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.
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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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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