7. Suppose that f (x, y) = e" cos TY and u? – v², y = u? + v?. x = Using the chain rule, compute the values of af af ди at the point (u, v) = (1, 1). Solution. • According to the chain rule af Əx af Əy af dy dy du' af ôf Əx fe ди Əx du dv Əx dv dy dv • We have af e cos TY, af = -Te" sin TY dy and ду dy 2и, -2v, 2и, = 2v. %3D ди du • At (u, v) = (1,1) we have (x, Y) (0, 2) and fe e° cos 27 = 1, af — те' sin 2т — ду ду 0, dy 2, du 2, -2, = 2. • It follows that af af = 1. 2+0· 2 = 2, = 1.(-2) +0 ·2 = -2. dv
7. Suppose that f (x, y) = e" cos TY and u? – v², y = u? + v?. x = Using the chain rule, compute the values of af af ди at the point (u, v) = (1, 1). Solution. • According to the chain rule af Əx af Əy af dy dy du' af ôf Əx fe ди Əx du dv Əx dv dy dv • We have af e cos TY, af = -Te" sin TY dy and ду dy 2и, -2v, 2и, = 2v. %3D ди du • At (u, v) = (1,1) we have (x, Y) (0, 2) and fe e° cos 27 = 1, af — те' sin 2т — ду ду 0, dy 2, du 2, -2, = 2. • It follows that af af = 1. 2+0· 2 = 2, = 1.(-2) +0 ·2 = -2. dv
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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Can you explain the steps in computing the circled part. Baby steps pls
Transcribed Image Text:7. Suppose that
f (x, y) = e® cos TY
and
x = u? – v?,
y = u? + v².
Using the chain rule, compute the values of
af
af
ди
at the point (u, v) = (1, 1).
Solution.
• According to the chain rule
af əx
dv
ôf ðy
+
af dy
af
af əx
Əx du
af
Əx dv
• We have
af
af
e cos TY,
-Te" sin TY
ду
and
ду
ду
2u,
2и,
-2v,
2v.
du
ди
dv
• At (u, v) = (1,1) we have (x, y)
(0, 2) and
af
e° cos 27
af
—те' sin 2т 3D0,
1,
ду
dy
ду
2,
ди
= 2.
2,
• It follows that
af
= 1. 2+0. 2 = 2,
ди
af
= 1. (-2) +0 · 2 = -2.
dv
||
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