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# CALCULUSI MATH150

SDSU

GPA 3.72

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This 17 page Class Notes was uploaded by Burnice Ratke on Tuesday October 20, 2015. The Class Notes belongs to MATH150 at San Diego State University taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/225277/math150-san-diego-state-university in Mathematics (M) at San Diego State University.

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Date Created: 10/20/15

Last Name First Name Instructor s Name Math 150 Group Final Spring 2008 You are not allowed to use notes books calculators personal stereos or cell phones You have exactly two hours Write clearly so that you can avoid mistakes and count on partial credits Carry out the obvious simpli cations so that you can display your answers in an easily readable manner The following list is for the recording of the points only Do not write your answers on this page Points 1 4 a 2 4 a 3 4 a 4 4 a 5 4 a 6 8 a 7 10 a 8 6 a 9 8 a 10 6 a 11 10 a 12 4 a 13 6 a 14 4 a 15 6 a 16 4 a 17 8 Total 100 1 4 pts Evaluate the limit the limit may be nite or in nite 1 371 1H1 941 902 75x4 2 4 pts Evaluate the limit by using L Hospital s rule ea 717 hm 7 xgt0 as In problems 375 nd the derivative do not simplify 3 4 pts 154sinx 4 4 pts d as a arctan 5 4 pts 6 Let 7 f as as x a 6 pts Find the linearization of the function f at a 9 b 2 pts Make use of the result of part a to approximate f 87 7 10 pts An inverted conical tank has depth of 10 m and radius at the top of 3 m If water is lling the tank at the rate of 8 mSmin7 determine the rate at Which the radius of the surface of the water is increasing at the instant When the depth of the water reaches 2 m You need not simplify your answer Hint The volume of a cone With base area A and height h is 1 7Ah 3 8 Assume that y is de ned implicitly as a function of at by the equation y379y73x0 dy 4 t E l t i a ps vauaedx d b 2 pts Calculate dig at at 0 if y 3 When at 0 ac 9 8 pts Let 1 4 1 3 2 ac ix 7 7x 7 as f 12 6 Determine the intervals on Which the graph of f is concave up or concave down7 and determine the xecoordinates of the points of in ection of the graph 0 10 6 pts Let 13 761291 Determine the absolute maximum and the absolute minimum of f on the interval 02 you need not determine the intervals on Which f is increasing or decreasing 11 Let 1 at 7 3 a 6 pts Determine the domain of f Make use of the rst derivative test to determine the intervals on Which f is increaslng or decreasing and the points at Which f has a local maximum or local minimum facx2 b 4 pts Determine the vertical asymptotes of the graph of fyou need not justify your assertions Showt at lirn 7 x2 0 x ioo Sketch the graph of f Indicate the vertical asymptotes and the local extrema clearly You need not nd the intersections With the axes 12 4 pts Evaluate sin12 dac V776 dac 13 6 pts Evaluate d 11 t i ed deco 14 4 pts Evaluate 15 6 pts Evaluate 16 4 pts Evaluate e 4xdac 17 8 pts Assume that the velocity of an object moving along a line is t 1 t2 at time i Let 5t be the position of the object at time t Determine 5t for any i gt 0 if s 0 4 Solutions 1 1 3796 7 1 3736 zir411136275x4 7 ILI 11CL 4 7 1 371 1 7 11 1 7 133114171 174 3xiTx74 0039 2 1 1717171 517171i71 136 12 1136 2x 11362quot 3 d 5 f 54 714 54 dag as s1nac 4x s1nacac cosac 4 i t 3 1 7i 1 7i dacarcan 3 711 3 7912 3 79 5 d x2 721x27167122ac72x3732x72x37 32x dac x2 7 16 x2 7 162 x2 7 162 x2 7 16239 6 1 I 7 1 for Wag Therefore7 1 7 977 17 f 6 6 7 L9f9f9912g9 b f 87 2 L9 87 12 g 703 7 1165 7 Let h denote the depth of the water and let 7 denote the radius of the surface We have h 10 10 gt Therefore7 7107 32d 7 1077 d Tdt 3 dt39 dt E 7 107772 Thus 87 107772 dr dr 7 24 3 When h 2 we have 3 3 3 T 131 1 213 Therefore at that instant dri 24 7 24 720g21 dz 10m2 32737quot 77 3 metersmin The rate at Which the water level is rising is 7 200 N MELE 771 dz 3 3777977quot E 3 metersmin 8 3 dy dy 32i7i730 ydac dac 239 d d 3 1 3279i 3i 77 11 dac dac 31279 1273 b11f211037 1 1 1 07 7 1 1 1273y3 973 6 9 4ilx3ix2gtx37x272x 1 7 f 19 dac 12x 6 d 1 1 11 7i737727 277 7 f 7 dag lt31 21 2x an at 2 x1ac 2 Therefore f 0 if at 71 or at 2 Here is the sign chart of f and the concavity of the graph of f 1 1 1 71 1 1 2 1 1 f as 1 1 0 1 7 1 o 1 1 in pt 1 concave up 1 1 1 concavity 1 concave up 1 in pt 1 concave down Thus the graph of f is concave up on 700 711 concave down on 171 21 concave up on 2 00 The xecoordinates of the points of in ection are 71 an 10 Let fxx3761291 f ac 31 712x93171173 Thus the only critical point of f in 0 2 is 1 We have f007 f223ndf14 Therefore the absolute maximum of f on 0 2 is f 1 4 and the absolute minimum of f on 02 is 0 11 a The domain of f consists of all at 7E 3 We have 1 ac 7 3 7 1 I 1 7 f x as is as 73 Therefore f x0ltgtxi321ltgtx73i1ltgtac2orac4 l x l l 2 l l 3 l l 4 l l f ac 0 7 doesnotexist7 0 f incr loc max decr does not exist decr locmin incr The function is increasing on 7002 decreasing on 23 decreasing on 34 increasing on 4 00 Thus f has a local maximum at 2 and a local minimum at b The line at 3 is the vertical asymptote We have lim fac7ac2 lim 0 raioo zeiooxi3 13 1 i Iet1dtel11 7i dac 0 2 14 We set u 1 sin so that du cos dac Thus7 cos 1 712 u12 dgp 7 u dui2 u2 x1sinac udu 12 W 15 We set u 352 so that du den2 xE 2 71 day 2 771 du 0 I1 902 0 1 u2 7 I 2 arcsin 33 xg 2 arcsln 7 arcsln 1 3 16 We set u 7435 so that du 74dac e 4xdac iieudu fie 17The position of the object at t is t 739 1 2 40 dT 1nt 14 1sinac

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