1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
belka [17]
3 years ago
13

(a) (i) Find the gradient of f. (ii) Determine the direction in which f decreases most rapidly at the point (1, −1). At what rat

e is f decreasing? (b) (i) Find the gradient of F. (ii) Find the directional derivative of F at the point (1, 1, −5) in the direction of the vector a = 2 i + 3 j − √ 3 k.
Physics
1 answer:
vitfil [10]3 years ago
8 0

Question:

Problem 14. Let f(x, y) = (x^2)y*(e^(x−1)) + 2xy^2 and F(x, y, z) = x^2 + 3yz + 4xy.

(a) (i) Find the gradient of f.

(ii) Determine the direction in which f decreases most rapidly at the point (1, −1). At what rate is f decreasing?

(b) (i) Find the gradient of F.

(ii) Find the directional derivative of F at the point (1, 1, −5) in the direction of the vector a = 2 i + 3 j − √ 3 k.

Answer:

The answers to the question are

(a) (i)  the gradient of f =  ((y·x² + 2·y·x)·eˣ⁻¹ + 2·y² )i + (x²·eˣ⁻¹+4·y·x) j

(ii) The direction in which f decreases most rapidly at the point (1, −1), ∇f(x, y) = -1·i -3·j is the y direction.

The rate is f decreasing is -3 .

(b) (i) The gradient of F is (2·x+4·y)i + (3·z+4·x)j + 3·y·k

(ii) The directional derivative of F at the point (1, 1, −5) in the direction of the vector a = 2 i + 3 j − √ 3 k is  ñ∙∇F =  4·x +⅟4 (8-3√3)y+ 9/4·z at (1, 1, −5)

4 +⅟4 (8-3√3)+ 9/4·(-5) = -6.549 .

Explanation:

f(x, y) = x²·y·eˣ⁻¹+2·x·y²

The gradient of f = grad f(x, y) = ∇f(x, y) = ∂f/∂x i+  ∂f/∂y j = = (∂x²·y·eˣ⁻¹+2·x·y²)/∂x i+  (∂x²·y·eˣ⁻¹+2·x·y²)/∂y j

= ((y·x² + 2·y·x)·eˣ⁻¹ + 2·y² )i + (x²·eˣ⁻¹+4·y·x) j

(ii) at the point (1, -1) we have  

∇f(x, y) = -1·i -3·j  that is the direction in which f decreases most rapidly at the point (1, −1) is the y direction.  

The rate is f decreasing is -3

(b) F(x, y, z) = x² + 3·y·z + 4·x·y.

The gradient of F is given by grad F(x, y, z)  = ∇F(x, y, z) = = ∂f/∂x i+  ∂f/∂y j+∂f/∂z k = (2·x+4·y)i + (3·z+4·x)j + 3·y·k

(ii) The directional derivative of F at the point (1, 1, −5) in the direction of the vector a = 2·i + 3·j −√3·k

The magnitude of the vector 2·i +3·j -√3·k is √(2²+3²+(-√3)² ) = 4, the unit vector is therefore  

ñ = ⅟4(2·i +3·j -√3·k)  

The directional derivative is given by ñ∙∇F = ⅟4(2·i +3·j -√3·k)∙( (2·x+4·y)i + (3·z+4·x)j + 3·y·k)  

= ⅟4 (2((2·x+4·y))+3(3·z+4·x)- √3∙3·y) = 4·x +⅟4 (8-3√3)y+ 9/4·z at point (1, 1, −5) = -6.549

You might be interested in
Why are frogs said to have two lives?
Stells [14]
They live half their lives in water and most of their life on land
8 0
3 years ago
What girl plays on xbox 14+
elena-s [515]

Answer:

This kinda sus

Explanation:

maybe not on brainly?

8 0
2 years ago
Read 2 more answers
Which of the following statements is TRUE? The side of the Moon facing away from the Earth is in perpetual darkness. The Moon is
Ivanshal [37]

Answer:

The Moon's distance from the Earth varies during its orbit

Explanation:

The correct statement is ,The Moon's distance from the Earth varies during its orbit.

Important point regarding moon:

1 .Moon is a natural satellite of the earth.

2. Moon is the fifth largest satellite in solar system.

3.Second densest  satellite in solar system.

4.Moon rotates about earth.

5.Moon is an astronomical body .

5 0
3 years ago
Stars form from clouds of gas and dust. As a protostar gravitationally contracts within its parent cloud, "conservation of angul
vredina [299]

Answer:

D) True. the protostar rotates more quickly.

Explanation:

If the system is isolated, the angular momentum must be retained.

Initial

        L₀ = I w₀

Final

       L_{f} = I_{f}  w_{f}

      L₀ = L_{f}

      I w₀ = I_{f}w_{f}

     w_{f}  = I /I_{f} w₀

In general, the radius of the cloud decreases significantly to form the star, the moment of inertia must decrease, so the angular velocity must increase

Let's examine the answers

A) False. The opposite happens

B) False. Speed ​​changes

C) False. For this there must be an external force, which does not exist

D) True. You agree with the above

5 0
3 years ago
An uncharged metal sphere, A, is on an insulating base. A second sphere, B, of the same size, shape, and material carrying charg
solong [7]

Answer:

0

Explanation:

  • If we bring the charged sphere B close to, but not touching it , to the uncharged sphere A, as charges can move freely on the conductor, a charge -Q will be built on the outer surface of the sphere A, facing to sphere B.
  • As the sphere A must remain neutral, a charge Q will be built on the surface, on the side farther to the sphere B, as the following condition must be met:

       Q +(-Q) =0.

  • If we now remove sphere B, and place it far away, there will be a charge redistribution within sphere A, making to disappear the separation between Q and -Q.
  • The total charge on sphere A must be 0, as there is no charge transfer from sphere B to sphere A.
3 0
3 years ago
Other questions:
  • A group of hikers hear an echo 3.3 s after they shout. The temperature is 20◦C.
    15·1 answer
  • What are (a) the kinetic energy, (b) the rest energy, and (c) the total energy of a 1.50 g particle with a speed of 0.600 c ?
    8·1 answer
  • Which of the following materials is a good conductor of electric current??? Anyone the options are A.) a rubber tire B.) a silve
    11·1 answer
  • A student drops a ball from the top of a 10-meter tall building. The ball leaves the thrower's hand with a zero speed. What is t
    13·1 answer
  • Spring #1 has a force constant of k, and spring #2 has a force constant of 2k. Both springs are attached to the ceiling. Identic
    6·1 answer
  • As the resistance increased under 25 V, the current blank. Compared to calculated currents, experimental currents proved to be b
    6·1 answer
  • Explica de que tipo es cada oración según la actitud del hablante
    14·1 answer
  • a ball rolls off the edge of a 2m high shelf at a speed of 5 m/s and hits the ground the taken to hit the ground is
    5·1 answer
  • A wave with a frequency of 5Hz travels a distance of 40mm in 2 seconds.What is the speed of the wave​
    5·1 answer
  • A machine does 50 J of Work on a 10 Kg Object. What is the Change in the Objects Total Energy?
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!