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# SEQ, SERIES, AND MULTIVAR CALC M 408D

UT

GPA 3.67

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This 9 page Class Notes was uploaded by Reyes Glover on Sunday September 6, 2015. The Class Notes belongs to M 408D at University of Texas at Austin taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/181460/m-408d-university-of-texas-at-austin in Mathematics (M) at University of Texas at Austin.

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Date Created: 09/06/15

UT Learning Center K Smwve rauccead Excel Calculus Refresher for M408D Day 3 Day 3 7 Integration Review Topics Basic Integration Rules U Substitution Area Between Curves Volume of Revolved Solids Integration by Parts Trig Substitutions Inverse Trig Function Integration Formulas Partial Fraction Decomposition Powers of Trigonometric Functions Evaluate the following integrals 2 l Ix de x2 2 Isinz 3x cos 3xdx 439 Jx22xl 5 J sin2 x dx 6 j xex2 1dx 7 A Write TWO de nite integrals one with respect to X the other with respect to y which represent the area Q bound between the graphs of f and g 1 fx 6 gm x B Find the volume of the solid generated by rotating Q about the XaXis C Determine a de nite integral which represents the volume of the solid generated by rotating the area Q about the yaXis 8 Find the volume obtained by rotating the region bounded by ylnx y2 x20 y0 about the yaXis Ix cos 3xdx 23x sin xdx 53x J1x2 1 d I 2x2 x x 15 dx 3 x2 2x 3 JZx 3dx x3x UT Learning Center SurviverstmceauExcm M408D Final Exam Review Day 3 1 True or False Lb Ub E H a NH 17 H 2 b There is only one and only one vector lto aand 1 namely axb c The dot product is cumulative and yields a scalar d TWo vectors are E if they are scalar multiples of each other a If ab0then cost9 2Assume aEFJE and 177 a Find ab b Find the angle between a and b 3 Assume ax273I and b772x339713 Findallvalues ofx forWhich all 4 Let a lt 1 21 gt b lt 011gt and c lt 130 gt Find the volume ofthe parallelpiped that these three vectors span 5 Find all vectors oflength 3lto alt121gt and b lt 011 gt x 2 t 6Let l y12t z 37 a Find the equation of the plane Which is lto and passes through 122 b What point do the plane and line intersect at 7 Aline is parameterizedby x1t y372t and z717t Anotherline is parameterized by x 27 v y v and z 71 2 These tWo lines intersect at the point 371 73 Find the equation of the plane that contains both lines 8 A certain curve is parameterized by x 2t2 y cos7rt and z 6 Find a parametric equation for the line that is tangent to the curve at the point 2 1 ez 9 The surfaces x2 3y2 522 9 and xyz 1 intersect at 111 Find the angle between the two surfaces at this point 10 Suppose w f x yz where f is some function Suppose that x yand z are functions of t Further suppose that j yz2x fy 26 xz 2y and fz ye xy If x0 2 y0 1 20 1 x390 3 y390 4 and z390 5 nd the derivative of w with respect to I when t 0 11 Find a unit vector in the direction in which f x y yze2x increases most rapidly at P0 1 and give the rate of change of f in that direction 12 Find the equation of the plane tangent to the surface of x2y xyzz 0 at the point 11 1 13 Use traces to sketch a graph of the following surfaces A z 4x2 y2 B zx2 y2 t 6 E199 E2x 18 Any funetion de ned over a symmetric domain can be written as the sum of an Even function and an 5191 function How 7 ij Od1x 19 Can a function de ned over a symmetric domain be 8 Odlgx a Even c Neither Ex b Odd d Both Give an example for each ES answer Od11x1 Od2x 20 E1E2X 10 1 21 E1Odx 1310 22 Od10d2x 11 1 0d1X 23 0dE1X a O 12 I Exdx 2 I Exdx 2 I Ex dx 24 Iff is de ned over the range of E1 393 393 then fE1x NB E1 is de ned over a symmetric domain The END

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