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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(x1)(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— 0:3694:79)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. 7377, 147148. V7. Determine the distance, in feet, the tourist should stand from the wall to maximize her
viewing angle of the painting. Show your work! {(WLAé) ' ;?<[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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 Summer '06
 JOHNSON
 Calculus

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