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# MULTIVARIABLE CALC MATH 2500

UGA

GPA 3.56

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This 5 page Class Notes was uploaded by Joanne Bergnaum on Saturday September 12, 2015. The Class Notes belongs to MATH 2500 at University of Georgia taught by Jes Ratzkin in Fall. Since its upload, it has received 9 views. For similar materials see /class/202076/math-2500-university-of-georgia in Mathematics (M) at University of Georgia.

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Date Created: 09/12/15

Practice Problems Math 2500 April 15 2008 These problems are not in any particular order The exam will be shorter The last 8 problems form the nal exam from the last time I taught this course 1 E0 9 F 5 533 5 90 For each of the following pairs of vectors compute 13 v 13 X 17 and the orthogonal projection of 13 onto 17 a alt1o1gtvlt11ogt b 12 01016 014 Let H1 be the plane through 111 with normal vector TF1 1710 and let H2 be the plane de ned by z y 2 11 a Write down a linear equation for 1111 b Find the cosine angle between H1 and 1121 c Parameterize the line 1 of intersection between H1 and 1121 d Find the distance between H2 and 111l Consider the curve Ct cos2tsint for 0 S t S 27L a Sketch this curve b Write down the tangent line to c at the point 7111 This point corresponds to the parameter value it 7r2l c Is the tangent line to this curve ever parallel to the line y 71 Be sure to explain your answer d Set up but do not evaluate the integral to compute the arclenth of this curve Consider the function 1 fzy 1t2dtl y a Compute the partial derivatives of b Does f have any critical points Be sure to explain your answer Consider the function fz y 12y 7 zygl a Does f have an upper or lower bound Explain your answer b Compute the partial derivatives of 0 Compute the directional derivative of f in the 1 il direction at the point 111 d Find the direction of steepest ascent for f starting at 1 1 Make sure to write down a unit vector e Notice that f1 1 01 Write down the equation of the tangent line to the f 0 level set at the point 111 Consider the function 3 2 ay 6W 7 a Find and classify all the critical points of b Find the absolute minimum of f restricted to the square 0 S I S 1 0 S y S 11 1 Recall that two tangent directions to the graph of f are given by the vectors 5f 10 01il Max may a Compute a normal vector for the graph b Can the tangent plane of the graph ever be parallel to the z 7 2 plane Explain your answer Find the absolute maximum and minimum of the function f my on the ellipse 4x2 y2 4 Evaluate fDx47127y2dAwhereDzy l 1 S12y2 S4ygt 01 H 53 HH OH H 9 H F H 5 H on H 7 H 00 H 50 to C Set up but do not evaluate D zemzydA where D is the region bounded by the curves y 13 and y I Be careful of signs Evaluate f D xl 7 12dA where D is the triangle with vertices 00 l 0 and 01 Consider the domain D z y l lz yl S l a Sketch D b If we change variables by u y 0 71 9 what is D in the uv coordinate system c Set up but do not evaluate the integral cos 7m 7 Try dA in the u v coordiante system D Consider the vector eld F 7yz a Sketch F b Com ute d where t cost sint for 0 S t S 27f p 7 7 Consider the vector eld F Zrexz z 2yexzy2 cos a Show that D Vf for some b Find a function f such that D Vf c Compute qu d where 7t cos tsin t for 0 S t S 7r2 Hint you donlt need to actually do an integral Consider 7y 12 7 ycoszyz 7 y3 7 zcoszy a Is Vf for some f Be sure to explain your answer b Compute 7 d Hint don7t actually compute the line integral Consider the vector eld F 2 13 7 yzexyz y 7 zzemyz 7x 22 H xyefyyl a Compute V and V X b Is Vf for some f Be sure to explain your answer c Compute f f2 V X D dA where E is the upper unit hemisphere centered at 000 with the outward un1t norm Consider the surface 77u v u cosv u sinv v a Compute the tangent vectors BFBu and BFBv b Verify