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mt02solfall04

# mt02solfall04 - MATH 19 A MIDTERM 2 Instructor Frank...

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Unformatted text preview: MATH 19 A MIDTERM 2 11/19/2004 Instructor: Frank Béiuerle, Ph.D. Your Name: Your TA: re - ___________>_O_Lu_)‘_<r£‘_j _______________ _______ 51 051149171________________ Max Your score Problem 1: 20 Problem 2: 10 Problem 3: 10 Problem 4: 10 Problem 5: 10 Problem 6: 20 Problem 7: 20 TOTAL: 100 Good Luck! 1. (20 points) Compute the derivatives 3 of the following functions: 031 (a) y = arctanh/E) I H660L ””72 CW) 7”? (b) y = 6‘2 1 3": e": - 71% (C) y =1n(x + sin 1:) + tan(ta.n:c) + 5““ 9:1": '_ »C!+wsx)+ Strc'lxdcwfﬁbclx +0 XfSMX «j: 8M X“ : e [,l ~ 5;,"wa ‘ Com —- (j (—ganX"{\/\74+ \$15) COW : >< ’ mm (m 9):) 7C 2. (10 points) Find the equation of the tangent line to the curve eﬂ = my at the point (e l) Ugﬁm W’” 3.: . a 27/3 ﬁrm 9 Whit-{‘4 \$13 1 “L g1! €___ : I+€le ' 2. ‘Q' r 7.. _e I“ .-— ,/:...’2~ 59—69:! —-‘> 29"” a M‘j”‘j Lmaﬁ ~79) We“ m:~% )¥,:C,‘/,:I *l t—E—Ub—e) 'ﬂ’ﬂ :1 61 f—v 3r—‘3w’5! :2) b]:‘~é-)<+2_+l J E j 3. (10 points) Show that tan 3"“ ~ :2: for a: VERY close to 0 S'“& hm“ H CIW‘WM ‘5‘ij ML A OJ "£1ka “MNMMF O Mini/1A LCX) if M W“%+L|¢k£ 1L?) YI-‘frwx ml ”(:0 4. (10 points) Compute the derivatives f’(:c), f”(:r:) and ﬁgmlm) of the function ﬁr) = sinh 2:5 ‘, f 1 ; wr- (ﬂ : 'Z cosh 2X 3: “ (ﬂ, : 4 sinh 2% (an194 r in (3&le 4 (X) :2 gnu/12y (:00 :J) 2013!; .{ClMHE-ﬂ : OZLLH - 3L1 2y SO ‘JCC 2 WWX. 5. (10 points) (a) State the Extreme Value Theorem. l+ .F ”,5 CK erdm mot H commands an [ab] (m 4% LL“) ﬂux at 3 LOW WU: Luau/L (MC-l 0L 97(20er mgmmum (M [Gib]. l (b) The function f (1:) = — does not have an absolute max nor an absolute min on [—131]. :1: Why does this not contradict the extreme value theorem? (PK CU”? —% lL Mi” Cowl/itulﬂdr 074 [-310 (C’USCO‘tilknmu‘ qli’ K30) (c) Suppose we know that f”(:c) < 0 for l S 3: S 3, f’(2) = 0, and f(2) = 3. Give a rough sketch of f (at) of the graph of f locally around 1.- = 2. mtg“ (inﬁll/L Edi {- mar 3:2. ”X {-0-} —o 1% ’ :? lLOLQwM "l-(bexul’LW-ﬁ R41 {“0040 :9 co quljﬁ 6. (20 points) After heavy rainfall it is observed that the depth of water in a conical reservoir of radius 10 meters and height 30 meters is increasing at 5 meters/ hour when the depth is 5 meters. How fast is the water ﬁlling the reservoir at this instant? Recall that the volume 1 . of a cone is given by V = Err-2h. Draw a picture and show all your work. Ores)“ yacht-4t Mum h(H:5 m '50 Eva Xiwlcmbj ﬁlm 1‘: 1 L9 »:) r -: E Wham-x M}: 5m 5 so *5 f U‘NWZCQB'CWU Wm M w‘ :99 _._ J. 2 m- M Mt) 3 3 VIM dWH ~ (IT thti 7. (20 points) Let f(::) = 3:55 — 53:3 + 1. Find the following: (a) Find the critical numbers of f. l its)“ 1533+“ “57‘3“ : O 15ft” 73-0 -‘- f5 Klf¥ﬂflfv<+ll :0 3&0) i) ”I (b) Classify the critical numbers of f (i.e. determine whether they correspond to a local min, local max or neither.) :l 10C ml M in (c) ive the In erval(s) where f is increasing. ((1) Give the interval(s) where f is decreasing. 'W—m‘ {”40 Gulf—1,0 l (e) Give the absolute max and absolute min of f on the interval [— 1 2] Can-Tami gc—l)i§(m1acr] Md £0) u u __‘ :qév‘w“ 3 1 z: '37 7‘ , ’1‘ ...
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mt02solfall04 - MATH 19 A MIDTERM 2 Instructor Frank...

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