Lesson test: Tangent planes and linear approximation

Tangent planes and linear approximation

1.[2p]
The tangent plane to $z = x^{2} y$ at the point above $(2, 1)$ can be written $z = a x + b y + c$ . What is $c$ ?

CorrectNot quite: -8

2.[2p]
Use the linear approximation of $f (x, y) = \sqrt{x^{2} + y^{2}}$ at $(6, 8)$ to estimate $\sqrt{{6.1}^{2} + {7.9}^{2}}$ . Give the estimate to two decimal places.

CorrectNot quite: 9.98

3.[2p]
Use the linear approximation of $f (x, y) = \ln (x + 2 y)$ at $(1, 0)$ to estimate $f (1.05, 0.02)$ . Give the estimate to two decimal places.

CorrectNot quite: 0.09

4.[2p]
For $f (x, y) = x y$ , the linear approximation at $(3, 2)$ is used to estimate $f (3.1, 1.9)$ . What is the true value minus the estimate?

CorrectNot quite: -0.01

5.[2p]
For $z = x^{2} y^{3}$ at $(2, 1)$ , compute the total differential $d z$ when $d x = 0.01$ and $d y = - 0.02$ .

CorrectNot quite: -0.2

6.[3p]
A cylinder has measured radius $10 \pm 0.2$ cm and height $20 \pm 0.1$ cm. Using the total differential, what is the maximum relative error in its volume $V = π r^{2} h$ , as a percentage? Give it to one decimal place.

CorrectNot quite: 4.5

7.[3p]
A pendulum gives $g = 4 π^{2} L / T^{2}$ . The length is $L = 1.000 \pm 0.005$ m and the period is $T = 2.00 \pm 0.02$ s. What is the maximum relative error in $g$ , as a percentage? Give it to one decimal place.

CorrectNot quite: 2.5

8.[2p]
Why is $f (x, y) = \frac{x y}{x^{2} + y^{2}}$ , with $f (0, 0) = 0$ , not differentiable at the origin, even though $f_{x} (0, 0) = f_{y} (0, 0) = 0$ ?

The only candidate plane is $z = 0$ , and at $(h, h)$ its error stays $\frac{1}{2}$ while the distance shrinks to $0$

Correct

Its partial derivatives at the origin are not equal to each other

The answer is: The only candidate plane is $z = 0$, and at $(h, h)$ its error stays $\tfrac{1}{2}$ while the distance shrinks to $0$

Its mixed partials disagree at the origin

The answer is: The only candidate plane is $z = 0$, and at $(h, h)$ its error stays $\tfrac{1}{2}$ while the distance shrinks to $0$

Its linear approximation at the origin has an error that shrinks faster than the distance, but only along the axes

The answer is: The only candidate plane is $z = 0$, and at $(h, h)$ its error stays $\tfrac{1}{2}$ while the distance shrinks to $0$

9.[3p]
Which of these statements about a function of two variables are true?

Select all that apply

If $f$ is differentiable at a point, it is continuous there

Correct

If $f_{x}$ and $f_{y}$ are continuous near a point, $f$ is differentiable there

Correct

If $f_{x}$ and $f_{y}$ both exist at a point, $f$ is differentiable there

The answer is: If $f$ is differentiable at a point, it is continuous there, If $f_x$ and $f_y$ are continuous near a point, $f$ is differentiable there, If $f$ is differentiable at a point, the coefficients of its tangent plane are $f_x$ and $f_y$ at that point

If $f$ is differentiable at a point, its partial derivatives must be continuous there

If $f$ is differentiable at a point, the coefficients of its tangent plane are $f_{x}$ and $f_{y}$ at that point

Correct

1.[2p] The tangent plane to z=x2y at the point above (2,1) can be written z=ax+by+c. What is c?

2.[2p] Use the linear approximation of f(x,y)=x2+y2 at (6,8) to estimate 6.12+7.92. Give the estimate to two decimal places.

3.[2p] Use the linear approximation of f(x,y)=ln(x+2y) at (1,0) to estimate f(1.05,0.02). Give the estimate to two decimal places.

4.[2p] For f(x,y)=xy, the linear approximation at (3,2) is used to estimate f(3.1,1.9). What is the true value minus the estimate?

5.[2p] For z=x2y3 at (2,1), compute the total differential dz when dx=0.01 and dy=-0.02.

6.[3p] A cylinder has measured radius 10±0.2 cm and height 20±0.1 cm. Using the total differential, what is the maximum relative error in its volume V=πr2h, as a percentage? Give it to one decimal place.

7.[3p] A pendulum gives g=4π2L/T2. The length is L=1.000±0.005 m and the period is T=2.00±0.02 s. What is the maximum relative error in g, as a percentage? Give it to one decimal place.

8.[2p] Why is f(x,y)=xyx2+y2, with f(0,0)=0, not differentiable at the origin, even though fx(0,0)=fy(0,0)=0?

9.[3p] Which of these statements about a function of two variables are true?

1.[2p]
The tangent plane to $z = x^{2} y$ at the point above $(2, 1)$ can be written $z = a x + b y + c$ . What is $c$ ?

2.[2p]
Use the linear approximation of $f (x, y) = \sqrt{x^{2} + y^{2}}$ at $(6, 8)$ to estimate $\sqrt{{6.1}^{2} + {7.9}^{2}}$ . Give the estimate to two decimal places.

3.[2p]
Use the linear approximation of $f (x, y) = \ln (x + 2 y)$ at $(1, 0)$ to estimate $f (1.05, 0.02)$ . Give the estimate to two decimal places.

4.[2p]
For $f (x, y) = x y$ , the linear approximation at $(3, 2)$ is used to estimate $f (3.1, 1.9)$ . What is the true value minus the estimate?

5.[2p]
For $z = x^{2} y^{3}$ at $(2, 1)$ , compute the total differential $d z$ when $d x = 0.01$ and $d y = - 0.02$ .

6.[3p]
A cylinder has measured radius $10 \pm 0.2$ cm and height $20 \pm 0.1$ cm. Using the total differential, what is the maximum relative error in its volume $V = π r^{2} h$ , as a percentage? Give it to one decimal place.

7.[3p]
A pendulum gives $g = 4 π^{2} L / T^{2}$ . The length is $L = 1.000 \pm 0.005$ m and the period is $T = 2.00 \pm 0.02$ s. What is the maximum relative error in $g$ , as a percentage? Give it to one decimal place.

8.[2p]
Why is $f (x, y) = \frac{x y}{x^{2} + y^{2}}$ , with $f (0, 0) = 0$ , not differentiable at the origin, even though $f_{x} (0, 0) = f_{y} (0, 0) = 0$ ?

9.[3p]
Which of these statements about a function of two variables are true?