{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Math13S09final_sol

# Math13S09final_sol - Tufts University Department of...

This preview shows pages 1–3. Sign up to view the full content.

Tufts University Math 13 Department of Mathematics Spring 2009 Solutions to Final 1. True or False? (a) F: cylinder (b) T (c) F (d) F: maybe saddle (e) T 2. (a) The vector < 3 , 0 , 2 > , from the origin to the point (3 , 0 , 2), as well as the direction vector < 1 , 2 , 1 > of the line are in the plane, so a normal is given by vn = < 3 , 0 , 2 > × < 1 , 2 , 1 > = < 4 , 5 , 6 > . Therefore, we have line given parametri- cally by x ( t ) = 3 4 t , y ( t ) = 5 t , z ( t ) = 2+6 t , (b) Let f ( x,y,z ) = e x ln y + z 3 , then the gradient is normal to level surfaces so we have v f = < e x ln y, e x y , 3 z 2 > | (3 , 1 , 2) = < 0 ,e 3 , 12 > is normal to the surface and tangent plane. Thus an equation for the tangent plane is e 3 y + 12 z = e 3 + 24. 3. x 2 y 2 z 2 = 1 is a hyperboloid of one sheet with the x -axis as the axis of symmetry. x 2 y 2 z 2 = 0 is a cone with the x -axis as the axis of symmetry. x 2 y 2 z 2 = 1 is a hyperboloid of two sheets with the x -axis as the axis of symmetry. x y z x 2 y 2 z 2 = 1 x y z x 2 y 2 z 2 = 0 x y z x 2 y 2 z 2 = 1 1

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

View Full Document
4. For double integrals we think of the integrating the height of boxes over a region in the plane.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

Math13S09final_sol - Tufts University Department of...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online