Lesson test: Partial derivatives

2.[2p]
Two moles of an ideal gas occupy $0.05$ cubic metres. Using $p = n R T / V$ with $R = 8.314$ , find $\partial p / \partial T$ in pascals per kelvin, rounded to one decimal place.

CorrectNot quite: 332.6

3.[3p]
One mole of an ideal gas is at $T = 300$ K in $V = 0.025$ cubic metres. Using $R = 8.314$ , find $\partial p / \partial V$ in pascals per litre, rounded to the nearest whole number.

CorrectNot quite: -3991

4.[2p]
For the production function $Q = 1.01 L^{3 / 4} K^{1 / 4}$ , find the marginal product of capital $Q_{K}$ at $L = 16$ , $K = 1$ , to two decimal places.

CorrectNot quite: 2.02

5.[2p]
For $f (x, y) = x^{3} y^{2} + x \sin y$ , evaluate the mixed partial $f_{x y} (2, 1)$ , rounded to three decimal places.

CorrectNot quite: 24.540

6.[3p]
For $f (x, y) = \frac{x y (x^{2} - y^{2})}{x^{2} + y^{2}}$ with $f (0, 0) = 0$ , what is $f_{x y} (0, 0)$ , differentiating first in $x$ and then in $y$ ?

CorrectNot quite: -1

7.[2p]
Which condition does Clairaut's theorem need for $f_{x y} (a, b) = f_{y x} (a, b)$ ?

Both mixed partials exist and are continuous on an open disc around $(a, b)$

Correct

Both first partials $f_{x}$ and $f_{y}$ exist at $(a, b)$

The answer is: Both mixed partials exist and are continuous on an open disc around $(a, b)$

The function $f$ is continuous at $(a, b)$

The answer is: Both mixed partials exist and are continuous on an open disc around $(a, b)$

The pure second partials $f_{x x}$ and $f_{y y}$ are equal at $(a, b)$

The answer is: Both mixed partials exist and are continuous on an open disc around $(a, b)$

8.[3p]
Which of these functions satisfy Laplace's equation $u_{x x} + u_{y y} = 0$ ?

Select all that apply

$x y$

Correct

$e^{x} \cos y$

Correct

$x^{3} - 3 x y^{2}$

Correct

$x^{2} + y^{2}$

The answer is: $xy$, $e^x \cos y$, $x^3 - 3xy^2$

$x^{3} - y^{3}$

The answer is: $xy$, $e^x \cos y$, $x^3 - 3xy^2$

$e^{x + y}$

The answer is: $xy$, $e^x \cos y$, $x^3 - 3xy^2$

9.[2p]
The function $g (x, y) = \frac{x y}{x^{2} + y^{2}}$ with $g (0, 0) = 0$ has which property at the origin?

Both partial derivatives exist and equal $0$ , yet $g$ is not continuous there

Correct

Both partial derivatives exist, so $g$ is continuous there

The answer is: Both partial derivatives exist and equal $0$, yet $g$ is not continuous there

Neither partial derivative exists, because $g$ has no limit there

The answer is: Both partial derivatives exist and equal $0$, yet $g$ is not continuous there

The partial derivatives exist but are different, $g_{x} = 0$ and $g_{y} = \frac{1}{2}$

The answer is: Both partial derivatives exist and equal $0$, yet $g$ is not continuous there

Sign in

Partial derivatives

1.[1p]
For $f (x, y) = x^{3} y - 2 x y^{2} + 5$ , evaluate $f_{x} (2, 1)$ .

2.[2p]
Two moles of an ideal gas occupy $0.05$ cubic metres. Using $p = n R T / V$ with $R = 8.314$ , find $\partial p / \partial T$ in pascals per kelvin, rounded to one decimal place.

3.[3p]
One mole of an ideal gas is at $T = 300$ K in $V = 0.025$ cubic metres. Using $R = 8.314$ , find $\partial p / \partial V$ in pascals per litre, rounded to the nearest whole number.

