Interpreting directional derivatives A function f and a point P are given. Let θ correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles θ ( with respect to the positive x-axis ) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of θ; call this function g. d. Find the value of θ that maximizes g ( θ ) and find the maximum value. e. Verify that the value of θ that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient . 35 . f ( x , y ) = 2 + x 2 + y 2 ; P ( 3 , 1 )
Interpreting directional derivatives A function f and a point P are given. Let θ correspond to the direction of the directional derivative. a. Find the gradient and evaluate it at P. b. Find the angles θ ( with respect to the positive x-axis ) associated with the directions of maximum increase, maximum decrease, and zero change. c. Write the directional derivative at P as a function of θ; call this function g. d. Find the value of θ that maximizes g ( θ ) and find the maximum value. e. Verify that the value of θ that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient . 35 . f ( x , y ) = 2 + x 2 + y 2 ; P ( 3 , 1 )
Interpreting directional derivativesA function f and a point P are given. Let θ correspond to the direction of the directional derivative.
a. Find the gradient and evaluate it at P.
b. Find the angles θ (with respect to the positive x-axis) associated with the directions of maximum increase, maximum decrease, and zero change.
c. Write the directional derivative at P as a function of θ; call this function g.
d. Find the value of θ that maximizes g(θ) and find the maximum value.
e. Verify that the value of θ that maximizes g corresponds to the direction of the gradient. Verify that the maximum value of g equals the magnitude of the gradient.
Let g(x, y) = 3 + x3 - 6 . x2 - 5 . y2 +1.
a.
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Hampton is a small town on a straight stretch of coastline running north and south. A lighthouse
is located 3 miles offshore directly east of Hampton. The light house has a revolving searchlight
that makes two revolutions per minute. The angle that the beam makes with the east-west line
through Hampton is called o. Find the distance from Hampton to the point where the beam
strikes the shore, as a function of o. Include a sketch.
Determine whether y is a function of x.
y2 =6x
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