fx265s2009

# fx265s2009 - δ x,y = y occupying the triangle with...

This preview shows pages 1–4. Sign up to view the full content.

Math 265 Final Exam S2009 Instructor: Answer each question completely. Show all work. No credit for mere answers with no work shown. Show the steps of calculations and give exact answers. State the reasons that justify conclusions. 1. A particle moves in space with position vector r ( t ) = cos(2 t ) i + 3 t j + sin(2 t ) k . a) Show that the velocity vector v ( t ) and the acceleration vector a ( t ) have constant length. b) Show that v ( t ) and a ( t ) are orthogonal for each t . c) Find the distance the particle moves between t = 0 and t = 2 π . 2. Find an equation of the plane containing the point (2 , 1 , 3) and the line x = 1 + 3 t,y = - 2 - t,z = 3 t .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
3. Find an equation of the tangent plane to the surface x 2 + xy + y 2 + z 2 = 16 at the point (1 , 2 , 3). 4. Find all of the critical points of the function f ( x,y ) = x 3 - 9 xy + y 3 , and classify each one as local maximum, local minimum, or saddle point.
5. Find the mass and the center of mass of a thin plate with density function

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: δ ( x,y ) = y occupying the triangle with vertices (0 , 0), (1 , 1), and (-1 , 1). 6. Convert the integral below from cylindrical coordinates to an equivalent integral in a) Cartesian, b) spherical coordinates. DO NOT EVALUATE. I = Z π Z 1 Z √ 3 r r 2 sin θ d z d r d θ. 7. Let S be the solid sphere centered at the origin with radius 2, let F = (3 xz 2 + 2 yz 2 ) i + (3 x 2 z 3-yz 2 ) j + (3 x 2 y 2 + z 3 ) k , and let n be the outward pointing unit normal vector on the boundary of S . Calculate RR ∂S F · n dS . 8. Consider the vector ﬁeld F ( x,y ) = h 3 x-y, 2 z, 4 x-z i . a) Calculate curl F and div F . b) Use Stokes’ Theorem to calculate R C F · T d s , where C is the triangular path from (2 , , 0) to (0 , 3 , 0) to (0 , , 2) and back to (2 , , 0)....
View Full Document

## This note was uploaded on 10/04/2010 for the course MATH 166 taught by Professor Hotlz during the Spring '08 term at HCCS.

### Page1 / 4

fx265s2009 - δ x,y = y occupying the triangle with...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online