ду Assume that the relation In (x°z°12) + 6(x – 1)y + 512xz + 729y°z=0 defines y implicitly as a function of x and z; that is y = y(x,z). Compute at dz (x,Z) = (1, – 1). ду (Simplify your answer.) %3D dz
ду Assume that the relation In (x°z°12) + 6(x – 1)y + 512xz + 729y°z=0 defines y implicitly as a function of x and z; that is y = y(x,z). Compute at dz (x,Z) = (1, – 1). ду (Simplify your answer.) %3D dz
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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Transcribed Image Text:3
Assume that the relation In (x°z°12) +6(x - 1)y + 512xz + 729y°z=0 defines y implicitly as a function of x and z; that is y = y(x,z). Compute
at
(x,z) = (1, – 1).
ду
(Simplify your answer.)
dz
II
Transcribed Image Text:Assume that the relation In (x°z"28) + 4(x - 1)y + 1728xz + 216y°z=0 defines y implicitly as a function of x and z;
ду
that is y = y(x,z). Compute
at (x,z) = (1, – 1).
dz
If the equation F(x,y,z) = 0 defines y as a function of x and z, say y = y(x,z) near a point Po (Xo-Yo.Zo) at which
= 0, and if F(x,y,z) has continuous first order partial derivatives near Po, then the partial derivatives
Fx (*0,Yo Zo)
F2 (*o,YoZo)
ду
ду
and
at (Xo,Zo) exist and are given by
ду
ду
and
dz
provided
%3D
дх
dz
dx
Fy (Xo Yo-Zo) #0.
+216y°z. Let Po (Xo Yo.Zo) = (1.Yo- - 1) be the
point at which F (Xp,Yo,zo) = 0. Substitute (x,z) = (1, - 1) into In (x°z'28) + 4(x - 1)y + 1728xz + 216y°z=0. Then
Define a function F(x,y,z) = In (x°z"28) + 4(x - 1)y + 1728xz +
solve the resulting equation to find the y-coordinate.
Yo = -2 (Simplify your answer.)
Therefore, Po is the point (1, – 2, – 1). Find F,(x,y,z) and F_(x,y,z).
Fy(x.y.z)
= 4(x– 1) + 648y²z
F2(x.y.z) =
1728
+ 1728x + 216y°
Find Fy(1, - 2, - 1).
Fy(x.y,z) = 4(x- 1) + 648y°z
Fy(1, - 2, – 1) :
= 4(1- 1) + 648( – 2)(- 1)
= - 2592
(Simplify your answer.)
Find F2(1, – 2, – 1).
1728
F¿(x,y,z) =
+ 1728x +
216y3
1728
F2(1, – 2, – 1) =
+ 1728(1) +216(- 2)°
- 1
- 1728
(Simplify your answer.)
ду
at (x,z) = (1, - 1) exists. Find
dz
ду
Since F,(1, - 2, – 1) = - 2592 +0, the partial derivative
dz
ду
F2(1 - 2, – 1)
2
(Simplify your answer.)
dz
Fy(1, - 2, – 1)
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