Gradients in three dimensions Consider the following functions f, points P, and unit vectors u . a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 61. f ( x , y , z ) = ln ( 1 + x 2 + y 2 + z 2 ) ; P ( 1 , 1 , − 1 ) ; 〈 2 3 ′ 2 3 ′ − 1 3 〉
Gradients in three dimensions Consider the following functions f, points P, and unit vectors u . a. Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector. 61. f ( x , y , z ) = ln ( 1 + x 2 + y 2 + z 2 ) ; P ( 1 , 1 , − 1 ) ; 〈 2 3 ′ 2 3 ′ − 1 3 〉
Gradients in three dimensionsConsider the following functions f, points P, and unit vectorsu.
a.Compute the gradient of f and evaluate it at P
b.Find the unit vector in the direction of maximum increase of f at P.
c.Find the rate of change of the function in the direction of maximum increase at P.
d.Find the directional derivative at P in the direction of the given vector.
61.
f
(
x
,
y
,
z
)
=
ln
(
1
+
x
2
+
y
2
+
z
2
)
;
P
(
1
,
1
,
−
1
)
;
〈
2
3
′
2
3
′
−
1
3
〉
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.
Find the derivative of the function.
F(x) = -1/12/2
x2
f'(x) =
Q4: Write the parametric equation of revolution surface in matrix
form only which generated by rotate a Bezier curve defined by the
coefficient parameter in one plane only, for the x-axis [0,5, 10,4],
y-axis [1,4,2,2] respectively, for u-0.5 and 0 = 45° Note: [the
rotation about y-axis].
Suppose that a parachutist with linear drag (m=50 kg, c=12.5kg/s) jumps from an airplane flying at an altitude of a kilometer with a horizontal velocity of 220 m/s relative to the ground.
a) Write a system of four differential equations for x,y,vx=dx/dt and vy=dy/dt.
b) If theinitial horizontal position is defined as x=0, use Euler’s methods with t=0.4 s to compute the jumper’s position over the first 40 s.
c) Develop plots of y versus t and y versus x. Use the plot to graphically estimate when and where the jumper would hit the ground if the chute failed to open.
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