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.
(I) Determine the gradient of f.
(II) Calculate the gradient at point P.
(III) Determine a rate of change of f in P in the direction of vector u.
a) f (x, y, z) xe2yz , P(3, 0, 2), u = (2/3, - 2/3, 1/3)
b) f (x, y, z) √x + yz, P(1, 3, 1), u = (2/7, 3/7, 6/7)
Using Properties of the Derivative In Exercise 26, use the properties of the derivative to find the following. (a) r′(t)
(b) d dt [u(t) − 2r(t)] (c) d dt [(3t)r(t)]
(d) d dt [r(t) ∙ u(t)] (e) d dt [r(t) × u(t)]
(f) d dt [u(2t)]
26. r(t) = sin ti + cos tj + tk, u(t) = sin ti + cos tj + 1 t k
* only d ,e, f *
Using the Hough transform i) Develop a general procedure for obtaining the normal representation of a line from its slope-intercept form, y = ax + b. ii) Find the normal representation of the line y = – 2x + 1.
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