4.[2p]
For the production function $Q = 1.01 L^{3 / 4} K^{1 / 4}$ , find the marginal product of capital $Q_{K}$ at $L = 16$ , $K = 1$ , to two decimal places.

5.[2p]
For $f (x, y) = x^{3} y^{2} + x \sin y$ , evaluate the mixed partial $f_{x y} (2, 1)$ , rounded to three decimal places.

6.[3p]
For $f (x, y) = \frac{x y (x^{2} - y^{2})}{x^{2} + y^{2}}$ with $f (0, 0) = 0$ , what is $f_{x y} (0, 0)$ , differentiating first in $x$ and then in $y$ ?

7.[2p]
Which condition does Clairaut's theorem need for $f_{x y} (a, b) = f_{y x} (a, b)$ ?

8.[3p]
Which of these functions satisfy Laplace's equation $u_{x x} + u_{y y} = 0$ ?

9.[2p]
The function $g (x, y) = \frac{x y}{x^{2} + y^{2}}$ with $g (0, 0) = 0$ has which property at the origin?

1.[1p] For f(x,y)=x3y-2xy2+5, evaluate fx(2,1).

2.[2p] Two moles of an ideal gas occupy 0.05 cubic metres. Using p=nRT/V with R=8.314, find ∂p/∂T in pascals per kelvin, rounded to one decimal place.

3.[3p] One mole of an ideal gas is at T=300 K in V=0.025 cubic metres. Using R=8.314, find ∂p/∂V in pascals per litre, rounded to the nearest whole number.

4.[2p] For the production function Q=1.01L3/4K1/4, find the marginal product of capital QK at L=16, K=1, to two decimal places.

5.[2p] For f(x,y)=x3y2+xsiny, evaluate the mixed partial fxy(2,1), rounded to three decimal places.

6.[3p] For f(x,y)=xy(x2-y2)x2+y2 with f(0,0)=0, what is fxy(0,0), differentiating first in x and then in y?

7.[2p] Which condition does Clairaut's theorem need for fxy(a,b)=fyx(a,b)?

8.[3p] Which of these functions satisfy Laplace's equation uxx+uyy=0?

9.[2p] The function g(x,y)=xyx2+y2 with g(0,0)=0 has which property at the origin?

1.[1p]
For $f (x, y) = x^{3} y - 2 x y^{2} + 5$ , evaluate $f_{x} (2, 1)$ .

2.[2p]
Two moles of an ideal gas occupy $0.05$ cubic metres. Using $p = n R T / V$ with $R = 8.314$ , find $\partial p / \partial T$ in pascals per kelvin, rounded to one decimal place.

3.[3p]
One mole of an ideal gas is at $T = 300$ K in $V = 0.025$ cubic metres. Using $R = 8.314$ , find $\partial p / \partial V$ in pascals per litre, rounded to the nearest whole number.

4.[2p]
For the production function $Q = 1.01 L^{3 / 4} K^{1 / 4}$ , find the marginal product of capital $Q_{K}$ at $L = 16$ , $K = 1$ , to two decimal places.

5.[2p]
For $f (x, y) = x^{3} y^{2} + x \sin y$ , evaluate the mixed partial $f_{x y} (2, 1)$ , rounded to three decimal places.

6.[3p]
For $f (x, y) = \frac{x y (x^{2} - y^{2})}{x^{2} + y^{2}}$ with $f (0, 0) = 0$ , what is $f_{x y} (0, 0)$ , differentiating first in $x$ and then in $y$ ?

7.[2p]
Which condition does Clairaut's theorem need for $f_{x y} (a, b) = f_{y x} (a, b)$ ?

8.[3p]
Which of these functions satisfy Laplace's equation $u_{x x} + u_{y y} = 0$ ?

9.[2p]
The function $g (x, y) = \frac{x y}{x^{2} + y^{2}}$ with $g (0, 0) = 0$ has which property at the origin?