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1 Common Exam 1 Math 170, Fall, 2014 Name: Instructions For Part I. The first six (6) pages are short answer. You don t need to show work. Partial credit will be rare. 1. (10 pts.) Compute the derivatives. Assume that any letter other than t is a constant. (a) ) d (Pe rt = (b) d (h 0 + v 0 t + 12 ) at2 = 2. (8 pts.) The height of an object is a function of time, h(t), with h in meters and t in seconds. You will not be given a formula or any numerical data for h. However, a person who does know the formula for h has figured out a formula for average velocity on the interval [0.7,u]. It also works for [u, 0.7]. h = 4 sin(2u) 4 sin(1.4) u 0.7 (a) Complete the following table of average velocities. Each answer should be accurate to 2 decimal places and must include correct units. Interval Average velocity: h [0.69, 0.7] [0.7, 0.71] [0.7, 0.72] (b) Estimate the velocity of the object at the instant t = 0.7 seconds. Include correct units. 1
2 3. (8 pts.) For each quantity in the left column, sketch the corresponding secant or tangent line in the graph to the right. (a) h on [1, 3] (b) h (3) (c) Average velocity on the interval [0, 2] (d) dh t=2 2
3 4. (10 pts.) The figure at right shows the electrical potential, V, in a circuit as a function of time, t. V is measured in volts; t is in milliseconds. Answer the questions below. (a) At what instant(s) in time is dv = 0? Include correct units. (b) At what instant(s) in time is V (t) = 0? Include correct units. (c) Which is larger? Circle one: dv t=0.5 dv t=9.0 (d) Estimate the instant in time when voltage is changing at the fastest possible rate. Include correct units. (e) What are the units on dv? 5. (10 pts.) Given the same voltage function from Problem 4, sketch a graph of dv. Use the blank axes below or a full sheet of graph paper. Your graph must be properly labeled. 6. (8 pts.) Suppose that a word problem gives a formula, P(t), for the pressure in a cylinder. A student working on this problem finds a formula for the derivative of P. Then he substitutes 2 for t in his derivative formula. Which of the following are correct notational or English descriptions of what he computed? Circle all that apply. (a) The slope of the tangent line to P(t) at t = 2. (b) P(2) (c) The slope of the secant line on the interval [0, 2]. 3
4 (d) P (2) (e) The rate of change of P at t = 2 minutes. (f) P on [0, 2] (g) dp t=2 Instructions For Part II. Show all work. Unsupported answers will not receive credit. Present your work cleanly and clearly. Neatness counts. Any calculator use other than arithmetic must be described in your work. At the point in your solution where you used your calculator, indicate: What features or functions you used, What you entered into your calculator (exactly as it appeared on the screen), and What information you got from your calculator. All answers must include correct units. For each problem you must write a final sentence that states your answer and explains what it means in the context of the problem. 7. (12 pts.) The pressure in a cylinder is given by P(t) = t with P in kilopascals (kpa) and t in seconds. At the instant when P = 185 kpa, how fast is the pressure changing? 8. (12 pts.) The voltage in a circuit is given by V (t) = 12 12e kt where t is time in seconds, V is in volts, and k is an unknown constant. At the instant t = 0 the rate of change of voltage is 20 volts per second. When will the voltage reach 10 volts? 4
5 9. (12 pts.) A model rocket is launched straight up. During the first two seconds its height is h(t) = 100t 2 12t 3 ; 0 t 2 where h is in feet and t is in seconds. How fast is it going at the instant t = 2 seconds? 10. (10 pts.) In the previous problem the rocket is powered for 2 seconds, with height given by h(t) = 100t 2 12t 3 ; 0 t 2 At the instant t = 2 seconds the fuel runs out. After that, the height is given by h(t) = a + bt 16t 2 ; 2 t where a and b are constants. You don t know a or b, but you do have a graph of h at right. Find the maximum height. 5
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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