Calculus Volume 3 · free preview
Functions of Several Variables
Functions of several variables assign outputs to pairs or triples of inputs, graphed as surfaces over the plane, with level curves and traces revealing the geometry and the domain questions paralleling the single-variable case.
1. Functions of two variables
Functions of two variables
2. Level curves and traces
Level curves and traces
3. Domains in the plane
Domains in the plane
4. Functions in applications
Functions in applications
5. Function evaluation
For f(x, y) = x^2 y + ln y, what is f(1, 2)?
6. Level curves of the paraboloid
The level curves of z = x^2 + y^2 are which family of curves?
7. Traces
The trace of the surface z = f(x, y) in the plane y = 0 is which curve?
8. Domain restriction
The function f(x, y) = root of y minus x squared has which domain?
9. Level curves versus traces
What distinguishes a level curve from a trace of a surface z = f(x, y)?
10. Three variables name
A function of three variables assigns values to triples, and its level sets are surfaces. As two words, what are they called? Answer like answer keys.
11. Saddle trace direction
The saddle z = x^2 - y^2 has a trace in y = 0 opening upward and a trace in x = 0 opening which direction? Answer with one word.
12. Domain boundary inclusion
The domain y at least x squared includes its boundary. As one word, what kind of region is it? Answer in lowercase.
13. Contour map spacing
Closely spaced level curves on a contour map indicate which surface feature?
14. Range of the paraboloid
The range of z = x^2 + y^2 is z at least which number? Answer with a number.
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