The direction of the gradient vector $$\nabla f at $$each point.

Question 1

What does the image above represent in the context of multivariable calculus?

Accepted Answer

A contour plot representing level curves of a function $$z = f(x, y).$$

Question 2

If you move perpendicular to the contour lines in the image, what are you following?

Accepted Answer

The direction of the gradient vector $$
abla f at $$each point.

Question 3

What does the spacing between the contour lines indicate about the surface $$z = f(x, y)$$?

Accepted Answer

Close spacing indicates a steep slope; wide spacing indicates a gentle slope.

Question 4

Suppose you are at a point where the contour lines form closed loops. What type of critical point could this be?

Accepted Answer

A local maximum or minimum.

Question 5

If you were to calculate the gradient $$
abla f at a $$point on this map, in which direction would it point?

Accepted Answer

Perpendicular to the contour line, in the direction of greatest increase.

Question 6

If you walk along a contour line on this map, what happens to the value of $$f(x, y)$$?

Accepted Answer

It stays constant.

Question 7

What real-world application does this type of plot have?

Accepted Answer

It is used in topographic maps to represent elevation.

Question 8

If the contour lines are very close together in a region, what can you say about $$\|
abla f\|$$ there?

Accepted Answer

$$\|
abla f\| is $$large.

Question 9

If you wanted to find the highest point on this surface using the contour plot, what would you look for?

Accepted Answer

The innermost closed contour with the highest value.

Question 10

Suppose you want to estimate the volume under the surface $$z = f(x, y)$$ above a certain region using this plot. Which calculus tool would you use?

Accepted Answer

A double integral.

Multivariable Calculus: Surfaces, Partial Derivatives, and Vector Calculus

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