{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Math 23 final2002 - Mathematics 23 Final Exam Answer all...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Mathematics 23 Final Exam, December 14, 2002 Answer all questions and show all work . Simplify your answers. 1. Find a vector perpendicular to the plane containing the points (1 , 2 , 1) , (2 , 4 , 2), and (1 , 0 , 1). 2. For the function f ( x, y ) = xe - 2 y + 6 y , find a. the maximum rate of change of f at (1 , 0). b. the direction in which the maximum rate of change at (1 , 0) occurs. 3. Let E be the solid above z = x 2 + y 2 and below x 2 + y 2 + z 2 = z . Set up an integral (in spherical coordinates) which gives the volume of E . 4. Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 3 y + 2 z = 6. 5. a. Find the tangent plane to the ellipsoid x 2 + 2 y 2 + 3 z 2 = 1 at the point (1 , 0 , 0). b. Find the points on the ellipsoid x 2 + 2 y 2 + 3 z 2 = 1 where the tangent plane is parallel to the plane 3 x - y + 3 z = 1. 6. a. Find the length of the curve C given by r ( t ) = h cos t, sin t, t i , 0 t 1. b. What is the curvature of C at r ( t o ), where t o is an arbitrary point 0 t o 1?
Background image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}