2A_Smidterm2_sol - Math 2A Sample Midterm 2 This exam...

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Unformatted text preview: Math 2A : Sample Midterm # 2 This exam consists of4 questions worth 10 points each and 4 questions worth 15 points each. Read directions for each problem carefiilly. Please show all work needed to arrive at your solutions. Label all graphs. Clearly indicate your final answers. 1.) Mark each of the following statements as true 01~ false. No work needs to be shown. a.) If f I (c) = 0 then c is cal maximum or minimum off. True False [Thing 6;. 1;? :01“ C :03 b.) If fry has an absolutem mi - um value at c, then f (c) = 0 or f does not exist at c [e coulcl been 03%:th c.) If f I (x) = g, (x) for a in (a, b), then f (x) - g(x) 1n (a, b). d.) There exists a function such that f (1) —— 2, f(2)= 5 and f I (x) < 2 for all x. ‘ \ True @ gfiq‘i; 2L1} ,_ Eva :: 3' $02,) .4 2, ngSSSElQ . . . 1 e.) l1mx.,w x 5111; = O ' 7 1 5m“; - Que: 3.3.2.2. ‘5‘ 1 h —- ’ True M )6 5% x ,1 “Qu‘ag ch \g 2.) Sketch a possible graph of a function f that satisfies the following conditions: rm) = 0. f(2) = 4. f(5) = 7. f’m) =f'(5) = o f'(x)>0for0<x<5 , f’(x)<0 for x<0 and x>5 fll(x)>0forx<2 , f"(x)<0forx>27 3.) Find 3-11 by implicit differentiation for 2x3 + xzy + y3 = 2 . (You must solvefor 3-1:.) 2 ‘2 '2 wx x +Eé320 23+25+x§§ $0“ (5242+ 2X35 *' §Q+3® ii. 3Q 4.) a.) Fill in the blanks in the following statement of the Mean Value Theorem. MEAN VALUE THEOREM: Let f be a function that satisfies the following hypotheses: 1. f is (:0 “‘5 {\UO 1} 5 on the closed interval [a, b]. 2. f is diwg (KbXQJ on the open interval (a, b). Then there is a number c in (a, b) such that f’ (c) = WE ” $03“) b.) Find the number c that satisfies the conclusion of the Mean Value Theorem for the function f (x) = 39::2 + 2x + 5 on the interval [0,1]. fibE—vfik‘g .3 1m ~QCQ} 2: <3§=1+E§>~ S a 5 WW a @th gwi 5.) Find the derivative f, (x) for each of the following. (You need not simplijfv answers.) 3.) f(x) = sin(1 + x2) 2* @3st 25‘} b-) f (X) = (5'63 - X)12 \2 (Ks-4‘)“ * (3543 c.) f (x) = sin(cos§7€sinx)) C93 (Cos‘ifi‘mfl i: as S {ngB (£33 ((3:33 (gmxfi fim (31‘91) (:an -an. a... 6.) A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3/min . How fast is the height of the water increasing? \f- “will“: 9‘33““. 3““Bmm c3}, Ah 2:? 7.) Consider the function f (x) = 2 + 8x2 — x4. a.) Find the intervals on which f is increasing and intervals on which f is decreasing. {Awasihji 0‘?" d 3 X4 wfl. Clioewms ; .. 23; x49 ><>2 b.) Find the local maximum and minimum values. Lowest mmfimm \loipg$t £C‘133 2+3‘leéa V8 QC'Z‘): \% [mall miszmm mg»? 1 $833 *4 2 c.) Find the intervals on which f is concave up and intervals on which f is concave down. 2 O ) A a... :3... a... Dix): \Qw £2.98 :7. Ll ($3333 :2 oz...» “”3“”‘x t3" 2 *2 «a a (use; .— 3) v (’1 ‘WB Gamma $03.” is 3% .. 2 2 ' ma {5‘ ( T? 1%) mate {i (1.) Find the points of inflection. ,, a. .m‘tézzW M X,-%w§~f,2e8%? a q x343; a...) 32+3‘gwlgzm {5; Y 5 a! C? 8.) Find the linearization of the function f (x) = x/x + 99 at a = 1. Use this to approximate V 102 . ...
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