Problem Possible Score Number Points Total 140

Transcription

1 Name: SOLUTIONS 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. This function is not invertible because, for instance, there will be two different times at which they airplane is at a height of 1000 feet. (b) The total distance travelled at time t. This function is invertible. The total distance travelled is a strictly increasing function, thus it is not possible for the total distance travelled to be the same at two different times. (c) The speed of the airplane at time t. This function is not invertible because there will be two or more times at which the airplane will be at the same speed. (d) The total number of mini pretzels consumed by the passengers at time t. (Mini pretzels are handed out 30 minutes after take-off.) This function is not invertible because the total number of pretzels consumed will be zero for any time 0 t < 30. (e) The amount of fuel in the airplane at time t. This function is invertible. The amount of fuel in the airplane is always decreasing, so the amount of fuel cannot be the same at two different times. 2

3 5 n 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. One can draw a picture of the problem as below. Then the total cost will be proportional to c = Cost under water + Cost on land = 1.5y + (13 x) I s l a n d 1 3 T o w We need to make this into a function of one variable so that we can take a derivative. We do this using the pythagorean theorem y 2 = x 2 Thus and so y = 25 + x 2 c(x) = x x To minimize the cost, we find the critical points of c(x) and then test to see which of those are minima. c (x) = 1.5x 25 + x 2 1 = 1.5x 25 + x x 2 The function c(x) has critical points for points x in the domain where c (x) is zero or undefined. There are no values of x where c(x) is undefined and we find the other critical points by solving for when the numerator above is zero. 1.5x 25 + x 2 = 0 1.5x = 25 + x 2 (1.5x) 2 = 25 + x x 2 x 2 = x 2 = 25 Thus x = 25/ (we disregard the negative square root since x > 0). Our domain in this case is a closed interval, 0 x 13 (we get the upper bound by using the pythagorean theorem). Since we have a closed interval we can find the absolute max and min on this interval by merely testing the value of c(x) at all of our endpoints and critical points. We find that c(0) = 20.5, c( 25/1.25) 18.59, and c(13) Thus x 4.47 and the cable should be run 13 x 8.53 feet along the shore. (Note that we could also have used the first or second derivative to test for the minimum point. These tests will work even when you don t have a closed interval). 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? Roger started to slow down at t = 6, or 1:06pm, so this is when he saw the police car. (b) For how long was he stopped by the side of the road? He was stopped between t = 8 and t = 12, so for 4 minutes. (c) How fast was he driving at 1:16 p.m? At t = 16 the graph appears to be linear (and we are told the speed is contant), so the velocity is equal to the slope of the line. Looking at this graph, the slope is approximately 7/6 or 1.17 miles per minute, which is 70 miles per hour. (d) How many miles per hour above the speed limit was he driving before he got pulled over? This is equal to the slope of the line in the first part of the graph. Looking at the graph, this is 8/ miles per minute, or 80 miles per hour. Since he was driving 5 miles over the speed limit after he got pulled over, the speed limit must be 75 miles per hour, and thus he was driving 5 miles over the speed limit when 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. Between 1 and 2 hours the temperature increases = 36 degrees, thus if T (t) is the temperature at time t, we estimate T (2) 36. We are also given that T (2) = 129. We can thus find the tangent line y 129 = 36(t 2) y = 36t + 57 Thus our linear approximation is T (t) 36t + 57 (for t near 2). After 2 hours and 15 minutes, t = 2.25 and our linear approximation gives T (2.25) 36(2.25) + 57 = 138 F. 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? The original function is increasing where f (x) > 0, thus on the interval (4, 9) (using the interval (4, ) would be acceptable as well). (b) On what interval(s) is the original function concave up? The original function is concave up where f (x) is increasing, thus on the intervals (2, 4) and (8, 9) (or (8, )). (c) For what values of x is f (x) = 0? f (x) = 0 where the graph of the derivative has a horizontal slope. x = 2, 4, and 6. This happens at (d) Suppose that f(3) = 1. Is f(4) positive, negative, or zero? Briefly explain why. Since f(3) = 1 and f(x) is decreasing between 3 and 4, this means that f(4) must be negative. 