that this is a good parameterization by checking that BFBu and BFBv are never parallel c Find the equation of the tangent plane to 397 for the parameter values u l v 7r d Is the tangent plane ever parallel to the z 7 y plane Be sure to explain your answer e What is this surface Can you draw a sketch of it Hint x a value of u for instance u 1 or u 0 and draw the resulting curve Compute f EV X D dA where D 7yz0 and E is the upper unit hemisphere centered at the origin with the outward unit normal Compute f f2 dA where D 1y27cos y y7e 22 27 z cos12y and E is the unit sphere centered at the origin with the outward unit normal Compute f E dA where D 1zey2z y7z coszz2 2 and E is the upper unit hemisphere centered at the origin with the outward unit normal Hint what is D restricted to the plane 2 0 to H to to to CAD to g to 01 to on to N to 00 to to Let 111 be the plane through 101 with normal TF1 1 710 and let 112 be the plane given by z y 2 a 3 points Find cos 9 where 9 is the angle between 111 and 112 b 4 points Find a linear equation for the plane 111 c 4 points Parameterize the line 1 of intersection between 111 and 112 d 4 points Find the distance between 112 and 1 12 Consider 12y7 zy7 12 7 6y 7 z 6 a 5 points Verify that the only critical points of f are 3 75 and 7235 b 5 points Classify these critical points as local maxima local minima saddle points or none of the above c 5 points Observe that f1 1 f 72 at the point 11 72 Write down an equation for the tangent line to the level set Recall that two tangent directions to the graph of a function are 19f 9f 1 0 7 0 1 7 lt BI lt By a 5 points Write down a normal vector to the tangent plane of the graph of b 5 points ls it ever possible that the tangent plane to the graph of f is parallel to the plane I y 0 e sure to exp ain your answer Consider the composition gt fztyt where 11 1 z 1 71 y1 2 yl 3 and Vf12 3 72 a 3 points What is gl b 4 points Suppose zZ 0 and yQ 0 ls it necessarily true that gQ 0 Explain your answer c 3 points Suppose g71 0 ls is necessarily true that z71 0 and y71 0 Explain your answer a 5 points Evaluate f D exg sz where D is the domain 1 y l 1 S 12 y2 S 4 I S 0 b 5 points Set up but do not evaluate the integral ffD e123 siny3 zdA where D is the region bounded by the line y 1 z and the curve y 12 It might help to draw a picture Consider the vector eld E y coszy z coszy y2 in the plane a 5 points Verify that E is the gradient of a function b 5 points Find a function f such that E Vf c 5 points Evaluate d where 7t t 1 7 t2 for 0 S t S 1 Hint you do not need to compute an integral Consider the vector eld E y yzemyz 7x 1267 zyexyz in threespace a 5 points Compute the curl of b 5 points ls E the gradient of a function Be sure to explain your answer c 5 points Evaluate f f2 V X E dA where E is the hemisphere z yz l 12 y2 221 2 0 oriented with the normal FL pointing away from the origin Hint you have to compute an integral but not necessarily over 2 Consider the vector eld E cosy2213 7zln y y7e 3 2 2sin12y5ln 1 12 y2 in threespace a 5 points Compute the divergence of b 5 points Evaluate dA where E is the unit sphere centered at the origin oriented with the outward normal FL Hint think before you compute Consider the curve Ct cos t sin t t a 4 points Find the equation of the tangent line to c at time t 7r b 3 points ls the tangent line to 5 ever parallel to the zy7plane Be sure to explain your answer c 3 points Find the length of the section of 5 corresponding to 0 S t S 27L 30 Consider the function fz y 13 7 12y 31y a 5 points Verify that the only critical points of this function are 00 39 b 5 points Classify these critical points as local minima local maxima saddle points or none of the above c 5 points Observe that fl l 3 Write down an equation for the tangent line to the f 3 level set of at l 1 31 Consider the composition gt fztyt Where 11 l yl 2 zl 