Lesson test: Functions of several variables

Functions of several variables

1.[2p]
Two moles of an ideal gas at $T = 300$ K occupy $V = 0.05$ cubic metres. Using $p = n R T / V$ with $R = 8.314$ , find the pressure in kilopascals, rounded to one decimal place.

CorrectNot quite: 99.8

2.[2p]
Use the wind chill formula $W = 13.12 + 0.6215 T - 11.37 v^{0.16} + 0.3965 T v^{0.16}$ to find the wind chill at $T = - 10$ °C and $v = 40$ km/h, rounded to one decimal place.

CorrectNot quite: -20.8

3.[1p]
What is the domain of $g (x, y) = \ln (x - y)$ ?

Every point strictly below the line $y = x$

Correct

Every point strictly above the line $y = x$

The answer is: Every point strictly below the line $y = x$

Every point on or below the line $y = x$

The answer is: Every point strictly below the line $y = x$

The whole plane except the line $y = x$

The answer is: Every point strictly below the line $y = x$

4.[2p]
The graph of $f (x, y) = \sqrt{25 - x^{2} - y^{2}}$ is a dome. Its level curve at height $3$ is a circle centred at the origin. What is its radius?

CorrectNot quite: 4

5.[3p]
A factory follows the Cobb-Douglas function $Q = 1.01 L^{3 / 4} K^{1 / 4}$ . Labour is cut to one quarter of its current value. By what factor must capital be multiplied to keep output unchanged?

CorrectNot quite: 64

6.[2p]
On a 1:50,000 map with contours every $10$ metres, a route crosses six contour gaps in $3$ mm of paper. What is its average gradient, as a decimal?

CorrectNot quite: 0.4

7.[2p]
For $f (x, y) = x^{2} + y^{2}$ at the point $(1, 2)$ , take a step of length $h$ due north, to $(1, 2 + h)$ . What number does the difference quotient $(f (1, 2 + h) - f (1, 2)) / h$ approach as $h$ shrinks to zero?

CorrectNot quite: 4

8.[2p]
Match each family of level curves to the quantity held constant along it.

Isotherms
Isobars
Isoquants
Contours

pressure
height
output
temperature

Show the answer

Isotherms: temperature Isobars: pressure Isoquants: output Contours: height

9.[2p]
Why can the derivative of one variable calculus not be reused unchanged for a function $f (x, y)$ ?

A step in the input plane has a direction as well as a length, and the rate of change depends on the direction

Correct

Functions of two variables are never continuous, so no limit exists

The answer is: A step in the input plane has a direction as well as a length, and the rate of change depends on the direction

The graph of $f (x, y)$ is a surface, and surfaces have no slope in any direction

The answer is: A step in the input plane has a direction as well as a length, and the rate of change depends on the direction

Difference quotients of $f (x, y)$ always tend to zero, so every derivative would vanish

The answer is: A step in the input plane has a direction as well as a length, and the rate of change depends on the direction

1.[2p] Two moles of an ideal gas at T=300 K occupy V=0.05 cubic metres. Using p=nRT/V with R=8.314, find the pressure in kilopascals, rounded to one decimal place.

2.[2p] Use the wind chill formula W=13.12+0.6215T-11.37v0.16+0.3965Tv0.16 to find the wind chill at T=-10 °C and v=40 km/h, rounded to one decimal place.

3.[1p] What is the domain of g(x,y)=ln(x-y)?

4.[2p] The graph of f(x,y)=25-x2-y2 is a dome. Its level curve at height 3 is a circle centred at the origin. What is its radius?

5.[3p] A factory follows the Cobb-Douglas function Q=1.01L3/4K1/4. Labour is cut to one quarter of its current value. By what factor must capital be multiplied to keep output unchanged?

6.[2p] On a 1:50,000 map with contours every 10 metres, a route crosses six contour gaps in 3 mm of paper. What is its average gradient, as a decimal?

7.[2p] For f(x,y)=x2+y2 at the point (1,2), take a step of length h due north, to (1,2+h). What number does the difference quotient (f(1,2+h)-f(1,2))/h approach as h shrinks to zero?

8.[2p] Match each family of level curves to the quantity held constant along it.

9.[2p] Why can the derivative of one variable calculus not be reused unchanged for a function f(x,y)?

1.[2p]
Two moles of an ideal gas at $T = 300$ K occupy $V = 0.05$ cubic metres. Using $p = n R T / V$ with $R = 8.314$ , find the pressure in kilopascals, rounded to one decimal place.

2.[2p]
Use the wind chill formula $W = 13.12 + 0.6215 T - 11.37 v^{0.16} + 0.3965 T v^{0.16}$ to find the wind chill at $T = - 10$ °C and $v = 40$ km/h, rounded to one decimal place.

3.[1p]
What is the domain of $g (x, y) = \ln (x - y)$ ?

4.[2p]
The graph of $f (x, y) = \sqrt{25 - x^{2} - y^{2}}$ is a dome. Its level curve at height $3$ is a circle centred at the origin. What is its radius?

5.[3p]
A factory follows the Cobb-Douglas function $Q = 1.01 L^{3 / 4} K^{1 / 4}$ . Labour is cut to one quarter of its current value. By what factor must capital be multiplied to keep output unchanged?

6.[2p]
On a 1:50,000 map with contours every $10$ metres, a route crosses six contour gaps in $3$ mm of paper. What is its average gradient, as a decimal?

7.[2p]
For $f (x, y) = x^{2} + y^{2}$ at the point $(1, 2)$ , take a step of length $h$ due north, to $(1, 2 + h)$ . What number does the difference quotient $(f (1, 2 + h) - f (1, 2)) / h$ approach as $h$ shrinks to zero?

8.[2p]
Match each family of level curves to the quantity held constant along it.

9.[2p]
Why can the derivative of one variable calculus not be reused unchanged for a function $f (x, y)$ ?