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Exam 3 Key - Calculus 1 Exam#3 Spring 2011 Name Show all of...

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Unformatted text preview: Calculus 1 Exam #3 ) Spring 2011 Name Show all of your work on this paper in the space provided. Solutions without correct supporting work will not earn credit. All answers must be exact unless stated otherwise in the problem. Only your best 10 attempts will count towards your grade. © 1. Find the absolute extreme values of f (x) 2 3+ In x2 on the interval [1,4]. §’(X3:-r_§l€+_:-2_ .. “tux 42m: 22 +o34a> labia X ' x2 —Ll+zx:0 ¥(I):4+014~1> NOS. 94:4 QM): Hanna CM “=9— 2. The first derivative of a fllllCtiOfl is given by f ’(x) = xx/ 16 — x2 . Find: a. The interval(s) where f (x) is increasing. 13/001) -: 341/912— M - - 4 we) 4 “We ‘4 b. The x-coordinate(s) of the inflection point(s) for y = f (x). 3. Find the relative (local) extrema: f (x) = (x — 3)Zex 9’01): (x~ 5316‘ + 2003369" M :(X-5)ex(X‘-5'lr2> | 2‘ .3 Ll refocravtw) Rail W: it He) CLY'5MX7'3 MMWSQ) 4. Below you see the graphs of the first and second derivatives of a function y = f (x). Add to the graph a Sketch of the approximate graph of f, given that the graph passes through the point P. (Just draw your graph over the given graphs. - ' ' ' U9 1/0 5 . Sketch the graph 0ft! lxvice-diflitrenliable function y w [(x) with the foiiowing properties. Label coordinates where possible. x y Derivatives x<2 ,\"<O. y">0 2 l y’ = 0, y" > 0 2<.\'<£4 y’>0, y">0 4 4 y' :3- 0. y” = 0 4<x<6 y‘>0.y”<{l 6 7 y‘ = 0, y" <: 0 x>6 into. £4.30 6. Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 10 cm. What is the maximum volume? 7. You are designing a rectangular. poster to contain 50 square inches of printing with a 4-in. margin at the top and bottom and a 2-1n margln at each side. What overall dimensions will minimize the amount of Paperused? __> R :x& R :Y +g) : X< 5g_;+8x—32j Q; 9)?ng x—Ll P" : r/(gfl+l3’X)L~L\\ * (9:81 in ot—Lm W: (“the 51+ -qzeW-a/ng’ lawn; 9';— S’xa- (04x40? _ 8 xz—sx—a fiCX-‘MH W” \bbt‘H/E: " (1102. C“ \Z:C] 3 l _ : “l as a ‘° ‘ 3W 8. a. Evaluate: limxaoSinxi-x= v I; Cl H X H Arse me M .J. 1 3 a“ pie/+5 : 6/ 76/ ._—___— i ‘ 9 a ftfifixzuadx - 'J—X‘l’C/ Mrj:6 5(1’245'l53dx ' ‘ 5x f b. fies" + 3csc2 x) dx= #— ‘fl 3 10. Suppose that x = 0 is a critical number of a function y = f (x). a. How can you determine from y = f ’ (x) whether there is a horizontal tangent line or vertical tangent line at x = 0? If there is a relative extremum at x = 0, how would this information about the tangent line affect how you sketch the graph at that oint? *1? into) :0 Jrhsm We. 18 a. hols-I20“ inmgmi flux/NU2 which mans Wm tau lost, 0. “Sn/100W Ct»th 60" W Poll/Cl: >kj§ Silo) Due w—Hw/ «‘5 a Verficafl Wm 80 unuflDhmre aJCLLSP oi—Mamewn. b. Is it possible that the graph would not have a relative extremum at x = 0? If so, provide an example. You can either give an equation for your example or sketch a graph. its 3H: )8 P0531912 ‘lrl'ldi 130 a) rip/EM a, {WWW 11. Giveny = 2x — 3x2/3, find: a. Relative extrema and intervals where the graph of function is increasing/ decreasing. %I__2__Q-x~‘l3 _. Q-u—ig— -- — 37C ' qr b. Inflection points and intervals where the graph of the function is concave up/down. Z bah:%x"4l3 : 5m PIPi MLO . VOW ' c. " etc the graph using the information above. ...
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