For each of the following pairs of functions f and g, determine if the level curves of the functions cross at right angles, and find their gradients at the indicated point. (a) f(x, y) = 5x +5y, g(x, y) = 5x - 5y. Do the level curves of f and g cross at right angles?? Vf(2, 1) = Vg(2, 1) = (b) f(x, y) = x² - y. g(x, y) = 2y + ln(x). Do the level curves of f and g cross at right angles? ? Vf(2, 1) = Vg(2, 1) =
For each of the following pairs of functions f and g, determine if the level curves of the functions cross at right angles, and find their gradients at the indicated point. (a) f(x, y) = 5x +5y, g(x, y) = 5x - 5y. Do the level curves of f and g cross at right angles?? Vf(2, 1) = Vg(2, 1) = (b) f(x, y) = x² - y. g(x, y) = 2y + ln(x). Do the level curves of f and g cross at right angles? ? Vf(2, 1) = Vg(2, 1) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![For each of the following pairs of functions \( f \) and \( g \), determine if the level curves of the functions cross at right angles, and find their gradients at the indicated point.
**(a)**
\[ f(x, y) = 5x + 5y, \quad g(x, y) = 5x - 5y. \]
Do the level curves of \( f \) and \( g \) cross at right angles?
Gradient at point (2, 1):
\[
\nabla f(2, 1) =
\]
\[
\nabla g(2, 1) =
\]
**(b)**
\[ f(x, y) = x^2 - y, \quad g(x, y) = 2y + \ln(|x|). \]
Do the level curves of \( f \) and \( g \) cross at right angles?
Gradient at point (2, 1):
\[
\nabla f(2, 1) =
\]
\[
\nabla g(2, 1) =
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f252638-0865-4e10-b890-ee416dfe8cab%2F99361bd9-9f50-4c2b-80c0-8e6581364b56%2Fjr23dg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For each of the following pairs of functions \( f \) and \( g \), determine if the level curves of the functions cross at right angles, and find their gradients at the indicated point.
**(a)**
\[ f(x, y) = 5x + 5y, \quad g(x, y) = 5x - 5y. \]
Do the level curves of \( f \) and \( g \) cross at right angles?
Gradient at point (2, 1):
\[
\nabla f(2, 1) =
\]
\[
\nabla g(2, 1) =
\]
**(b)**
\[ f(x, y) = x^2 - y, \quad g(x, y) = 2y + \ln(|x|). \]
Do the level curves of \( f \) and \( g \) cross at right angles?
Gradient at point (2, 1):
\[
\nabla f(2, 1) =
\]
\[
\nabla g(2, 1) =
\]
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