Looking ahead—tangent planes Consider the following surfaces f ( x, y, z ) = 0, which may be regarded as a level surface of the function w = f ( x, y z ) . A point P ( a. b, c ) on the surface is also given. a. Find the ( three-dimensional) gradient off and evaluate it at P b. The set of all vectors orthogonal to the gradient with their tails at P form a plane. Find an equation of that plane ( soon to be called the tangent plane ) . 73. f ( x , y , z ) = e x + y − z − 1 = 0 ; P ( 1 , 1 , 2 )
Looking ahead—tangent planes Consider the following surfaces f ( x, y, z ) = 0, which may be regarded as a level surface of the function w = f ( x, y z ) . A point P ( a. b, c ) on the surface is also given. a. Find the ( three-dimensional) gradient off and evaluate it at P b. The set of all vectors orthogonal to the gradient with their tails at P form a plane. Find an equation of that plane ( soon to be called the tangent plane ) . 73. f ( x , y , z ) = e x + y − z − 1 = 0 ; P ( 1 , 1 , 2 )
Solution Summary: The author explains that the gradient of f(x,y,z)=ex+y-z
Looking ahead—tangent planesConsider the following surfaces f(x, y, z) = 0, which may be regarded as a level surface of the function w = f(x, y z). A point P(a. b, c) on the surface is also given.
a.Find the (three-dimensional) gradient off and evaluate it at P
b.The set of all vectors orthogonal to the gradient with their tails at P form a plane. Find an equation of that plane (soon to be called the tangent plane).
73.
f
(
x
,
y
,
z
)
=
e
x
+
y
−
z
−
1
=
0
;
P
(
1
,
1
,
2
)
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
1. Find the gradient of the function.
Reminder that the gradient is a vector. Review the notation requirements below regarding vectors and be sure to use correct notation so that you are not marked down.
f(x,y) = x^2 + y^2 - 10
upside down triangle f(x,y) = ??
2. Given a point P on the plane and a normal vector n , find the equation of the plane.
Review the notation requirements below regarding planes and be sure to use correct notation so that you are not marked down.
P(3,0,1) n = (-2,-4,6)
3. Describe or sketch the level curve z=4 for the function below.
z = sqrt (x^2 + y^2 - 9)
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