ex2sol-b - MTH 133 m Section 1—4-—— Test Two...

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Unformatted text preview: MTH 133 m Section 1—4 -—— Test Two DEPARTMENT OF MATHEMATICS Michigan State University October 13, 2005 NAME: <3 “i” :53?“ ID NUMBER: SIGNATURE: SECTION: 1. This exam has 5 pages including this cover. There are 6 questions. 2. Include units in your answers whenever appropriate. 3. You may NOT use your calculator. 4. Sines, cosines, and tangents of simple angles are to be written out. For example, if you have a cos(%), you should write 5. Present your solutions to each problem in a clear and orderly fashion. Include enough steps, including diagrams, to permit the grader to follow your method. Your method must justify your answer. PROBLEM POINTS SCORE 1 20 2 20 3 40 4 5 5 10 6 5 TOTAL 100 Present your solutions clearly. 2 1. (20 points) Find the following limits. Mark each instance in which L’Hépital’s rule is used. (a) hm arctansc {if 3 1—4) CC \V I! ‘ § ; . Mr Lag}? by?“ X4?a ; 1:, 5 M Q - 1/<1—x) f g 3" 03) $13?er g 5 > “316;? a g "m if}: X/m (j 1% .XUi; g’? ’X v, far-x *2 r “3:; o. , :3 ‘ $0 59'! «if/m >< GK” ’2? {fiv‘sfé’ {M} g 53 Cofiyénhusgflag : (I’M i 12“ >4 Y 4} g ~§ if” X J : ff” «jg/V1 5;: 3 ga gr? " >5 O x Lfi’i ,L w I 3% X w” { (W Present your solutions clearly. 2. (20 points) Differentiate, but DO NOT SIMPLIFY each of the following functions. (a) f (a?) = 35m , 33%»; 5 4:;{75‘} 7" g " {Cg fix *‘ lagio‘kj Xmekfii/M? { if Ffi “L + ‘0 (is ” g x («a {E g k is a? :. :1 {EMF X5,” ?“ w i“ 3"”: / 3. (40 points) Find each of the following indefinite integrals. (20/ 1 d3: . V1—2w2 : i 32:. g: ‘{ z/‘jz‘w’? 31;? X LA ‘22 Ea)" E ~ V ’ xi 3"” 2 a?“ :3 y; :: .3; i M kg“ 6 ii; a; lwifl “Z; s “f, M?! < (3: mi {3; m {/i M“ i y l, ’W “W” (jay/93m ‘53,”?4 *(K Present your solutions clearly. $2}? 5a *7: {ff} (b) /tanha:d:c 5; {{E‘Ss’o >< film i” 4‘ C 1 (d) /x(l+(lnm)2)d$ Leg: i fig: w ‘§ H m2 "3“ {31’1" 44% m4:- C :2. fife/Egg? C - 4. (5 points) gig/39.3» x Which one of the following two functions has a faster rate of growth? Justify your answer. (In 93)2 330‘1 /Cvki ; 4 wéx‘? WM ‘24 L g»? A?“ (:3; g X p «x990 ‘ {‘1 1:: A 'M ’ gag} Xxx/27$}; "$395 lfigflyaa :2 H I ;’ X'a‘gg / Z L (w , ~, m LA“ : g???“ 7({3 Mia» ‘ 2. “a? 7 a! égix ;/Q/O§>< $43 2. Present your solutions clearly. 5 ' 5. (10 points) Let f(.’13) = sinm, so that f‘1(:c) :— arcsincc. Note that f(»%) = -%. Find the derivative of f “1(93) at a: = —% using two different methods: (a) directly take the derivative of f '1 and evaluate; .., i (i <4“? {0; WM)! 1 3 L ’3 So (9;); Z I 35% i“ a (b) use the inverse function theorem. ~---- 5‘ ’ “l f N g {A \x f ’ targeted M"; 9 w we read +9 '9’“ i ( val e) t W _; ‘ m; (7c5 :— a: o S y: t ’c T" ~» + 235' «w - ca: . a i .«l 4/ a L > ; 'M" M as: {i1 l { a; e we 6. (5 points) a?" The graphs of f and g(a:) and their tangent lines at (4,0) are provided below. 9 f (93) Find lim —. 36-4 9(93) MI“? Wig “33 "€16 3%?" {we}; Cs :(opé: m MQ‘ ...
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