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Test1 - MATH 31L TEST#1 Little Spring 2009 Name 1(8 points...

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Unformatted text preview: MATH 31L TEST #1 Little, Spring 2009 Name: 1. (8 points) In this problem you are to construct a mathematical model of a table of data. The table is shown below, but only two data points are filled in. The following are descriptions of some plots made from the complete table of data: ° If we plot the points (Ly), we see a rising curve, which is concave up. ' If we plot the points (t, 1n 3;), we see a gradually rising curve, which is concave down. - If we plot the points (1n t, In y), we see a line with positive slope. State what type of function woiild best model the relationship between 3; and t, and state how you know. Construct an explicit function that expresses y in terms of t. 2. (8 points) Suppose that 20% of a certain radioactive substance decays every 5 hours. . Assume that the initial amount of the substance is Q0. (a) Find a function Q(t), which gives the amount of the substance present after t hours. (b) What is the half-life of the substance? (Give a numerical answer, rounded to one decimal Page 2 3. (7 points) Let 5(t) be a function which gives the position of a bug at time t. The position is expressed 111 centimeters and the time in seconds. Explain why we needed to introduce the cencept of a limit to be able to compute the velocity of the bug at a given time, say at time t = 4. At some point in your explanation, Show the mathematical definition of the instantaneous velocity of the bug at time t = 4. 4. (5 points) Compute the following" limit by hand [i.'e., no calculators] ' 5. (6 points) Let f(:1:)— -* mm .Make a numerical approximation of f’ (2 ), accurate to 3 decimal places (no rounding). Be sure you make it clear what you are doing, and State briefly why you think you have 3 decimal place accuracy. {You may NOT use differentiation theorems which we have not covered this semester] l ,, _ 7 Page 3 6. (10 points) Let f (x) x i + 53:. Use the definition of the derivative to show that f’ (513- — —1 + 5. There will be no credit for giving the answer without showing the derivation from the definition. Be sure to use proper mathematical notation. it's a major part of this question. Your work must also be orderly and clear. 7. (6 points) The value of a certain automobile purchased 111 2001 can be apprOximated by the function V(t)— — 25( 84), where V 1s given in thousands of dollars and t is the number of years elapsed since the year 20.01. ' (a) Find the function V’(t), and then use it to compute V’ (4) (b) Give the meaning of the number you computed 1n step (b). Express your answer in terms ‘ that a person without knowledge of calculus could understand (that 1s,don't use words like “rate” or L‘instantaneous” or “denvat1ve”) 8. (8 points) The graph at the right is the graph of f ’ (t). Among the graphs below are the graphs of f (t) and f ” (t) [The horizontalaxes are scaled the same on all of the graphs] Indicate which graph is f(t) and which is f_”(t). f'Ct). 9. (6 points) The diagram on the right shows the graph of y = f (t) Suppose that instead of having the graph, we knew the starting point, (to, yo) = (2, 1), and the value of % at any point. Show on the diagram the points (t1, yl) and (t2, yg) that Euler's method would produce. Use At 2 2. Leave some evidence as to why you chose the points that you indicate. ' ' . Page 5 10. (12 points) Suppose you put a potato in a hot oven, maintained at a constant temperature of 200 C. As the potato picks up heat from the oven, its temperature rises. Let H (t) denote the temperature of the potato t minutes after it was put in the oven. Suppose'that H(30) 2 120 and H’ (30) : 2. (a) Why will the graph of H over time be concave doWn? [Pick one answer] Because the temperature must be rising over the entire time interval. Because the temperature will rise less and less per minute as the temperature approaches 200 C. Because rate at which the temperature is increasing is itself 1ncreasing as the temperature approaches 200 C. None of these. (b) What units are attached to H’(30)? (c) Find L(t), the local linearization of H based at t = 30. (d) Use your local linearization to approximate H (34). Is this approximation an overestimate or an underestimate? Explain how you know. 11. (8 points) We have a car, which is moving on a straight road. Its position, infeet from some reference point, is given by the function 3(t), where t is the number of seconds since it started moving, and s has the usual orientation (i.e., right is the positive direction). (a) If the car were moving toward the right and braking at the same time, what would the curve look like during this event? Answer with a sketch drawn at the right. (b) Suppose now that 5(t)— 1153— 41:2 + 12: + 5, s’(t) = t2 — 8t + 12, and 5"(t) = 2t — 8. At what time(s) does the car stop? Answer: During what time interval(s) IS it moving toward the left and accelerating toward the right? You must show your work. ' Answer: Page 6 12. (8 points) For each part below sketch a curve with the indicated properties. If it‘s impossible to ' do, then write “Impossible. ” (a) f” < O for all :1: 7é 0, and f is continuous but not differentiable at x=0. (h) f is differentiable at a point where it is not continuous. (c) f”(:c) = 0 for all 2:, and'f’(:r) < O for all 3:. . (d) f’(1) does not exist, f has an inflection point at cc 2 1, and f has no cusps. l3. (8 points) Without using the Product Rule, show that the proposed differentiation “rule”, (f(:v)g(w))— “ f’(w )9 (m) is NOT true. ...
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