Question

Consider the function f : R 2 → R defined by f(x, y) = 4 + x 3 + y 3 − 3xy.

(a)Compute the directional derivative of f at the point (a, b) = ( 1 2 , 1 2 ), in the direction u = ( √ 1 2 , − √ 1 2 ). At the point ( 1 2 , 1 2 ), is u the direction of steepest ascent, steepest descent, or neither? Justify your answer.

(b)Must f attain an absolute minimum and an absolute maximum on the rectangle D = [0, 2] × [0, 4]? Justify your answer.

(c)Calculate the rate of change of f along the curve r(t) = (t, t2 ), at t = −1.

(d) Classify the critical points of f using the second derivative test.

Answer #1

] Consider the function f : R 2 → R defined by f(x, y) = x ln(x
+ 2y). (a) Find the gradient of f(x, y) at the point P(e/3, e/3).
(b) Use the gradient to find the directional derivative of f at
P(e/3, e/3) in the direction of the vector ~u = h−4, 3i. (c) Find a
unit vector (based at P) pointing in the direction in which f
increases most rapidly at P.

s] Consider the function f : R 2 → R defined by f(x, y) = x ln(x
+ 2y). (a) Find the gradient of f(x, y) at the point P(e/3, e/3).
(b) Use the gradient to find the directional derivative of f at
P(e/3, e/3) in the direction of the vector ~u = h−4, 3i. (c) Find a
unit vector (based at P) pointing in the direction in which f
increases most rapidly at P.

Consider the function
F(x,y)=e^((-x^2/4)-(y^2/4)) and the point P(−1,1).
a. Find the unit vectors that give the
direction of steepest ascent and steepest descent at P.
b. Find a vector that points in a direction of
no change in the function at P.

Consider the function F(x,y) = e^(((-x^2)/2)-((y^2)/2)) and the
point P(-3,3).
a. Find the unit vectors that give the direction of the steepest
ascent and the steepest descent at P.
b. Find a vector that points in a direction of no change at P.

For f(x,y,z) = sqrt(35-x^2-4y^2-2z) 1. Find the gradient of
f(x,y,z) 2. Evaluate delta f(x,y,z) 3. Find the unit vectors U+ and
U- , that give the direction of steepest ascent and the steepest
descent respectively.

f(x, y) = 4 + x^3 + y^3 − 3xy
(a,b)=(0.5,0.5)
u = ( √ 1 /2 , − √ 1 /2 )
a) Calculate the rate of change of f along the curve r(t) = (t,
t2 ), at t = −1
b)Classify the critical points of f using the second derivative
test.

6. Consider the function f defined by f (x, y) = ln(x
− y). (a) Determine the natural domain of f. (b) Sketch the level
curves of f for the values k = −2, 0, 2. (c) Find the gradient of f
at the point (2,1), that is ∇f(2,1). (d) In which unit vector
direction, at the point (2,1), is the directional derivative of f
the smallest and what is the directional derivative in that
direction?

For f(x, y) = x2 + 4xy - y2 at the point P(2, 1), a) find the
unit vector u in the direction of steepest ascent; b) find the unit
vector u in the direction of steepest descent; and c) find a unit
vector u that points in the direction of no change in the function.
show all work please
a)
b)
c)

Consider the function f(x, y) = sin(2x − 2y) (a) Solve and find
the gradient of the function.
(b) Find the directional derivative of the function at the point
P(π/2,π/6) in the direction of the vector
v = <sqrt(3), −1>
(c) Compute the unit vector in the direction of the steepest
ascent at A (π/2,π/2)

If f(x, y) = 4 + x^3 + y^3 − 3xy
Must f attain an absolute minimum and an absolute maximum on the
rectangle D = [0, 2] × [0, 4]?

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 5 minutes ago

asked 9 minutes ago

asked 15 minutes ago

asked 18 minutes ago

asked 36 minutes ago

asked 53 minutes ago

asked 58 minutes ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago