Question 1

Given the function $$P(t) = \frac{1}{1 + Ae$$^{-kt}$$}$$, at what value of $$P$$ does the maximum value of $$\frac{dP}{dt}$$ occur?

Accepted Answer

$$P = \frac{1}{2}$$

Question 2

For the function $$f(x, y) = x^2$ + y^2$, what is the equation of the contour line at height c$?

Accepted Answer

$$x^2$$ + $$y^2 = c$

Question 3

Which of the following best describes the surface $$g(x, y) = x^2$ - y^2$?

Accepted Answer

A saddle-shaped surface

Question 4

Suppose you have the point $$A = (1, -1, 0). $$Which coordinate is closest to the $$xz-$$plane?

Accepted Answer

The $$y-$$coordinate

Question 5

What is the distance between the points $$A = (1, -1, 0)$$ and $$B = (4, 2, 0) in 3D $$space?

Accepted Answer

$$\sqrt{(4-1)^2$ + (2-(-1))^2$ + (0-0)^2$} = \sqrt{9 + 9} = \sqrt{18}$$

Question 6

Which equation represents a plane parallel to $$z=0$$ but three units above it?

Accepted Answer

$$z = 3$$

Question 7

For the function $$f(x, y) = \sqrt{x^2$ + y^2$}$$, what is the radius of the contour at height $$c$$?

Accepted Answer

$$c$$

Question 8

Which of the following is a function of two variables?

Accepted Answer

Area of a rectangle as a function of length and width

Question 9

If $$f(x, y) = 2x + 3y + 1$$, what is the equation of the contour line at $$c$$?

Accepted Answer

$$2x + 3y + 1 = c$$

Question 10

On a topographic map, what do closely spaced contour lines indicate?

Accepted Answer

A steep slope

Question 11

For the function $$f(x, y) = x^2$ + y^2$, what is the radius of the contour corresponding to c = 9$?

Accepted Answer

$$3$$

Question 12

Which of the following best describes a contour line?

Accepted Answer

A curve connecting points of equal value of a function

Question 13

If you fix $$x = 2 in$$ $$f(x, y) = x^2$ + y^2$, what is the resulting cross-section as a function of y$?

Accepted Answer

$$f(2, y) = 4 + y^2$

Question 14

Which of the following is true about the graph of $$z = x^2$ + y^2$?

Accepted Answer

It is a paraboloid opening upward

Question 15

For the function $$g(x, y) = x^2$ - y^2$, what is the equation of the cross-section when x = 1$?

Accepted Answer

$$g(1, y) = 1 - y^2$

Question 16

Which of the following is a real-world example of a function of two variables?

Accepted Answer

BMI as a function of height and weight

Question 17

What is the equation for the contour line of $$f(x, y) = \sqrt{x^2$ + y^2$} at $$height $$c$$?

Accepted Answer

$$x^2$$ + $$y^2$$ = $$c^2$$

Question 18

Which of the following statements about contour lines is correct?

Accepted Answer

Contour lines are obtained by taking horizontal slices of a surface

Question 19

For the function $$f(x, y) = 2x + 3y + 1$$, what is the contour line when $$c = 4$$?

Accepted Answer

$$2x + 3y + 1 = 4$$

Functions of Several Variables: Graphs, Surfaces, and Contour Diagrams

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