# Quiz 4 - Math 53: Multivariable Calculus September 26, 2007...

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

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: Math 53: Multivariable Calculus September 26, 2007 Quiz 4 Lecturer: Prof. Michael Hutchings GSI: Gary Sivek Name: Answers 1. (10 pts) For each of the given surfaces in three dimensions, identify the traces for x = k , y = k , and z = k , then identify and sketch the surface. (a) x 2 + 4 y 2 z 2 = 4 The trace x = k gives k 2 + 4 = z 2 4 y 2 , which is a hyperbola. The trace y = k similarly gives 4 k 2 + 4 = z 2 x 2 , which is also a hyperbola, so we have some sort of hyperboloid. The trace z = k gives x 2 +4 y 2 = k 2 4, which is an ellipse if k 2 4 0, or | k | 2, but when | k | &lt; 2 then there are no solutions. This then divides the surface in two, so we have a hyperboloid of two sheets. The graph is below on the left. (b) y = x 2 The trace z = k yields the standard parabola y = x 2 . The trace x = k gives y = k 2 ; since x and y are constant but z varies freely, this is a line. For the trace y = k , we have x 2 = k ; if k &lt; 0 this has no solutions, and if...
View Full Document

## Quiz 4 - Math 53: Multivariable Calculus September 26, 2007...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online