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# Calculus II MATH 116

KU

GPA 3.6

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This 8 page Class Notes was uploaded by Edgar Jacobi on Monday September 7, 2015. The Class Notes belongs to MATH 116 at Kansas taught by Jeff Mermin in Fall. Since its upload, it has received 13 views. For similar materials see /class/182360/math-116-kansas in Mathematics (M) at Kansas.

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Date Created: 09/07/15

1 F Math 116 Je Mermin7s sections Test 1 Review True False questions There will be ten truefalse questions taken verbatim from the quizzes See the quizzes and solutions Indicate the rst step you would take in evaluating each of the integrals below Or if they re easy enough to require no intermediate steps eval uate themi Otherwise do not actually evaluate the integrals The rst ve are done as examples Several of these have more appropriate answers in addition to those given ere a f 2x3dxi This is x4 C 01 Factor out the 2 f 141am Use algebra to simplify the expression fxe dx Use integration by parts with u x and dv exdxi f 295952 420dxi Substitute u x2 4 952 e dxi Break it up into a sum of two integrals 95 7 95 955195 2 i Thzs zs E952 7 2 3 9 3x5 gx C g h k m n f 115196 2 Reunite the integrand as 2 7 214A or 7 7 1 Use integration by parts with u 7 4x and dv 7 211dx or Substitute u 2x 1 f dz Substitute u 7x f 952953 9dx Substitute u x3 9 f 95 de Substitute u 2x2 1 f In x2dx Use integration by parts with u In as and dv dag f xgexdx Use integration by parts with u x3 and dv e dx f xgexgdx Substitute u 952 f Substitute u In x 0 5195 Use algebra to simplify the expression 3 Evaluate the inde nite integrals a f ides We have b 6x2e e xdx We have 6x2e e dx f6x2e dx fe xclx To evaluate the second integral set u 795 so clx idu and fe xclxfi e C7e xCl To evaluate the rst integral set u 12 s0 clx ght and f6x2e dx f4euclu 4e C 4e Cl Thus we have 6x2e e xdx 6x2e dx e xdx a C f x In xdx 2 Set u lnx and clv xdx s0 clu idx and v x Then we have xlnxdxudv uv7vdu 7 1 2 1 2 1 7 maxim 7 ies was 1 2 1 ix lnx7ixdac 712 712 72xlnx 4xCi 4 Express the area bounded by the given curves in terms of a de nite integral or integrals Do not actually compute a numerical answer a acly952ancly807ac3 b The curves y x2 and y 807 x3 meet at x2 807 x3 ie ac 4 Each of these curves meets ac 1 when ac 1 Thus the region is bounded on the left at ac l and on the right by ac 4 For ac between 1 and 4 we take ac 2 as a sample point 22 4 lt 72 80 7 23 so y 80 7 x3 is the top boundary and y x2 is the bottom boundary Thus the area is given by 80 7 x3 7 daci yac473ac2 andy6x2i The curves intersect at x4 7 3x2 6x2 ie at ac 0 and ac i3 Thus there are two regions to consider the rst bounded on the left by ac 73 and on the right by ac 0 and the second bounded on the left by ac 0 and on the right by ac 3 For the rst region we use ac 7l as a sample point and compute 7l4737l2 72 lt 6 67l2 Thus the top boundary is given by y 6x2 and the bottom boundagy is given by y x4 7 3952 Thus an the area of the rst region is 6952 7 x4 7 3952 dac 173 For the second region we use as 1 as a sample point and compute 14 7 312 72 lt 6 61 Thus the top boundary is given by y 6x2 and the bottom boundary is given by y x4 7 3952 Thus 13 the area of the rst region is 6952 7 x4 7 3952 dost 10 The entire area is the sum of these two areas or 6952 7 x4 7 3952 doc 6952 7 x4 7 3952dx1 13 Or equivalently 6952 7 x4 7 3952 doc 73 Math 116 Je Mermin7s sections Final Exam Review You will be given the following tables 0 The values of the functions sin cos tan on certain important angles 0 The derivatives of the trigonometric funtions o The integrals of the trigonometric functions 1 TrueFalse questions There will be ten truefalse questions taken verbatim from the quizzes All quizzes are fair game including those given before Test 1 2 Evaluate the following 11 a 2x3 7 3x2 de 10 b emdx c x251dx d ltz Wm Book problems on this material 0 Chapter 61 pp 4077408 9750 0 Chapter 62 pp 4197420 1750 0 Chapter 64 p 439 17740 0 Chapter 65 p 449 1728 0 Chapter 6 review p 490 1732 0 Chapter 71 p499 1732 0 Chapter 7 review p 534 176 3 Compute the following or state that they diverge No justi cation is necessary for correct answers on this problem However incorrect answers may receive partial credit if they are well justi ed a ZN n1 n ltbgtn1320lt n gt 3 Inln2n 3 00 n 1 1 f E 27173 7 27171 n3 Book problems on this material 0 Chapter 74 p 532 15742 0 Chapter 7 review p 534 15730 0 Chapter 112 p 717 30744 0 Chapter 113 p 727 5726 0 Chapter 11 review p 767 1118 11 ye 4 Evaluate dydx 10 y1 5 Express each of the following in terms of one or more de nite integrals or double integrals with explicit bounds Do not actually compute explicit answers a 3cos xdydx where R is the region bounded by x 7 x val y0 andysecx b sinx cos ydydx where R is the region bounded by x 7r R y7r x0andy27r c The area of the region R bounded by y 4 7 295 y 2 x 0 and x 1 Book problems on this material 0 Chapter 66 pp 4617463 1742 0 Chapter 87 p 608 1725 0 Chapter 8 review p 619 43746 93 Solve the following 3 Find all the relative maxima and minima if any of the function fx y 295g 7 5x2 7 2y2 4x 4y 7 4 b Find the maximum and minimum of the function fx y 3xy7 6 subject to the constraint x2 y 4 Book problems on this material 0 Chapter 83 p 570 1720 0 Chapter 85 p 594 1716 0 Chapter 8 review p 618 29738 H W N tb 030 Math 116 Je Mermin7s sections Test 3 Review True False questions There Will be ten truefalse questions taken verbatim from the quizzes All quizzes are fair game including those given before Tests 1 and 2 Evaluate all the second partial derivatives of the function fx y 3x2 34 Find all the relative maxima and minima of the function fx y x2 y3 m or show that none exist You do not need to compute the partial derivatives 768 f5 2x 7 W 768 fy 3 2 W fans 2 Iligj 1536 M z w Find the maximum value of x y 5 subject to the constraint x2 2 y 3 1 Find d2 if 2 yer 1 Evaluate the double integrals 12 y1 a 962 96y y2dydx 10 y0 b 4xdydx Where R is the region bounded by y 952 y 0 and R 951

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