Find the point(s) on the surface at which the tangent plane is horizontal. z = 8 - x² - y² + 7y Step 1 The equation of the surface can be converted to the general form by defining F(x, y, z) as F(x, y, z) = 8 - x² - y² + 7y - z The gradient of F is the vector given by VF(x, y, z) = F Fx(x, y, z)= = II Step 2 Determine the partial derivatives Fx(x, y, z), F(x, y, z), and F₂ 11 əx = -2x X (8 - x² - y² + 7y - z) (x, y, z)i + F₂(x, y, z)=(8 - x² - y² + 7y − z) -2y -1 -2x Ə F₂(x, y, z) = (8 (8 - x² - y² + 7y - z) əz -2y -1 |(x, y, z)j + F₂(x, y, z)k. y +7 F₂(x, y, z).
Find the point(s) on the surface at which the tangent plane is horizontal. z = 8 - x² - y² + 7y Step 1 The equation of the surface can be converted to the general form by defining F(x, y, z) as F(x, y, z) = 8 - x² - y² + 7y - z The gradient of F is the vector given by VF(x, y, z) = F Fx(x, y, z)= = II Step 2 Determine the partial derivatives Fx(x, y, z), F(x, y, z), and F₂ 11 əx = -2x X (8 - x² - y² + 7y - z) (x, y, z)i + F₂(x, y, z)=(8 - x² - y² + 7y − z) -2y -1 -2x Ə F₂(x, y, z) = (8 (8 - x² - y² + 7y - z) əz -2y -1 |(x, y, z)j + F₂(x, y, z)k. y +7 F₂(x, y, z).
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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Question
Transcribed Image Text:Step 3
Hence, VF(x, y, z) = -2xi + (-2y + 7)j - k.
If F is differentiable at(x, y, z) and VF(x, y, z) = 0, then VF(x, y, z) is normal
For any points at which the tangent plane is horizontal, the gradient vectorVF(x, y, z) is parallel to the
Step 4
Therefore, VF will contain only a
Hence, the coefficients of i and j in the equation for VF will be equal to 0.
That is,
-2x = 0
Step 6
X =
and
-2y + 7 = 0
z = 8-
Step 5
Substitute the values of x and y in the equation for the surface to find z.
z = 8x² - y² + 7y
✓
0
z = 20.25✔
3.5
component at any such points.
0(3.5)2 +7(3.5✔ 3.5 )
20.25
The point at which the tangent plane is horizontal is (0,3,5.16,25
X
normal to the tangent plane through(x, y, z).
).
-axis.
Transcribed Image Text:Find the point(s) on the surface at which the tangent plane is horizontal.
z = 8 - x² - y² + 7y
Step 1
The equation of the surface can be converted to the general form by defining F(x, y, z) as
F(x, y, z) = 8 - x² - y² + 7y - z
The gradient of F is the vector given by
VF(x, y, z) = F
Fx(x, y, z)=
=
II
Step 2
Determine the partial derivatives Fx(x, y, z), F(x, y, z), and F₂
11
əx
=
-2x
X
(8 - x² - y² + 7y - z)
(x, y, z)i +
F₂(x, y, z)=(8 - x² - y² + 7y − z)
-2y
-1
-2x
Ə
F₂(x, y, z) = (8
(8 - x² - y² + 7y - z)
əz
-2y
-1
|(x, y, z)j + F₂(x, y, z)k.
y
+7
F₂(x, y, z).
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