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1. Given the function z = f(x, y) = -x3 + 4x²y - 3xy² +8: a. Find the directional derivatives at the domain point (xo, yo) = (2, 1) in the directions of the vectors v = <4, -3> and w<5, 1>. Clearly show all the key steps to produce the results! b. What is the highest value of the directional derivative for this function at this domain point? In what direction in the domain plane does it occur? c. What are the directions of the function's level contour at this location and what is its value? d. Plot the vectors v and w from part a, and the information found in parts b & c in the xy - plane provided above to represent the behavior of the function in the neighborhood of the given domain point.

1. Given the function z = f(x, y) = -x3 + 4x²y - 3xy² +8: a. Find the directional derivatives at the domain point (xo, yo) = (2, 1) in the directions of the vectors v = <4, -3> and w<5, 1>. Clearly show all the key steps to produce the results! b. What is the highest value of the directional derivative for this function at this domain point? In what direction in the domain plane does it occur? c. What are the directions of the function's level contour at this location and what is its value? d. Plot the vectors v and w from part a, and the information found in parts b & c in the xy - plane provided above to represent the behavior of the function in the neighborhood of the given domain point.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.3: Lines And Planes
Problem 18EQ
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1. Given the function z = f(x, y) = -x3 + 4x²y - 3xy² +8: a. Find the directional derivatives at the domain point (xo, yo) = (2, 1) in the directions of the vectors v = <4, -3> and w<5, 1>. Clearly show all the key steps to produce the results! b. What is the highest value of the directional derivative for this function at this domain point? In what direction in the domain plane does it occur? c. What are the directions of the function's level contour at this location and what is its value? d. Plot the vectors v and w from part a, and the information found in parts b & c in the xy - plane provided above to represent the behavior of the function in the neighborhood of the given domain point.
Transcribed Image Text:1. Given the function z = f(x, y) = -x3 + 4x²y - 3xy² +8: a. Find the directional derivatives at the domain point (xo, yo) = (2, 1) in the directions of the vectors v = <4, -3> and w<5, 1>. Clearly show all the key steps to produce the results! b. What is the highest value of the directional derivative for this function at this domain point? In what direction in the domain plane does it occur? c. What are the directions of the function's level contour at this location and what is its value? d. Plot the vectors v and w from part a, and the information found in parts b & c in the xy - plane provided above to represent the behavior of the function in the neighborhood of the given domain point.
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