71 yl 1 and Vfl2 a 4 points What is gl b 3 points Suppose zZ 0 and yZ 0 Is it necessarily true that gQ 0 Be sure to explain your answer c 3 points Suppose gO 0 Is is necessarily true that zO 0 and yO 0 Be sure to explain your sweri 32 Below is a sketch of some of the level curves of a function f fz 712 lt CAD to F 01 CAD 0 710 a 5 points In Which direction does Vf710 point Be sure to explain your answer b 5 points If 712 is a critical point of f What kind of critical point do you expect it to be Be sure to explain your answer i 5 points Set up but do not evaluate the integral f fD z ln1 12 y2dA as an iterated integral Where D is the domain bounded by z y2 7 2y and y z 7 1 It might help to draw a picture 10 points Evaluate f D 412 y2 7 ldA Where D l 1 S 12 y2 S 4 i Consider the vector eld F yew 21167 in the plane a 5 points Verify that Vf for some function f fz b 5 points Find a function f such that Vfi c 5 points Evaluate 7 d Where 7t cos t sin t for 0 S t S 7r2i Think before you computer i Consider the vector eld F 2 ln1 y2 7 I 2y 7 re 71 my sinl 12 y2 in three7spacei a 5 points Compute the divergence of b 5 points Evaluate dA where E is the unit sphere centered at the origin oriented with the inward normal Think before you compute 37 Consider the vector eld E yz coszy2 zz coszy2 y my coszyz in threePspace a 5 points Compute the curl of b 5 points Is E the gradient of a function fzyz Be sure to explain your answer c 5 points Evaluate ffEV X E dA where E is the upper unit hemisphere centered at the origin parammeterized by Wcosv Wsinv t l 0 S u S 10 S v S 27r oriented with the upward normal Hint you don7t have to integrate over 2 you can replace it with another surface which has the same boundary CA3 00 Consider the curve 7t t3t2 t a 4 points Find the equation of the tangent line to 7 at t l b 3 points The the tangent line to 7 ever parallel to the plane 2 1 Be sure to explain your answer c 3 points Set up but do not evaluate the integral to nd the length of the section of 7 for 0 S t S 2 CAD to Consider the function y W 1th a 5 points Compute the partial derivative BfBz and BfBy b 5 points Does f have any critical points Be sure to explain your answer lnlz2y2dA D where D l S 12 y2 S 4y S 0 Hint flntdt tlnt 7t const g D 10 points Evaluate 41 Consider the function ay W 7 I ey 1 a 5 points Show that 711 is the only critical point of b 5 points Classify the critical point as a local minimum local maximum or saddle point c 5 points Find the directional derivative of f in the U 12 direction at the point 10 yo l 2 d 5 points Observe that fl 2 l 6 Find the equation of the tangent line of the level set l e at the point 1 2 42 Suppose f fz y while both I and y are functions of t Let g be the composition gt fzt a 4 points If 10 l zO 71 y0 2 yO 4 and Vfl2 722 nd g 0 b 3 points If z l 0y 0 must it be true that g l 0 Be sure to explain your answer c 3 points If g 7l 0 must it be true that zil 0 and yil 0 Be sure to explain your answer 43 Consider the vector eld E F1F2 y yemywexy on the plane a 5 points Is E the gradient of a function Be sure to explain your answer b 5 points Compute the path integral LFa d where 7t 2 cos tsin t for 0 S t S 27f Hint the area of an ellipse with semimajor axis a and semiminor axis 12 is Nah 44 Consider the vector eld E 721 y21nl 22 26122 22 7 my cosz2 7 a 5 points Compute the divergence V b 5 points Evaluate the ux f f2 E dA where E is the boundary of the solid cylinder 12 y2 S l 71 S 2 S l oriented with the outward normal FL c 5 points Evaluate the ux f f5 E dA where S is the hemisphere 12 y2 22 l 2 2 0 oriented with the upward normal FL Hint you may want to use a theorem to avoid doing the ux integral but you would need to supplemen 77 S with another surface Choose the supplemental surface to t the problem It may also help to draw a picture

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