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1 Name: PID: Section: This practice exam is designed to help you prepare for the upcoming final exam. The types of questions seen in this practice exam are representative of what you will see on the actual exam. The practice exam covers most of the material. However, please keep in mind there could be content on the actual exam that is not included in the practice exam. For this reason it is important you study all the exam material and don t rely soley on the practice exam. You will benefit the most from this exam by treating it like a real exam. Study all the material before attempting the exam. When you do attempt the practice exam allow yourself 2 hours and only your calculator and a writing utensil. Always show your work and provide justification where necessary. Page: Total Points: Score:

2 True or False Answer the following questions by choosing true or false and clearly filling in exactly one choice. No credit will be given for choices not clearly marked. 1. (5 points) The indefinite integral of any constant function is (5 points) If f(2) > g(2), then it follows that f (2) > g (2). 3. (5 points) The function g(t) = 1.01e 0.99t is an example of an exponentially decaying function. 4. (5 points) If v(t) is the velocity of a car t hours after the start of a trip, then gives the total distance traveled by the car in the first 5 hours. 5. (5 points) The derivative of a position function is a velocity function. 6. (5 points) The derivative of g(x) = x 2 f(x) is given by g (x) = 2xf (x). 7. (5 points) If and t = v(t)dt dalways F (t)dt = 0 then F is always constant on the interval between t = 0 8. (5 points) If f (p) = 0 and f (p) > 0, then p is local maximum of f(x). 9. (5 points) A left hand sum of B(x) over [2, 6] gives an under estimate of 6 2 B(x)dx d. Page 2

3 Multiple Choice Answer the following questions by choosing a single answer and clearly filling in that choice. No credit will be given for choices not clearly marked. 10. (5 points) Let W (c) be the average weight, in pounds(lbs), of an adult where c is the average number of Calories per day consumed. Suppose W (2000) = 155, and W (2000) = 0. Which of the following correctly interprets this information? At an average daily consumption of 2000 calories, the average person will have a weight of 155 lbs, and their weight will be decreasing. At an average daily consumption of 2000 calories, the average person will have a weight of 155 lbs, and their weight will be neither increasing or decreasing. At an average daily consumption of 2000 calories, the average person will have a weight of 155 lbs, and their weight will be increasing There is not enough information given to make the claims above. 11. (5 points) Which of the following is an antiderivative of 10x 4 3x x 4? 40x 2 6x 5 8x 10x5 3 5 x3 +2x x5 3x3 10 8x 3 2x x3 2 3 x (5 points) The figure below gives the growth rate, in feet per year, of two species of trees over the course of 10 years. Use the figure to choose the correct statement. After 10 years, species A has grown more than species B. After 10 years, species B has grown more than species A. After 10 years, both species grew the same amount. The given data is inconclusive. We cannot tell whether species A grew more or species B grew more. 13. (5 points) The derivative of q(x) = ln(x 2 + 1) is given by which of the following? 1 x 2x ln(x)2x 1 x 2 2x x x ln(x) Page 3

4 14. (5 points) The average rate of change (AROC) of f(x) = x 2 over [3, 3.01] is given by which of the following (5 points) Given g (x) below, which of the following statements is correct? g(0.4) < g(1.2) g(0.4) = g(1.2) g(0.4) > g(1.2) None of the above statements can be determined with knowing more information. Short Answer For the following question show all your work. Make sure your answers include any relevant units. No credit will be given for answers without the appropriate amount of work shown. 16. (12 points) This summer Borice decides to put a rectangular swimming pool in his back yard. The material he uses for the border of the length of the pool costs $10 per foot and the material he uses for the width costs $5 per foot. He would like spend $1,000 on the perimeter of the pool. Given these constraints, what are the dimensions of the pool (length and width) that will maximize the surface area of his pool? Page 4

5 17. The population of a colony of ants, in thousands of ants, can be modeled by the following function I(t) = 0.1t t + 50, where t is months since the start of (a) (5 points) Determine I (7). (b) (5 points) Explain the meaning of your answer from part (a) in the context of the problem. (c) (5 points) Determine I (t), what does this tell you about the rate of change of the colony over time? 18. Suppose h(t) = 100(2.05) t + ln(t 2 ). (a) (8 points) Determine h (t). You do not need to simplify the derivative. (b) (4 points) Determine the equation for the line tangent to h(t) at t = 1. Approximate values to two decimal places. Page 5

6 19. The total value of Jenae s Amazon stock, S(t) dollars, is given by the plot below, where t is time in days since the beginning of the month. (a) (6 points) Estimate S (4.5). Give units. (b) (4 points) Interpret the meaning of your answer from part (a). Include units. (c) (6 points) Is S (9) positive, negative, or zero? Use your your answer to explain what is happening to rate of change of the stock value at this time. 20. Determine the following limits. Show all your work. If a limit does not exist indicate why with your work and write DNE. (a) (5 points) lim x 500e 0.03x (b) (5 points) lim x 4 g(x) given that g(x) = { x, x < 4 0, x = 4 x 2 5, x > 4 Page 6

7 21. The half-life of Cobalt-60 is 5 years. Recall radioactive decay happens at a constant percent change. (a) (5 points) Given a sample of Cobalt-60 that currently has a mass of 100mg, determine a formula that gives the amount of the sample remaining Cobalt-60 after t years. (b) (5 points) Use your model to predict the time when 40% of the sample will remain. 22. The figure to the right gives the rate of income of Mindy and Kora, in thousands of dollars per year, where t is years since (a) (4 points) from 2010 to 2015 who had the greater change in income? (b) (4 points) Estimate how much more income the person you chose in part (a) made than their counterpart from 2010 to Page 7

8 (8 points) Determine the derivative of Q(t) = e 0.2t (3t 2 + 1) 5. You do not need to simplify the derivative. 25. (6 points) The function A(t) models the number of thousands of cases of influenza in a small country where t is the time in days since the start of influenza season. Does this function have any point(s) of inflection? If so, list any points of inflection and explain their significance in the context of this problem. If not, explain what this means in the context of the problem. Page 8

9 26. The following figure gives the plot of P (x). (a) (4 points) Sketch a plot of the second derivative in the template below. (b) (4 points) At x 3.8 the function P (x) attains a local maximum. (c) (4 points) List any intervals where P (x) is concave up. (d) (4 points) List any critical points of P (x) (not P (x)). (e) (4 points) Estimate P (3). Show your work. Page 9

10 27. The table below gives values of the derivative f (x), of a function f(x). You may assume that f (x) is strictly increasing or strictly decreasing between the given points. x f (x) (a) (2 points) Not including endpoints, list any maxima of f(x) over [ 6, 6]. If a maximum lies between two x values in the table use interval notation. For example if you decide there is a maximum somewhere between x = 4 and x = 2 then list your answer as ( 4, 2). (b) (2 points) List any values of x given in the table where f(x) is increasing. (c) (2 points) Not including endpoints, list any points of inflection of f(x) over [ 6, 6]. (d) (2 points) Estimate f(4) f( 2). Hint, use Riemann sums. Page 10