Find the gradient of the function f and evaluate it at the given point Po, f(x, y, z)=e+y cos z + (y + 1) arcsin x, Po= (0,0, π/6).
Find the gradient of the function f and evaluate it at the given point Po, f(x, y, z)=e+y cos z + (y + 1) arcsin x, Po= (0,0, π/6).
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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![**Problem 6.1 - Gradients and Directional Derivatives**
a) Find the gradient of the function \( f \) and evaluate it at the given point \( P_0 \).
\[
f(x, y, z) = e^{x+y} \cos z + (y+1) \arcsin x, \quad P_0 = (0, 0, \pi/6).
\]
b) In which direction does the function \( f(x, y) = x^2 y + e^{xy} \sin y \) decrease most rapidly at the point \( P_0 = (1, 0)? \)
c) Find the derivative of the function \( f(x, y, z) = \cos (xy) + e^{yz} + \ln (zx) \) at \( P_0 = (1, 0, 1/2) \) in the direction given by \( \mathbf{u} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} \).
d) In which directions (if any) is the derivative of
\[
f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}
\]
at \( P_0 = (1, 1) \) equal to zero?
e) The derivative of \( f(x, y, z) \) at a point \( P_0 \) is greatest in the direction of \( \mathbf{v} = \mathbf{i} + \mathbf{j} - \mathbf{k} \). In this direction, the value of its derivative is \( 2\sqrt{3} \). What is \( \nabla f \) at \( P_0 \)? Justify your answers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77ccc228-61ae-45bc-bff3-36e89ce5abe9%2F8f915921-d1da-4cb5-bf2a-aa498e6e0474%2Fhm53g19_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 6.1 - Gradients and Directional Derivatives**
a) Find the gradient of the function \( f \) and evaluate it at the given point \( P_0 \).
\[
f(x, y, z) = e^{x+y} \cos z + (y+1) \arcsin x, \quad P_0 = (0, 0, \pi/6).
\]
b) In which direction does the function \( f(x, y) = x^2 y + e^{xy} \sin y \) decrease most rapidly at the point \( P_0 = (1, 0)? \)
c) Find the derivative of the function \( f(x, y, z) = \cos (xy) + e^{yz} + \ln (zx) \) at \( P_0 = (1, 0, 1/2) \) in the direction given by \( \mathbf{u} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} \).
d) In which directions (if any) is the derivative of
\[
f(x, y) = \frac{x^2 - y^2}{x^2 + y^2}
\]
at \( P_0 = (1, 1) \) equal to zero?
e) The derivative of \( f(x, y, z) \) at a point \( P_0 \) is greatest in the direction of \( \mathbf{v} = \mathbf{i} + \mathbf{j} - \mathbf{k} \). In this direction, the value of its derivative is \( 2\sqrt{3} \). What is \( \nabla f \) at \( P_0 \)? Justify your answers.
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