Question 1

Find  if z =  .

Accepted Answer

The answer of Find  if z =  .
...

Question 2

Find  (x, y) and  (x, y) if f(x, y) = ln  .

Accepted Answer

The answer of Find  (x, y) and  (x,...

Question 3

Find the value of the positive constant real number k such that the function  is harmonic for all points in 3-space.

Accepted Answer

The answer of Find the value of the positive constant...

Question 4

Compute and if f(x, y) = xy + sin xy + x² + 5xln y.

Accepted Answer

The answer of Compute  and  if f(x, y)...

Question 5

Calculate and simplify  +  if z =  .

Accepted Answer

The answer of Calculate and simplify  +  if...

Question 6

Let F(x,y) =  (a) Use the definitions of partial derivatives to calculate  (0, 0) and  (0, 0).
(b) Calculate  and  for (x, y) ≠ (0, 0).
(c) Use part (b) and the definitions of partial derivatives to calculate the mixed partial  .

Accepted Answer

The answer of Let F(x,y) =  (a) Use the...

Question 7

Evaluate  (3, 2, 1) if f(x, y, z) = x  (yz).

Accepted Answer

The answer of Evaluate  (3, 2, 1) if f(x,...

Question 8

Find  if w =  .

Accepted Answer

The answer of Find  if w =  .
...

Question 9

Calculate and simplify u_t - u_xx - u_yy for the function u = .

Accepted Answer

The answer of Calculate and simplify u_t - u_xx -...

Question 10

For what value(s) of the constant k is the function f(x, y, z) =  sin(kz) a harmonic function in 3-space? Note that a harmonic function f in 3-space satisfies  +  +  = 0.

Accepted Answer

The answer of For what value(s) of the constant k...

Question 11

Find if z = x sin y, where x = t² + 2t + 1 and y = ln t.

Accepted Answer

The answer of Find  if z = x sin...

Question 12

Find if z = , where x = ln t and y = ln t².

Accepted Answer

The answer of Find  if z =  ,...

Question 13

Find if z = sin(uv)cos(uv), where u = e^t and y = e^2t.

Accepted Answer

The answer of Find  if z = sin(uv)cos(uv), where...

Question 14

Accepted Answer

The answer of 
...

Question 15

Let h(t) = f(x, y), where f(x, y) = + , x = t, y = t². Find (2).

Accepted Answer

The answer of Let h(t) = f(x, y), where f(x,...

Question 16

Let z = f(x, y), where x = u  , and y =  . Use the chain rule to find  at (u, v) = (2,- 1) given that  (2,- 1) = - 7,  (2,- 1) = 5,  (2,- 2) = 3, and  (2,- 2) = -2.

Accepted Answer

The answer of Let z = f(x, y), where x...

Question 17

If W =  , where x = t + sin(t) and y = 2t -1, then the value of  at t = 0 is equal to:

Accepted Answer

The answer of If W =  , where x...

Question 18

Let g be a function of a single variable having continuous second order derivatives, and letu(x, y) = g(y + mx) , for some constant real number m. Determine all values of m such that u(x, y) satisfies the partial differential equation  - 10  + 24  = 0.

Accepted Answer

The answer of Let g be a function of a...

Question 19

Use the chain rule to find the values of and . at (u, v) = (0, 1), where z = x³y⁵,x = u - v, and y = u + v.

Accepted Answer

The answer of Use the chain rule to find the...

Question 20

Assuming that the function f has continuous first partial derivatives and , calculate and simplify f(x²y, x ).

Accepted Answer

The answer of Assuming that the function f has continuous...

Quiz 13: Partial Differentiation

Find if z = .

Find (x, y) and (x, y) if f(x, y) = ln .

Find the value of the positive constant real number k such that the function is harmonic for all points in 3-space.

Compute and if f(x, y) = xy + sin xy + x² + 5xln y.

Calculate and simplify + if z = .

Let F(x,y) = (a) Use the definitions of partial derivatives to calculate (0, 0) and (0, 0). (b) Calculate and for (x, y) ≠ (0, 0). (c) Use part (b) and the definitions of partial derivatives to calculate the mixed partial .

