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2. For the function z = f(x, y) is known f(2, -1) = -2, f.(2, –1) = -1, f,(2, –1) = 2. Find the following (a) Gradient of f at the point (2, -1) :grad f(2, -1) (b) Directional derivative of f at the point (2, –1) in the direction of the vector -27+37: fa(2, -1). (c) Estimate f(1.99, –1.02) =

2. For the function z = f(x, y) is known f(2, -1) = -2, f.(2, –1) = -1, f,(2, –1) = 2. Find the following (a) Gradient of f at the point (2, -1) :grad f(2, -1) (b) Directional derivative of f at the point (2, –1) in the direction of the vector -27+37: fa(2, -1). (c) Estimate f(1.99, –1.02) =

Advanced Engineering Mathematics
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 the function \( z = f(x, y) \) is known \( f(2, -1) = -2 \), \( f_x(2, -1) = -1 \), \( f_y(2, -1) = 2 \). Find the following: (a) Gradient of \( f \) at the point \( (2, -1) \) : \( \nabla f(2, -1) \) (b) Directional derivative of \( f \) at the point \( (2, -1) \) in the direction of the vector \( -2 \hat{i} + 3 \hat{j} \): \( f_{\vec{u}}(2, -1) \). (c) Estimate \( f(1.99, -1.02) \approx \) (d) Differential of \( f \) at the point \( (2, -1) \)
Transcribed Image Text:For the function \( z = f(x, y) \) is known \( f(2, -1) = -2 \), \( f_x(2, -1) = -1 \), \( f_y(2, -1) = 2 \). Find the following: (a) Gradient of \( f \) at the point \( (2, -1) \) : \( \nabla f(2, -1) \) (b) Directional derivative of \( f \) at the point \( (2, -1) \) in the direction of the vector \( -2 \hat{i} + 3 \hat{j} \): \( f_{\vec{u}}(2, -1) \). (c) Estimate \( f(1.99, -1.02) \approx \) (d) Differential of \( f \) at the point \( (2, -1) \)
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