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Calculus III

by: Felicita Lockman
Felicita Lockman
GPA 3.68

Yuri Godin

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About this Document

Yuri Godin
Class Notes
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This 2 page Class Notes was uploaded by Felicita Lockman on Sunday October 25, 2015. The Class Notes belongs to MATH 2241 at University of North Carolina - Charlotte taught by Yuri Godin in Fall. Since its upload, it has received 72 views. For similar materials see /class/228908/math-2241-university-of-north-carolina-charlotte in Mathematics (M) at University of North Carolina - Charlotte.


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Date Created: 10/25/15
Math 2241 t N 03 14gt 5 quot Q 1 9 Calculus Practice Problems for the Final Exam December 7 2010 For what values of a is the vector v 72 1 a a perpendicular to the plane 2 62 7 31 4 b parallel to the vector product 0 1 2 gtlt 1 03 c perpendicular to the vector 123 Find the angle between the diagonal of a cube and one of its edges Answer arccos 3 Solution Consider a unit cube in the first octant and find the angle between its diag d onald 111 andtheedgei 100 Since di cos 19 then COS L9 z i Therefore 19 mm L 121212 120202 Find a plane through the points A1 1 71 B2 0 2 and C0 721 Answer 72 7 5y 7 42 6 Find a vector parallel to the plane 22 7 y 7 2 4 and perpendicular to the vector 111 Answer 0373 Find an equation of the tangent plane to the surface 2 24 7 213 22 at the point 102 Answer 2 62 7 4 Find the arc length of the curve rt 3t 7 1 74 sint 4005 t 72 g t g 2 Answer 20 If f 2 y 22 13 find the directional derivative of f at the point 72 1 in the direc tion of the vector u 42 3 j Answer 714 d Let 2 12 7 22 cosy 2 P y cos It Find d7 by using the chain rule Answer 76219 cosy 2y 7 22 sin y sint 6 Let 2 z 7 y2 z u 21 y 2u 7 1 Find 672 by using the chain rule Answer 1 696 7 1 If w f z y z 7 y has continuous partial derivatives with respect to u z y 1 6106107 6f 2 6f 2 xiysh0wthataa7 75 Find and classify the critical points of the function f 2 y 24 y4 7 42y 2 Answer 00 saddle i1 i1 local minima Sketch the region of integration and change the order of integration for 1 1 1732 I 7 dy my dz 0 27y d d 1 Ziyz by using polar coordinates where R is the region in the upper half plane bounded by the semi circle y l 7 2 Answer 0571 ln 2 13 Evaluate exactly the integral I R 1 2mim2 14 Evaluate exactly the integral I m2 yz 12 dy dm by using polar coordi 0 0 nates Sketch the region of integration first Answer 2 lnl 7


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