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# Review Sheet for MATH 223 with Professor Reihani at KU 2

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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Kansas taught by a professor in Fall. Since its upload, it has received 19 views.

## Reviews for Review Sheet for MATH 223 with Professor Reihani at KU 2

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Date Created: 02/06/15

Preparation for the First Midterm Math 223 March 2009 Kamran Reihani 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 ii 731971 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 Describe the line 6 passing throug the point P 1 1 2 and perpendicular to the plane 73 z 7 y 7 z 2 by parametric equations Answer m1ty 17tz27t For each of the following limits either evaluate it or explain why it does not exist sinmy wygt7lto0gt z y l 90244 25 11m myz7l000 as2 y 222 a 0 Answer a Does not exist b O Let f R2 7 R be the function de ned piecewise by 2 2 sin for z 00 fmy 24 W 7y7 77 0 for x 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 Suppose that there exists a function g X Q 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 CZ Use the Chain Rule to compute Dgo f1711 where fz yz z 7 mzym yz 7 2 and gm yz 90242714223 074 000 Answer Viewing y as a function of m z implicitly by the equation 33422 1 3x21 7 lnmyz O and using the Chain Rule nd the partial derivative 87 in terms of m y 2 Answer 9 Let f R2 7 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 7 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 What does this tell you about the function 1 Answer a fm0 0 fy0 0 0 b Df o x0 1 c 0 d f is not differentiable at 00 10 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 0 ii V17 11 Let f R2 7 R be the function de ned piecewise by 2y2 for z 0 0 rm 2y232 w C for 9w 070 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 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 What does this tell you about the function 1 Answer a c 0 b Ma b 2552 332 leg11 c M0 0 0 ma 0 0 Vf00 0 d 0 e 1 f f is not differentiable at 00 my 221 12 Let f R3 7 R be the function given by fzyz and let a 2 5 3 E 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 10 7 b vaa 157014 v 7 3w c No d 5x 2y 7 62 72 l 8 90797 2 E R3 13 14 15 16 Find an equation for the plane P tangent to the surface arccosmy 7 24 7139 at the point 37 57 72 Answer 5x 1 3y 1 322 34 O 22 2 On the ellipsoid 2 yj 3 17 nd all points at which the tangent plane is parallel to a given plane Pambyczd 1 Answer imw 4b7 90 Let 1 32 ycosz 0 and 2 z 2342 7 422 0 be two surfaces passing through the point P 07 7r7 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 7139 Consider the surface S 2my22 7 223 25 2 passing through the points P 717 17 71 and Q 111 By parametric equations7 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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