1. (a) (4 points) Four students see this function: f(t) = 7 4t. Which student has written the derivative correctly? Circle the student s name.

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1 Math Spring Common Exam 1 Name: Part 1: Short Answer The first five (5) pages are short answer. You don t need to show work. Partial credit will be rare. When appropriate answers must include correct units. 1. (a) (4 points) Four students see this function: f(t) = 7 4t. Which student has written the derivative correctly? Circle the student s name. Alice writes df dt = 1 (7 4t) 1/ Bob writes df dt = 1 (7 4t) 1/ ( 4) Chris writes df dt = Dani writes df dt 1 (7 4t) 1/ 4 = (7 4t)/ ( 4) (b) (4 points) Four students see this derivative: dg dt = 7 4t. Which student has written a correct antiderivative? Circle the student s name. Alice writes g(t) = (7 4t)/ Bob writes g(t) = 1 (7 4t) 1/ ( 4) Chris writes g(t) = Dani writes g(t) = 4(7 4t)/ (7 4t)/ ( 4). (8 points) Suppose that a problem states the function T (t), which concerns an object s temperature T at different times t. A student working on this problem then evaluates the derivative of T at 7 minutes. Which of the following are correct notational or English descriptions of what the student computed? Circle all that apply. (a) T on [0, 7] (b) T (7) (c) T (7) 1

2 (d) T on [6.8, 7.] t (e) The rate of temperature change when t = 7. (f) dt dt t=7 (g) The slope of a secant line on [6.5, 7.5]. (h) The slope of the tangent line to the graph of T (t) when t = 7.. (10 points) An object is thrown straight up. Its height is given by with y in feet and t in seconds. y(t) = 50 0t (a) Write a formula for the average velocity on the interval [, x]. y (b) Compute lim x t. 4. (8 points) The graph represents the position, s, of a moving particle as a function of time, t. The particle has velocity v 1 at time t = 1, velocity v at time t = and velocity v at time t =. Arrange the values of 0, v 1, v, v from the least to greatest in the blanks below. < < <

3 5. (10 points) The graph of the electric potential in a circuit, V measured in volts (V), as a function of time, t measured in milliseconds (ms), is shown below. Sketch and label the graph of dv dt on the axes below. Be sure to show an appropriate scale.

4 Part : Full Work Show all work. Unsupported answers will not receive full credit. Present your work clearly with correct notation. Neatness counts. If you use your calculator beyond basic arithmetic you must State what calculator tool/function you used. Write what you entered into the calculator in calculator notation. State what information you got from the calculator. If you used a graph, recreate the graph with proper labeling. When appropriate answers must include correct units. Most problems must be answered with a final sentence that states or interprets the answer in the context of the problem. Read the specific instructions for each problem. 6. (1 points) This table provides the size of a growing rabbit population (P ) at different times (t). t (years) P (rabbits) (a) Find P on [6, 8]. Write your answer in a sentence to indicate its meaning in the context of this problem. (b) Estimate dp. Show work to indicate how you arrived at your estimate. Then, write your dt t=6 answer in a sentence to indicate its meaning in the context of this problem. 7. (10 points) The height of a skydiver is given by the function h(t) = t 5e 0.t where h is in meters and t is in seconds. At what rate is the skydiver s height changing when t = 10 seconds? 8. (1 points) The pressure in a cylinder is given by P (t) = 500 t + 1 with P in kilopascals (kpa) and t 0 in minutes. When is the pressure in the cylinder changing at 50 kpa/min? 4

5 9. (1 points) The potential in a circuit is given by V (t) = 1 1e 0.16t where t is time in seconds and V is the potential in volts. At the instant when the potential is 6 volts, how fast is the potential changing? Write your answer in a sentence to indicate its meaning in the context of this problem. 10. (10 points) The height of a ball thrown into the air is given by h(t) = 16t + v 0 t + 64 where h is in feet, t is in seconds, and v 0 is an unknown constant. If the ball reaches its maximum height at 1.5 seconds, when will it hit the ground? Write your answer in a sentence to indicate its meaning in the context of this problem. 5