Let z = f(x, y) = x^2 + 4y^2, and consider the z-value 16. Find the level set for the z-value 16, and describe it geometrically. (Recall that the level set such that z = 16 is a set of points in the domain.) Find a parametrization ~r(t) for the level set in the xy-plane. Hint: Use sine and cosine. Find the time t such that the parametrization gives the point ~r(t) = (0, 2), and find the derivative of ~r at that time. Find the gradient of z at the point (0, 2), and show that the gradient of z at (0, 2) is perpendicular to ~r 0 (t) at (0, 2).
Let z = f(x, y) = x^2 + 4y^2, and consider the z-value 16. Find the level set for the z-value 16, and describe it geometrically. (Recall that the level set such that z = 16 is a set of points in the domain.) Find a parametrization ~r(t) for the level set in the xy-plane. Hint: Use sine and cosine. Find the time t such that the parametrization gives the point ~r(t) = (0, 2), and find the derivative of ~r at that time. Find the gradient of z at the point (0, 2), and show that the gradient of z at (0, 2) is perpendicular to ~r 0 (t) at (0, 2).
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Let z = f(x, y) = x^2 + 4y^2, and consider the z-value 16. Find the level set
for the z-value 16, and describe it geometrically. (Recall that the level set
such that z = 16 is a set of points in the domain.) Find a parametrization
~r(t) for the level set in the xy-plane. Hint: Use sine and cosine. Find
the time t such that the parametrization gives the point ~r(t) = (0, 2), and
find the derivative of ~r at that time. Find the gradient of z at the point
(0, 2), and show that the gradient of z at (0, 2) is perpendicular to ~r 0
(t) at (0, 2).
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