Chapter 12 REVIEW -- Multivariable Calculus
The following questions are designed to get you thinking
about the material. These are NOT necessarily the type of questions
that you will see on the test; in fact, many probably cannot be
answered adequately using only pen and paper in a one-hour test.
None-the-less, these are the types of things that I'm hoping
you have learned. When I grade your test, I'll be looking for
evidence that you have thought about and deciphered these concepts.
- 1.
- Besides a symbolic formula, what other ways are there
to represent a function of two variables? Which representations
are more useful for which purposes?
- 2.
- Suppose you're looking at a table of values for a function of
two variables (like on page 611, 612, or 630).
As you read across the second row, what are
you looking at? How is this related to the graph of the function?
- 3.
- Given a set of level curves for a function, how can you tell if
the function is linear?
- 4.
- Given a table of values for a function, how can you tell if
the function is linear?
- 5.
- What is a ``cross-section'' (vertical slice)? How do you find
a cross-section from a formula? What does the cross-section
tell you about a graph?
- 6.
- What is a ``level curve'' (contour)? What does the spacing
of the level curves tell you about the graph?
- 7.
- Given a table of values, how can you determine what
cross-sections of the graph might look like?
- 8.
- Given a contour plot, how can you determine what
cross-sections of the graph might look like?
- 9.
- Suppose you are looking at the graph of a function of
two variables (assume it's a nice, differentiable function).
What can you say about the level curves near a maximum
(top of a ``hill'' on the graph)?
What can you say about the cross-sections through the maximum?
- 10.
- What equation describes the x-z plane? the x-y plane?
- 11.
- What does the equation for a horizontal plane look like?
A vertical plane?
- 12.
- Identify at least two forms for the equation of a
(non-vertical) plane. What are the meanings of all the
constants in these equations?
- 13.
- How do you find the distance from a point (x,y,z) to
one of the coordinate planes? How do you find the distance
to one of the coordinate axes?
- 14.
- What is the distance equation in 3D?
What is the equation for a sphere? How are these related?
- 15.
- If you have equations for a sphere and for a plane which is
parallel to some coordinate plane, how would
you find the equation for their intersection (if any)?
What should the intersection look like?
- 16.
- Suppose an equation in 3D (x-y-z-space) does not involve y.
What does this say about the graph? What does it say about the
contour plot?
- 17.
- Suppose you have an equation and its graph. How do you find an
equation whose graph is the same, but shifted?
- 18.
- Is it possible for level curves to cross, if they correspond
to different levels?
- 19.
- Explain why it is impossible to draw the graph of a function
of three variables. If we have a function f(x,y,z)
of three variables and look at contours by setting
f= (const.),
what are we drawing?
- 20.
- What does the formula for a linear function of three variables
look like? What does the formula for a linear function of two
variables look like? Which of these function would have a graph
which is a plane (in 3D)?
- 21.
- Given three points on a plane (in 3D), how could you find an
equation for the plane?
Tamara Olson