Lesson test: The chain rule

The chain rule

1.[2p]
Let $z = x^{2} + 3 x y$ with $x = 2 t$ and $y = t^{2}$ . Use the chain rule to find $d z / d t$ at $t = 1$ .

CorrectNot quite: 26

2.[3p]
One mole of ideal gas, $P = R T / V$ with $R = 8.314$ J/(mol K), is at $350$ K in $0.03$ m³. The temperature rises at $1$ K per second and the volume at $0.0001$ m³ per second. At what rate is the pressure changing, in pascals per second? Round to the nearest whole number.

CorrectNot quite: -46

3.[2p]
Let $z = x y + y^{2}$ with $x = s + t$ and $y = s - t$ . Find $\partial z / \partial t$ at $(s, t) = (3, 1)$ .

CorrectNot quite: -6

4.[3p]
For $f (x, y) = x y^{2}$ in polar coordinates $x = r \cos θ$ , $y = r \sin θ$ , find $f_{θ}$ at the point $(x, y) = (3, 4)$ .

CorrectNot quite: 8

5.[2p]
For the same $f (x, y) = x y^{2}$ , find $f_{r}$ at the point $(x, y) = (3, 4)$ . Give it to one decimal place.

CorrectNot quite: 28.8

6.[2p]
The point $(1, 2)$ lies on the curve $x^{2} y + y^{3} = 10$ . Use $d y / d x = - F_{x} / F_{y}$ to find the slope of the curve there. Give it to three decimal places.

CorrectNot quite: -0.308

7.[2p]
The sphere $x^{2} + y^{2} + z^{2} = 14$ passes through $(1, 2, 3)$ . Treating $z$ as a function of $x$ and $y$ near that point, find $\partial z / \partial x$ there. Give it to three decimal places.

CorrectNot quite: -0.333

8.[2p]
For $f (x, y) = x^{2} y / (x^{2} + y^{2})$ with $f (0, 0) = 0$ , both partials at the origin are $0$ , yet along the path $x = y = t$ the derivative of $f$ at $t = 0$ is $\frac{1}{2}$ . Why does the chain rule not apply?

$f$ is not differentiable at the origin, so there is no tangent plane for the derivation to use

Correct

$f$ is not continuous at the origin

The answer is: $f$ is not differentiable at the origin, so there is no tangent plane for the derivation to use

The path $x = y = t$ is not differentiable at $t = 0$

The answer is: $f$ is not differentiable at the origin, so there is no tangent plane for the derivation to use

The chain rule only applies to paths along the coordinate axes

The answer is: $f$ is not differentiable at the origin, so there is no tangent plane for the derivation to use

9.[3p]
Which of these statements are true?

Select all that apply

If $w$ depends on three intermediate variables that each depend on $s$ and $t$ , then $\partial w / \partial s$ is a sum of three products

Correct

If $z = f (x, y)$ with $x$ and $y$ functions of $s$ and $t$ , then $\partial z / \partial s = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s}$ , since the $\partial x$ cancels

The answer is: If $w$ depends on three intermediate variables that each depend on $s$ and $t$ , then $\partial w / \partial s$ is a sum of three products, On the folium $x^{3} + y^{3} = 6 x y$ , both $F_{x}$ and $F_{y}$ vanish at the origin, where the curve crosses itself, Where $F_{y} = 0$ but $F_{x} \neq 0$ on a curve $F (x, y) = 0$ , the tangent is vertical

On the folium $x^{3} + y^{3} = 6 x y$ , both $F_{x}$ and $F_{y}$ vanish at the origin, where the curve crosses itself

Correct

For a curve $F (x, y) = 0$ , the slope is $d y / d x = F_{x} / F_{y}$ wherever $F_{y} \neq 0$

Where $F_{y} = 0$ but $F_{x} \neq 0$ on a curve $F (x, y) = 0$ , the tangent is vertical

Correct

1.[2p] Let z=x2+3xy with x=2t and y=t2. Use the chain rule to find dz/dt at t=1.

2.[3p] One mole of ideal gas, P=RT/V with R=8.314 J/(mol K), is at 350 K in 0.03 m³. The temperature rises at 1 K per second and the volume at 0.0001 m³ per second. At what rate is the pressure changing, in pascals per second? Round to the nearest whole number.

3.[2p] Let z=xy+y2 with x=s+t and y=s−t. Find ∂z/∂t at (s,t)=(3,1).

4.[3p] For f(x,y)=xy2 in polar coordinates x=rcosθ, y=rsinθ, find fθ at the point (x,y)=(3,4).

5.[2p] For the same f(x,y)=xy2, find fr at the point (x,y)=(3,4). Give it to one decimal place.

6.[2p] The point (1,2) lies on the curve x2y+y3=10. Use dy/dx=−Fx/Fy to find the slope of the curve there. Give it to three decimal places.

7.[2p] The sphere x2+y2+z2=14 passes through (1,2,3). Treating z as a function of x and y near that point, find ∂z/∂x there. Give it to three decimal places.

8.[2p] For f(x,y)=x2y/(x2+y2) with f(0,0)=0, both partials at the origin are 0, yet along the path x=y=t the derivative of f at t=0 is 12. Why does the chain rule not apply?

9.[3p] Which of these statements are true?

1.[2p]
Let $z = x^{2} + 3 x y$ with $x = 2 t$ and $y = t^{2}$ . Use the chain rule to find $d z / d t$ at $t = 1$ .

2.[3p]
One mole of ideal gas, $P = R T / V$ with $R = 8.314$ J/(mol K), is at $350$ K in $0.03$ m³. The temperature rises at $1$ K per second and the volume at $0.0001$ m³ per second. At what rate is the pressure changing, in pascals per second? Round to the nearest whole number.

3.[2p]
Let $z = x y + y^{2}$ with $x = s + t$ and $y = s - t$ . Find $\partial z / \partial t$ at $(s, t) = (3, 1)$ .

4.[3p]
For $f (x, y) = x y^{2}$ in polar coordinates $x = r \cos θ$ , $y = r \sin θ$ , find $f_{θ}$ at the point $(x, y) = (3, 4)$ .

5.[2p]
For the same $f (x, y) = x y^{2}$ , find $f_{r}$ at the point $(x, y) = (3, 4)$ . Give it to one decimal place.

6.[2p]
The point $(1, 2)$ lies on the curve $x^{2} y + y^{3} = 10$ . Use $d y / d x = - F_{x} / F_{y}$ to find the slope of the curve there. Give it to three decimal places.

7.[2p]
The sphere $x^{2} + y^{2} + z^{2} = 14$ passes through $(1, 2, 3)$ . Treating $z$ as a function of $x$ and $y$ near that point, find $\partial z / \partial x$ there. Give it to three decimal places.

8.[2p]
For $f (x, y) = x^{2} y / (x^{2} + y^{2})$ with $f (0, 0) = 0$ , both partials at the origin are $0$ , yet along the path $x = y = t$ the derivative of $f$ at $t = 0$ is $\frac{1}{2}$ . Why does the chain rule not apply?

9.[3p]
Which of these statements are true?