Using a FunctionIn Exercises 67 and 68, (a) find the gradient of the function at P , (b) find a unit normal vector to the level curve f ( x , y ) = c at P , (c) find the tangent line to the level curve f ( x , y ) = c at P , and (d) sketch the level curve, the unit normal vector, and the tangent line in the x y -plane. f ( x , y ) = 4 y sin x − y c = 3 , P ( π 2 , 1 )
Using a FunctionIn Exercises 67 and 68, (a) find the gradient of the function at P , (b) find a unit normal vector to the level curve f ( x , y ) = c at P , (c) find the tangent line to the level curve f ( x , y ) = c at P , and (d) sketch the level curve, the unit normal vector, and the tangent line in the x y -plane. f ( x , y ) = 4 y sin x − y c = 3 , P ( π 2 , 1 )
Solution Summary: The author explains the formula for the gradient of a function f(x,y) at the point
Using a FunctionIn Exercises 67 and 68, (a) find the gradient of the function at
P
, (b) find a unit normal vector to the level curve
f
(
x
,
y
)
=
c
at
P
, (c) find the tangent line to the level curve
f
(
x
,
y
)
=
c
at
P
, and (d) sketch the level curve, the unit normal vector, and the tangent line in the
x
y
-plane.
f
(
x
,
y
)
=
4
y
sin
x
−
y
c
=
3
,
P
(
π
2
,
1
)
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.
Using a Function, (a) find the gradient of the function at P, (b) find a unit normal vector to the level curve f(x, y) = c at P, (c) find the tangent line to the level curve f(x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. f(x, y) = 9x2 − 4y2,
c = 65, P(3, 2)
(a) Find the gradient of f .(b) Evaluate the gradient at the point P.(c) Find the rate of change of f at in the direction of the vector u
(a) Find the gradient of f .(b) Evaluate the gradient at the point P.(c) Find the rate of change of f at P in the direction of the vector u.
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