1. Consider the function f : R² → R such that f(x, y) = 2xy² + x² − 2xy = y + 1.(a) Write down the gradient of f, Vf(x, y).(b) Determine the equation of the tangent plane to f at the point (1,1). Express yourfinal answer in the form z = ax + by + c.(c) Calculate the directional derivative of f at (1,1) in the direction of (2, -1)T.

Answered: Image /qna-images/answer/9eab60f4-baa7-46f5-8f8b-d5ad9a559d89.jpg

Answered: 1. Consider the function f : R² → R…

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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Similar questions

Find the gradient of the function w = 1/(√1 − x2 − y2 − z2), and the maximum value of the directional derivative at the point (0, 0, 0).
Find the gradient of the function f(x, y) = (x + y)/(y + 1),and the maximum value of the directional derivative at the point (0, 1).
Given f(x,y,z) = x2 + y2 + z2, find the maximum value of the directional derivative (df/ds) at the point (3, 0, 4) by using the gradient of f.
Find the gradient of f(x, y) = y ln x + xy2 at the point (1, 2).
Compute the directional derivative of the function f(x, y, z) = 3xyz at the point P = (1, −3, −1) in the direction directly towards Q = (0, −1, −3).
Let f(x,y)=ln(5x+4y). Find the directional derivative of f at (1,0) in the direction of the line y=2x−2.
Let f(x,y) = 2x^2 + xy −y^2. Prove that the directional derivative of f(x,y) at point x = (3,−2) in the direction v = i − j is 3 √2
Find the gradient of the function z = cos(x2 + y2), at the given point (3, −4)
Let f(x,y)=x3- x - 3x2y + y3 (a) At the point (x, y) = (1, 2), in what direction does f increase at the greatest rate? (b) Calculate the directional derivative of f (x, y) at (1, 2) in the direction ⟨3, -1⟩. (c) Find the equation of the tangent plane to f at the point (1, 2). (d) Use linear approximation at (1,2) to estimate f(1.2, 2.1).
Prove that the directional derivative of f(x,y) = y^4+2xy^3+x^2y^2 at point (0,1) in the direction v = i +j is 3√2.

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