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# Vector Calculus MATH 223

KU

GPA 3.6

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This 5 page Class Notes was uploaded by Edgar Jacobi on Monday September 7, 2015. The Class Notes belongs to MATH 223 at Kansas taught by Kamran Reihani in Fall. Since its upload, it has received 25 views. For similar materials see /class/182384/math-223-kansas in Mathematics (M) at Kansas.

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Date Created: 09/07/15

H to 03 F U a T 00 to O Find the best quadratic approximation to the function gz y Preparation for the Second Midterm Math 223 April 2009 Kamran Reihani The motion of a moving particle in R3 is described by the parametrized curve rt 3 sin 252 2253 3 cos 252 Find the total distance 6 traveled by this particle from time t 0 to t 1 Answer 6 2g71 Find the ow line xt of the vector eld Fm yz 7m y 5 1 with x0 5231 Answer xt 524 3t 1 2713 lnt 5 z2 y2 22 171 yj 2k Evaluate divF Consider the vector eld Fm y z Answer div F 4xz2 y2 22 Either nd or prove that there does not exist a a function f on R3 such that Vf y i m zcosyj sinyk b a vector eld F on R3 such that curl F 21 yj mk Answer a m yz my 1 zsiny b F does not exist 4m2 7 5m 3 2 1 near 00 39 Answer p2zy 3 7 5x 4x2 7 3342 Determine all possible local maxima local minima and saddle points of the function fx yz em22m y2 22 3 Answer A local minimum at 70 0 and a saddle point at 7100 Let B be the region in R2 bounded by the curves y and y i Express the area of B as iterated integrals in two different ways ii Calculate the area of B by evaluating one of the iterated integrals you found in part Answer 1 A f0 ff dy dz f5 fy y dz dy ii A g 4 1 Pl3W3 y17x313dmdy 0 0 Evaluate the iterated integral 3 A 7 nswer 8 Let R be the region 9x2 4342 g 1 in the my plane Evaluate the following double integral 9352 4m M R 7139 A 7 nswer 2 1 Let R be the triangle with vertices 00 10 and Evaluate the following double integral by an appropriate change of variables awry 1A 4 my 5 7 Answer H to DJ F Let P be the parallelogram with vertices 00 1 1 3 1 and 4 2 Evaluate the following double integral by an appropriate change of variables m2x3ydA R 5 9 A 7 nswer 3 Let S be the region bounded by the planes z 0 z y y 1 z O and the surface 2 2 1 Evaluate the following triple integral Szdv 7 A 7 nswer 2 4 Find the volume of the solid bounded by the paraboloids z 2 y2 and z 1 7 m2 yz Answer Find the volume and the geometric center of the solid bounded by the sphere 2 y2 22 9 and the cone 22 3m2 3342 Answer V 187r17 G 00 g Preparation for the First Midterm Math 223 March 2009 Kamran Reihani 1 Find the shortest distance from the point Q 12 71 to the plane 73 2z 1 y73 7 2271 0 ii Find the point B on the given plane 73 that is closest to the given point Q Answer lt1 11 731971 3 2 Let A 1 23 B 111C 7210 What is the area of the triangle AABC ii What is the volume of the parallelepiped spanned by the position vectors of A B and C Answer ii 4 3 Describe the line 6 passing throug the point P 112 and perpendicular to the plane 73 z 7 y 7 z 2 by parametric equations Answer m1ty 17tz27t 4 For each of the following limits either evaluate it or explain why it does not exist 1m s1nmy wygt7lto0gt z y 4 5 b hm amp myz7l000 as2 y 222 a Answer a Does not exist b 0 5 Let f R2 7 R be the function de ned piecewise by 1 962 2408111 for 967 24 7g 07 0 1 00 24 902 y 0 for z y 00 a Find the partial derivatives of f at any point z y E R2 b ls f of class Cl Explain why or why not c Show that f is differentiable on R2 6 Suppose that there exists a function g X g R4 7 R of class C1 such that gwwx yz wz 7 m y2 and gyw my z 72wy m 2 Explain why 9 cannot be of class OZ 7 Use the Chain Rule to compute Dgo f1711 where fz yz z 7 mzym yz 7 2 and gm yz 90242714223 4 0 74 Answer 0 0 0 J 8 Viewing y as a function of m z implicitly by the equation 33422 1 3x21 7 lnmyz 0 and using the Chain Rule nd the partial derivative 8737 in terms of m y z 2 Answer xsyz 7 l 7 1 2x3yz 1 3m2 7 7 9 H O Let f R3 a R be the function given by fzyz Let f R2 a R be given by fzy zy a Use the de nition of the partial derivative as a limit to nd fm0 0 and fy0 0 b Let x R a R2 be de ned by xt 25252 Show that f o x is differentiable and nd Df o x0 directly c What does the Chain Rule suggest about Df o x0 d You should have gotten different answers for parts b and function 1 What does this tell you about the Answer a fm0 0 fy0 0 0 b Df o x0 1 c 0 d f is not differentiable at 00 Find the direction as a unit vector in which the function fzyz zzyz 7 Bzyzz 23 attains the maximum rate of increase at the point 1 71 1 ii What is the rate of change of the given function in the direction found in part i Answer V 7 ii Let f R2 a R be the function de ned piecewise by 2 2 2 y232 c for my for any 739 070 07 0 rm where c is some constant a Determine the value of c so that f is continuous on R2 b For ab 7 00 determine fmab and fyab c Determine fm00 fy0 0 and Vf00 You will need the de nition of the partial derivative as a A limit d Let V be the unit vector i What does your answer to part c suggect about va0 0 e For the vector V de ned in part d calculate va0 0 using the de nition of the directional derivative as a limit f You should have gotten different answers for parts d and e function 1 What does this tell you about the Ammnmwoomnmw d 0 e 1 2 1247 3172 52447235 27fya7b f f is not differentiable at 00 C fwlt07 0 07 fit07 0 07 Vf070 039 my 221 and let a 253 6 R3 a Find Vfa b Let V u v w be a unit vector in R3 Find the directional derivative vaa in terms of u v w c Does there exist a unit vector V for which vaa 1 Either nd such a vector V exactly or explain why no such V can exist d Let S be the level surface of 1 containing a Find an equation for the tangent plane to S at a e Determine the set of all points zy z E R3 where the corresponding level surface of 1 does not have a well de ned tangent plane Answer a H w mampw i e zyz E 7 c No d 5z2y76272 y0 13 H a Find an equation for the plane P tangent to the surface arccosmy 7 24 7139 at the point 3 5 72 Answer 5x 1 3y 1 322 34 O 2 2 On the ellipsoid 2 yj Z 1 nd all points at which the tangent plane is parallel to a given plane Pambyczd 1 Answer imQLAb c Let 1 32 ycosz my 0 and 2 z 2342 7 422 0 be two surfaces passing through the point Z P 0 7r 7 Describe the line 6 tangent to the curve of intersection of the given surfaces 51 and 2 at the given point P by parametric equations Answer x47r2t y7r77rt 77139 2t 272 7r Consider the surface S 2my22 7 223 25 2 passing through the points P 71 1 71 and Q 111 By parametric equations describe the line 6 tangent to the surface S at the point P with the property that Z is parallel to the tangent plane to the surface S at the point Q Answer m71t y1t 2717t

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