? (b) Using the value of k found in part (a), after how many hours will the temperature of the body be decreasing at a instantaneous rate of 1 F per hour? 15

16 (c) Using the value of k found in part (a), show that, 24 hours after death, the coroner s rule of thumb gives approximately the same temperature as the formula. 16

17 13. (10 points) Below are graphs of y = f(x) and y = g(x). Find constants A,B and C so that g(x) = Bf(x + A) + C f x x x g x 2 17