Math 170 - Fall 2015 - Common Exam 3
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. (8 points) The lines labeled A and B in the graph ds at right give the velocity, , for each of two cars dt A and B, in miles per hour (mph) on the 0 ≤ t ≤ 4 hour time interval. Circle the correct response to each item below.
miles in 0 ≤ t ≤ 4 hours.
(a) Car A travels 0
20
40
60
80
100
miles less than car A in 0 ≤ t ≤ 2 hours.
(b) Car B travels 0
20
40
60
80
100
(c) The distance that car A travels in 0 ≤ t ≤ 2 hours is equal to the distance that car B travels in hours. 0
1
2
3
4
5
miles farther than car B in 1 ≤ t ≤ 3 hours.
(d) Car A travels 0
20
40
1
60
80
100
2. (12 points) A function, g, is defined on the domain [a, ∞). This function has the following sign chart.
(a) Complete the sign chart above. (b) What type of point occurs at the location x = b? i. ii. iii. iv.
Local Maximum Local Minimum Inflection Point None of the above
(c) Select all locations where g 00 (x) = 0. i. ii. iii. iv.
x=a x=b x=c x=d
3. (8 points) The graph at the right shows an object’s rateof temperature change dT , in ◦ C/min at different times (t, in dt minutes).
Various statements related to the graph appear below. Circle all that are true. (a) The object is cooling on the (0, a) and (b, d) intervals. (b) The object is cooling on the (c, d) interval. (c) A local temperature maximum occurs at t = b. 2
(d) A local temperature maximum occurs at both t = a and t = c. (e) A local temperature maximum occurs at t = c. (f) The temperature function, T (t), is concave down on the interval (a, c). Z a dT 1 dt = ak (g) 2 0 dt d2 T k (h) On the interval [0, a], 2 = − . dt a 4. (8 points) Four students write guesses of the function y(t) based on the derivative dy = te2t dt Circle the name of each student whose guess provides the correct
dy . dt
Notes: (a) It is possible for more than one student to write a correct guess. (b) Shown work is not required, but incorrect responses without shown work will not receive partial credit. 1 2 2t Alice: Guesses y = 2 t e 2 1 2t 1 2t Bob: Guesses y = te − e + 10 2 4 1 2 1 2t e Chris: Guesses y = t 2 2 1 1 Dani: Guesses y = e2t t − 2 2 5. (8 points) Suppose the function, g, is defined as Z 2
with g measured in feet and t measured in seconds. (a) Find g 0 (2). (b) Find g 0 (4). (c) Find g(2). (d) Find g(4).
3
x
2tdt
g(x) =
Part 2: 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. (12 points) The table below gives the rate of change for V , the potential in a circuit, measured in volts. Time, t, is measured in minutes. t (min)
0
2
4
6
8
dV (volts/min) dt
1
4.5
7
8.5
9
(a) Estimate ∆V on the [0, 8] interval, using n = 4 subintervals. Make sure that your work clearly shows what you are doing. Write a sentence that states your answer in the context of the problem. (b) Graph the velocity data (use proper labeling) and shade an area that corresponds to your estimate in part (a). 7. (10 points) For t > 0.5 minutes, the pressure in a cylinder is changing at a rate of dP 20 =5− dt t
kPa/min
Find the change in pressure on the time interval [2, 3]. Write a sentence that states the meaning of your answer in the context of the problem. 8. (12 points) A rabbit population has 10,000 rabbits at t = 0 months. Because of a disease outbreak, the population size P changes at the rate dP = −430e−0.2t dt
rabbits per month
How many rabbits are present after six months? Write a sentence that states your answer in the context of the problem. 4
9. (12 points) A box has four sides and a square base. The material used for the four sides costs $0.05 per square inch and the material used for the square base costs $0.20 per square inch. The box is required to hold a volume of 128 cubic inches. Find the dimensions that minimize the cost of the box. For full credit you must: • Label the objective function and the constraint equation. • Use calculus and algebra to solve the problem and clearly show all steps leading to the answer. • Include evidence that confirms you have located a minimum – not some other critical point. • Write a final sentence that states your answer to the question in the context of the problem. 10. (10 points) An object bobs up and down on a spring. At time t = 0 seconds, the height is 100 centimeters(cm). The object’s velocity is dh = 20 cos(kt) dt
cm/s
(a) Find k so that the first time the object returns to its initial height of 100 cm is at t = 5 seconds. (b) Find the maximum height of the object.
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