MAC2311_04fa_3 - MAC 231 l Test 3 FALL 2004 A. Sign your...

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Unformatted text preview: MAC 231 l Test 3 FALL 2004 A. Sign your scantron sheet on the back at the bottom in ink. B. In pencil, write and encode in the spaces indicated: 1) Name (last name, first initial, middle initial) 2) UP ID Number 3) Discussion Section Number C. Under “special codes", code in the test ID number 3,2. l 2 o 4 5 6 7 8 9 O l - 3 4 5 6 7 8 9 0 D. At the t0p right of your answer sheet, for “Test Form C ode”, encode B. A o C D E E. This test consists of l l five-point multiple choice questions, five one-point bonus questions and two pages (both sides) of partial credit worth 25 points. The time allowed is 90 minutes. F. When you are finished: 1) Before turning in your test, check for transcribing errors. Any mistakes you leave in are there to stay. 2) You must turn in the scantron sheet and tear off sheets to your discussion leader. 3) Answers will be posted later this evening on the class website. r4 2 . 1‘ Find the value(s) ofx at which flx) :- e- "'2x has relative extremal relative minimum relative maximum atx=___ atx=__ a. 0 il b 0 none 0 none 0 d i1 0 2. The deflection of a beam of length 4 feet is given by the function D(x) : x4 — 4x3 + 4x2 for 0 S x S 4 where x is the distance from one end of the beam. Find the value of x which yields the maximum deflection. Hint: When is I) a maximum on [0.4]? a.x=:0 b.x:2 c.x=l d.x=4 e.x:3 3. The graph offlx) : 4x5 + 5x4 + 6 is both decreasing and concave up on __ ___. a. (—3/4, 0) only b. (~00, —3/4) U (0. 00) c. (—1,—3/4) only d. (—l,0) only e. (—60,—1) only 1B 4. Given the graph of f ’(x), which one of the following statements would be true for the graph affix)? a, fix) is concave down on (0,4) U (4.7). b. flx) has local maxima atx = 2 andx = 6. c. f(x) is increasing on (0.2) U (4, 6) U (8,00). d. _/(x) has local minima at x : 4 and x : 7. 0. fix) has inflection points atx : 2> x : 6. andx :2 7. 5, If _/(x) 2 (x2 — 4103/5, find/(x) and determine which of the following statements is/are true. P: The graph of/(x) has one horizontal tangent line. Qzflx) has exactly three critical numbers, R: .1: -— 0 is a vertical asymptote of the graph of fix). S: The graph offlx) has two vertical tangent lines. a. P and R only b. P and Q only c. P and S only (1. P, Q, and S only e. Q, R, and S only 6. Find the linearization of/(x) _— 1n(x2 ) nearx : e and use it to approximate the value of in 9. a. ln9e-4+2e b.1n9zg——l c.1n9~+ d. 1119226 0. ln9z8w2e 2B dy 7. Use logarithmic differentiation to find at x ~—— 2 if y 2 xWx). a.—2 b. 2~2h12 c. l—-lr12 d. 4 —41n2 e. 2 8. Find the value of C guaranteed by the Mean. Value Theorem for/(x) = (x— n3 on [0,3]. ' a. 0:2 b. 623/2 0. 6:0 (1. 021/2 e. Cal 9. The radius of a sphere is measured as 5 inches and this measurement is used to calculate the volume of the sphere. Use differentials to approximate the percentage error that could be made in the calculation if the radius measurement has an error ofi0.01inch. a. :15 % b. i075 % c. $0.6 % d. 14.2% e. i0.5% 10. Use L’Hospital’s rule, if possible, to find the following limit. 3&3 w (3 + x) lim x-~>0 352 a. 1/6 b. 0 c. 3/2 d. 2/3 c. 1/2 3B 2 . . . 11. The graph of fix) : L + cosx has an inflection pomt on 4 [0,270 atx =- a giandfl- b land—S—E— CZEand—H—II” ' 3 3 ' 3 3 I 6 6 d. % and ~56l e. f has no inflection points Bonus Questions W!!! (1 point each). Be sure to bubble (a) for True and (b) for False. 12. lfflx) has a critical number at x z 2. then/(2) must be a local or relative extreme value of the function. a. True b. False d 2 2x+2 l3. — log (x +2x) =— #——-——-w—-—. dx( 2 ) (x2+2x)in2 a. True b. False 14. The graph of/(x) : has an inflection point at x z 0. a. True b. False 15 . If g ’(x) h ’(x). then g(x) and 11(x) must be the same function. a. True b. False 16. Ifflx) is continuous on [—3, I], then/(x) must have a minimum and a maximum value on [T3, 1]. a. True I). False 4B MAC 23] i '1‘EST3B PART II FALL 2004 N A M E_ ____ _ _ SECTK)N__ MGNATURE_mHum_ UHD___ sum W an WQRK 191992355 FULL “FEE I. An inverted conc- shaped container is being filled with liquid at a rate ot‘37r cubic inches per second, If‘the height oi‘the cone is twice the radius. how fast is the height of the liquid rising when it is 6 inches high? Him: I7 — «é—m-Z/z. fl_ i n/sec u’r ' "—"'—'— 513 2. Consider the fimctionflx) : x3 3(x + 5) and its derivatives: z 'fiu(x) A Determine the following: (ifnone. write "nonc‘d x intercept(s): x —- y intereept(s)t y 1— vertical asymptote: x 2 horizontal asymptote: y — critical number('s): .r — __ _ horizontal tangent line(s) at x -—- _ vertical tangent linc(s) or cusp(s) at X = _ I increasing on decreasing on relative (local) maximum at x 7 relative (local) minimum at x — _ concave up on concave down on _ in ll ection point(s) at x r ()B NAMEW __ SECTION 3. Sketch the graph offlx) from problem #2. Use all the information you obtained in problem #2‘ Label all interceptsl local extrema and inflection points. Hint: I/(AZ) -= 4.76. 7B 4. A shipping crate with a square base is to be constructed so that it has a volume of 250 cubic meters. The material for the top and bottom costs $2 per square meter and the material for the sides costs $1 per square meter What dimensions will minimize the cost Ollcontruction‘? Function to be minimized; _ Constraint: V 2. "1 Use the Second Derivative Test to verify yoiir result-st 8B ...
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MAC2311_04fa_3 - MAC 231 l Test 3 FALL 2004 A. Sign your...

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