Solutions to Exam 2, version 2, Ma 123,F05

Calculus (With Analytic Geometry)(8th edition)

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Unformatted text preview: ’- 5 Math 123 Calculusl Name: m?» Fall 2005 RWJ No calculators allowed, one page of notes allowed Instructions: 0 Read questions carefully Help me award you (partial) credit by showing your work Note point values Check your answers if you finish early The exam ends promptly at 9:52 Good luck! Some Possibly Usefitl Formulas: ". n(n+l) ". ( +1)(2 +1) ". ( +1) 2 El: 2 i§12=nn 6 n §l3=[nn ] ’aifiéf$« P ‘Q l RESEAW /33/1’/"”"_ 2.!) . x')’ 1.». . , O s “Eroserflghlll” VL 1. (49 pts.) If f(x) =x4 —x3 —3x2 +5x—2 then it can be shown that @=4x3—3x2—6x+5 (x—1)2(4x+5) and f"( ) =12x2 —6x—6€6(x-1)(2x+1p g / a. Where is f decreasing, if anywhere? 4— a (“M/‘5/‘f/ X jg: ( b. Where is f increasing, if anywhere? (*5/4/1) M (1,“) 0. Where is f concave down, if anywhere? kF—H [" 1/1, l 3 2- d. Where is f concave up, if anywhere? («pa/r) M (We) continued on next page e. State any inflection points of f (give just the x coordinate value(s), if any). Wm: M): (-i, 40(4)) tr ( l/ LKI)) (f f. Where, if anywhere, does f take on relative maximum values (give just the x coordinate value(s), if any)? [Um g. Where, if anywhere, does f take on relative minimum values (give just the x coordinate value(s), if any)? cl’firwtkrvfi 2005 2. (11 pts.) Evaluate 2(71'2 — 31' + 5) using the results on the cover sheet of this exam. i=1 =7 ZZ“—3 21* 2:; Z?‘ ch (:¢’ ’ 7 (WWKWé’XW'Q -3 @5)(””‘) + was cs“) é ’L 3. (20 pts.) Find the following: a. 17x3 +53:2 +3x—4 dx c. [(7sec2x—2cosx)dx : 7 S $¢QF -2 S 5a,.wa 4. (10 pts.) Find :17: by implicit differentiation if (x+ y)2 = x3 cosy Also, evaluate the derivative at the point (1, O). A L z A 2:fo«+*9)) J?‘ (23600] '2-(KH7>’(I* 6&1): ngmg 4— 0:36-94:79)in Arc Ax £12 ( "LOG-#1) + «59417) -,- 3’0)sz ~20”) II (LC/*0) f’ [9:1‘0 5. (10 pts.) A painting is hung flat against an art museum wall, with its bottom and top edges at distances 7 ft and 12 ft, respectively, above the floor (so that the painting is 5 ft tall). The painting is viewed by a tourist whose eye level is 6 ft from the floor. The viewing angle of the painting by the tourist depends upon the/distance x she is from the wall. In particular, it can be shown1 that the viewing angle is biggest (making the painting appear as large as possible) when the quantity 5x x2+6’ x20 f(x)= is maximized. continued on the next page 1 Nahin, Paul (2004), When Least is Best, Princeton University Press, Princeton, NJ, pp. 73-77, 147-148. V7. Determine the distance, in feet, the tourist should stand from the wall to maximize her viewing angle of the painting. Show your work! {(WLA-é) -' ;?<[2-)c) @774)" : “§x"+3° T €(é._7(") “D Q H— ()1 (KL1"< )L C" x2JZ MXIW‘M @ QC¢Q (é)! acadtwfiue 1; ad’ék/ll)’ fin WXIMV": [malt Hp)=o avg zth—ao q; raw) WW imam Mun” 13 x42 L4: ...
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