Evaluate (3, 2, 1) if f(x, y, z) = x (yz) .

Find if w = .

Calculate and simplify u_t - u_xx - u_yy for the function u = .

For what value(s) of the constant k is the function f(x, y, z) = sin(kz) a harmonic function in 3-space? Note that a harmonic function f in 3-space satisfies + + = 0.

Find if z = x sin y, where x = t² + 2t + 1 and y = ln t.

Find if z = , where x = ln t and y = ln t².

Find if z = sin(uv) cos(uv) , where u = e^t and y = e^2t.

Let h(t) = f(x, y) , where f(x, y) = + , x = t, y = t². Find (2) .

Let z = f(x, y) , where x = u , and y = . Use the chain rule to find at (u, v) = (2,- 1) given that (2,- 1) = - 7, (2,- 1) = 5, (2,- 2) = 3, and (2,- 2) = -2.

If W = , where x = t + sin(t) and y = 2t -1, then the value of at t = 0 is equal to:

Let g be a function of a single variable having continuous second order derivatives, and letu(x, y) = g(y + mx) , for some constant real number m. Determine all values of m such that u(x, y) satisfies the partial differential equation - 10 + 24 = 0.

Use the chain rule to find the values of and . at (u, v) = (0, 1) , where z = x³y⁵,x = u - v, and y = u + v.

Assuming that the function f has continuous first partial derivatives and , calculate and simplify f(x²y, x ) .

Quiz 13: Partial Differentiation

Find if z = .

Find (x, y) and (x, y) if f(x, y) = ln .

Find the value of the positive constant real number k such that the function is harmonic for all points in 3-space.

Compute and if f(x, y) = xy + sin xy + x2 + 5xln y.

Calculate and simplify + if z = .

Let F(x,y) = (a) Use the definitions of partial derivatives to calculate (0, 0) and (0, 0). (b) Calculate and for (x, y) ≠ (0, 0). (c) Use part (b) and the definitions of partial derivatives to calculate the mixed partial .

Evaluate (3, 2, 1) if f(x, y, z) = x (yz) .

Find if w = .

Calculate and simplify ut - uxx - uyy for the function u = .

For what value(s) of the constant k is the function f(x, y, z) = sin(kz) a harmonic function in 3-space? Note that a harmonic function f in 3-space satisfies + + = 0.

Find if z = x sin y, where x = t2 + 2t + 1 and y = ln t.

Find if z = , where x = ln t and y = ln t2.

Find if z = sin(uv) cos(uv) , where u = et and y = e2t.

Let h(t) = f(x, y) , where f(x, y) = + , x = t, y = t2. Find (2) .

Let z = f(x, y) , where x = u , and y = . Use the chain rule to find at (u, v) = (2,- 1) given that (2,- 1) = - 7, (2,- 1) = 5, (2,- 2) = 3, and (2,- 2) = -2.

If W = , where x = t + sin(t) and y = 2t -1, then the value of at t = 0 is equal to:

Let g be a function of a single variable having continuous second order derivatives, and letu(x, y) = g(y + mx) , for some constant real number m. Determine all values of m such that u(x, y) satisfies the partial differential equation - 10 + 24 = 0.

Use the chain rule to find the values of and . at (u, v) = (0, 1) , where z = x3y5,x = u - v, and y = u + v.

Assuming that the function f has continuous first partial derivatives and , calculate and simplify f(x2y, x ) .

Compute and if f(x, y) = xy + sin xy + x² + 5xln y.

Calculate and simplify u_t - u_xx - u_yy for the function u = .

Find if z = x sin y, where x = t² + 2t + 1 and y = ln t.

Find if z = , where x = ln t and y = ln t².

Find if z = sin(uv) cos(uv) , where u = e^t and y = e^2t.

Let h(t) = f(x, y) , where f(x, y) = + , x = t, y = t². Find (2) .

Use the chain rule to find the values of and . at (u, v) = (0, 1) , where z = x³y⁵,x = u - v, and y = u + v.

Assuming that the function f has continuous first partial derivatives and , calculate and simplify f(x²y, x ) .