MATH
Math 1A - Fall 1993 - Ribet - Midterm1, 2, and Final

# Math 1A - Fall 1993 - Ribet - Midterm1, 2, and Final - THU...

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Unformatted text preview: 09/28/2000 THU 11:15 FAX 6434330 MOFFITT LIBRARY 001 Math 1A Professor K. Ribet Fall, 1993 First Midterm Examiseptember 23, 1993 80 minutes as a limit. Do not evaluate (1 1a {5 points). Express the derivative — {xsin(i)} x223 da: the limit. 1b (5 points). Use the deﬁnition of the derivative to calculate f’(1) When = The formula 03 — a3 = (b — (1)032 + at —|— 612) may be helpful. 2. In the problems on this page, you may use the differentiation formulas we have derived in class: 33.2 a (5 points Find the equation of the line tangent to the curve y = 3 Jr 1 at the point a: 1 l — . ( 7 b (5 points A sugar cube tumbles from a 98—meter tall campanile. How fast is it falling after 73 seconds? [If you use the formula 30\$) 2 4.9152, explain in a sentence or two What it 11163118.] c (2 points HOW fast is the cube falling when it hits ground? (t " 1X75 - 5) (t — 3)(t — 5) horizontal and vertical asymptotes of the curve u : 9(15). Make a crude sketch of this curve, showing the asymptotes. 4a (5 points). Find lim - | — ﬂ) , Where [ ] is the “greatest integer” function. tw~>5— — 4b (5 points). Find £131 (x/tz — t + — x/tz + t + 1). 3 (8 points). What is the domain of the function g(t) = ? Find the 5 (5 points each). 1,691 _ 2591 a. Evaluate —b—T by using rules of differentiation, first expressing the limit as _* _ a derivative. '3. Suppose that f(7T/2) : 12, f’(ﬁ'/2) = 3. Using the methods discussed in class, _ cos(9) calculate 61:11.32 09/28/2000 THU 11:16 FAX 6434330 MOFFITT LIBRARY 002 6 (5 points each}. 1 a. Find if = —Zsin(:c). .7: . _ sin(:c) _ _ ID. Find 11m . Explain your reasoning. 17—)00 :02 Second Midterm—October 28, 1993 80 minutes 1a ( 5 points A sample of chalk contains 0.3 grams of radioactive dwinellium, which has a halfslife of 18 years. How many years must expire before the sample contains only 0.01 grams of radioactive dwinellium? 1b {4 points). Find all possible values of cosh 3:, given that sinha: : 5/12. 2a ( 5 points Use diﬁerentials to ﬁnd an approximate value for sin 1°. 21) { 5 points Let g be the function inverse to f. Calculate g’(2) from the table 3. Find the derivative % (each part is worth four points): a. y : arcsinh/E) 3 b.y=e\$+1 C. y m logﬁcos r) (0 < a: < 77/2). 4 (8 points A slug and an ant left the base of the Campanile at midnight last night. The ant began moving directly north, toward Evans Hall. The slug moved east, toward a slugfest in Birge Hall. At 8AM this morning, the ant had traveled 60 feet and was moving north at 10 feet / hour. The Slug was 80 feet east of the Campanile, but had started moving west at the rate of 5 feet/hour. At what rate was the distance between the slug and the ant changing at 8AM? 5 Evaluate the limits (four points each): a 1' w o l — 9333+ .222 + 125 12 n b. lim (1+3—) 71—}00 n 09/28/2000 THU 11:16 FAX 6434330 MOFFITT LIBRARY 003 C 1. l_ l .t—Etll- t et—l . . dy . _ y 6251 pomts). Find a; 1fy —— a: . 6b (5’ points). Find the derivative y', given y2 + 6333; + 332 : 7. 6c (2 points). Find a formula for y” in terms of :r, y and y' if y2 + 6mg + 2:2 = 7. Final Exaln—Deceniber 15, 1993 1a (5 points). Differentiate with respect to 3:: V a: + 1b (5 points). Find L(—4) given that L(71) = 1 and that L’(w) = 32 2a (6' points). Find lim f(x)g(\$), Where = (0.4)(“7—2) and g(.r) = 3:132. :r—H'] 1 2b (5 points). Evaluate 3 {5 ‘ t) E 1 t f d”: a p03” 3 0 Va 113.. e + C1:)2]. #2 coszr ctr, simplifying your answer as much as possible. 3]) (6 points). Evaluate f _ fir/3 811153 4 (9 points). A rain gutter is to be constructed from a metal sheet of Width 30cm by bending up one—third of the sheet on each side through an angle 9. How should 6 be chosen so that the gutter Will carry the maximum amount of water? (A crude handvdrawn diagram was supplied.) 12 5a {6 points). Evaluate lim E 2 cos (n + 3k) . 11—)00 n 71 16:1 5b (5 points). Let c be a real number. Show that the equation 3:5 — 6:1: + c = 0 has at most one root in the interval [—1, 1]. 6a (6' points). Find all numbers a such that the line tangent to y = :32 —|— 1 at the point (a, a2 + 1) passes through (0, —8). 6b {5 points). Find the derivative of (sinmfanx with respect to 1'. Suppose that 7?, is the region lying between the graphs of y = 393 and y = 2753 in the part of the plane where :t: and y are positive. 73. (5 points Find a deﬁnite integral which represents the area of R. 7b (6 points Find a deﬁnite integral Which represents the volume of the ﬁgure generated by revolving 7?, about the y~axis. 09/28/2000 THU 11:16 FAX 6434330 MOFFITT LIBRARY 004 d COS Z‘ 83 (6 points). Find % dt. 8b (4 points Find the average value of sinx on the interval [0, 7r]. . h E h H . 1 Eﬁh 9a (5 points). Evaluate Hm h——>0 h i9 dx 10a (2 points). Bob and Jill lift a 90-pound Stairmaster over a distance of 30 feet. HOW much work do they perform? 9b (5 points). Find at the point (3, 1) on the curve 2(322 + y2)2 = 25(cc2 — yz). 10b (4 points). At 7PM, a. large pizza is taken from at 415°F oven to a 65"F dining room. At 7:08PM, the pizza. has cooled to 135°F. What is the temperature of the piece which remains at 7:16PM? (Assume the validity of Newton’s law of cooling— the pizza. cools at a, rate proportional to the difference of its temperature and that of the room.) ...
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