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. By the definition of the derivative f (x) = lim h 0 f(x + h) f(x) h (x + h) 2 + (x + h) 1 (x 2 + x 1 ) = lim h 0 h x 2 + 2xh + h x+h = lim x2 1 x h 0 h 2xh + h = lim h 0 h x+h 1 x x(x + h) x(x + h) (2xh + h 2 )x(x + h) + x (x + h) = lim h 0 hx(x + h) h(2x + h)x(x + h) h = lim h 0 hx(x + h) (2x + h)x(x + h) 1 = lim h 0 x(x + h) (2x + 0)x(x + 0) 1 = x(x + 0) = 2x3 1 x 2 = 2x 1 x 2 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? If we take out a square with sides of length x, then the length of the box will be 15 2x and the width of the box will be 10 2x and the height of the box will be x. Thus the volume of the box will be V (x) = (15 2x)(10 2x)x = 150x 50x 2 + 4x 3 To maximize or minimize the volume of the box, we first find the critical points of V (x). So V (x) = 0 when V (x) = x + 12x 2 x = 25 ± Or when x 1.96 or x = But note that the domain of x is (0, 5), so we discard the second solution. We use the second derivative test and see that V (x) = x > 0, so that V (1.96) 53 < 0 so x = 1.96 gives a minimum. Thus we remove a square with sides of length 1.96 inches to maximize the volume of the box, and we find that the volume of the box cannot be minimized (or you might argue that the minimum volume of the box is zero if you consider x = 0 and x = 0 to give boxes of zero volume). 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? = = 1084 (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. A(t) = { 700 (t 6) (t 6) (t > 6) = { 700 (t 6) t (t > 6) 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. B(t) = { t (t 6) (t 6) (t > 6) = { t (t 6) t (t > 6) (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? 5 < t < 14 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. (a) lim f(x) + g(x) x 1 Both f and g have limit 0 as x 1, so the answer is 0. (b) (f + g)( 1) Neither f nor g is defined at x = 1, so the expression cannot be evaluated. (c) lim f(x) + g(x) x 0 As x 0, the left hand limit of f is 1 and the left hand limit of g is 1, so the answer is 1 + ( 1) = 0. (d) lim x 0 f(x)g(x) The left hand limit of f(x)g(x) is 1, while the right hand limit of f(x)g(x) is +1. Since these numbers are different, the expression cannot be evaluated. (e) lim x 3 g(x) f(x) As x 3, g(x) is finite, while f(x) +. Therefore, g(x)/f(x) 0. 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. The average rate of change is calculated by: P (50) P (2) 50 2 = 9, 100, 000, 000 6, 200, 000, = 2, 900, 000, , 416, 667 people/year. (b) Is P (2) positive or negative? Why? P (t) is an exponential function, so it is always concave up. Therefore P (2) is positive. 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). 0 is the smallest, because P (t) is always positive (since the population is always growing). The data in the article suggest that the growth rate was largest in 1963 (at t = 37), and that it will be smaller in 2050 (at t = 50) than it is in 2000 (at t = 0). Therefore, the correct order is: 0, P (50), P (2), 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. The population is always growing, so P (t) is an increasing function. Therefore, its maximum is achieved at the right endpoint, t = 50. This maximum is P (50) = 9.1 billion. 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)

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? We want = 30.6e k. Thus so that k = ln e k = = , = (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? T (t) = 30.6 k e kt, so we wish to solve 1 = 30.6 k e kt for t. We get t = ln( 30.6k) k = ln(( 30.6)( )) hours 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. The formula gives T = 30.6e F. The coroner s rule gives T = = 73.6 F. The two are roughly the same. 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. The width of the broken segment is transformed from 1 to 2, and it is flipped, so there is a vertical stretch of 2. Hence B = 2. After applying this stretch, a shift upwards of 1 is needed to get the y-values of the breakpoints right. Thus C = 1. Finally, the graphs match after shifting left by 1. Hence A = 1. 17