1/20 2/17 3/14 4/29 5/20 Total/100. Exam II- VERSION I Spring 2011

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1 1/20 2/17 3/14 4/29 5/20 Total/100 Do not write in the spaces above. MATH Dr. Morton Exam II- VERSION I Spring 2011 Name: Directions: You have 50 minutes in which to complete this exam. Make sure that you show all work, or risk losing credit. Be sure to answer all questions asked. When entering data into your calculator, make sure that you carefully confirm that all data has been entered correctly. Round all decimals to 4 places (or as appropriate). Give all units of measurement, whenever appropriate! Good luck!

2 1. (20 points) Use the four-step method to find the (general) derivative of y = 4x 2 13x (Note: you are not finding the derivative at a point, but rather the general derivative.) To receive full credit you must carefully show all work (making sure you aren t skipping any steps) and you must label the four steps with a sentence in English (as we did in class). Also, notation does count here.

3 2. (17 points) The graph below gives the total number of prisoners (in thousands) under the jurisdiction of U.S. federal or state adult correctional authorities from 1970 through Note that the inputs are years since For the following, make sure to use four decimal places. a. Find the change in the number of prisoners between 1970 and 1990, showing all work and giving the correct units. b. Find the percentage change in the number of prisoners between 1970 and 1990, showing all work and giving the correct units. c. Find the average rate of change in the number of prisoners between 1970 and 1990, showing all work and giving the correct units. d. Carefully draw a secant line on the graph above to represent the average rate of change in the number of prisoners between 1970 and 1990.

4 3. (12 points) The body-mass index (BMI) for women weighing 120 pounds and ranging in height from 50 through 76 inches can be modeled by b(x) = 83700, where x is a woman s height in inches. We want to numerically estimate the slope of the tangent line at x=75. Please note that b(75)=14.88, so that the point of tangency is (75, 14.88). a. Plug b(x) into your calculator for Y1. b. What is the formula for the slope of the secant between a point (x,y) on the BMI curve and the point of tangency (75,14.88)? Write it here and then plug this into your calculator for Y2. x 2 c. Use the slope equation (i.e. Y2) to numerically estimate the slope of the tangent line at x=75, by filling in the table below. Note that I am not asking for the value of the function at each input point just the slope of the secant line (i.e., you are not reporting Y1 just Y2). Recall that you must go to Tblset (at the top of your calculator) and change the independent variable from auto to ask! First, from the left: Input of close point on left Secant line slope (use at least 8 decimal places) What is the trend of the slopes (to three decimal places) i.e., what are the slopes on the left of the point of tangency trending towards? Answer: Next, from the right: Input of close point on right Secant line slope (use at least 8 decimal places) What is the trend of the slopes (to three decimal places) i.e., what are the slopes on the right of the point of tangency trending towards? Answer: d. What is the slope of the tangent line at x=75?

5 4. (29 points) The following indicates the number of grocery bags used at a certain chain of stores. This function is denoted B(x) where x is the time since 10:00 am and B(x) is the number of bags used at time x in hundreds. a. What should be the units on the slope? b. What is B(5)? c. Carefully draw a tangent line at x=5. Make sure that your tangent line is slanted correctly and is not too short. d. Carefully find the slope of the tangent line at x=5. To do this, I want you to tell me which two points you are using below (and label these points on the tangent line). Make sure that I can tell that you understand where to place these points correctly. e. Write your findings from part d using derivative notation. You may use whichever derivative notation you would like. f. Interpret your findings from parts b and d in a sentence in English (including the time that these describe and if this indicates either an increase or decrease). g. Find all x values for which db dx = 0. Write them here.

6 5. (20 points) Short Answer: a. Suppose that F(t) represents the average size of a U.S. farm (in acres) at time t (in years). You are told that: F(1960)=300 F (1960) = 8.21 Using these two pieces of information, estimate the average size of a U.S. farm in 1963, showing all work (and giving an explanation as needed). b. Using the farm information from part a above, find the percentage rate of change in 1960, showing all work and giving the correct units. Use 4 decimal places. c. Suppose that T(f) is boiling point of water in degrees Celsius at altitude f feet. Give two notations for the derivative of T(f) What are the units on the derivative of T(f)? d. Find all x values at which the derivative of the graph below not exist. Write them here.