Question

Calc III Tutorial: Introduction to Multi-Variable Functions

Answer 1

\({\bf{Definitions:}}\)
real-value function: assigns a unique real # w = f(x1, x2...xn) to the set D(x1,x2...xn) where D is the domain (input variable, independent variable) and w is the range (output variable, dependent variable)
interior point: in 2 dimensions, a point (x0,y0) in a region R that is the center of a disc that lies entirely within R
basically, a point where you can draw a circle around it and not go outside R, keep in mind the radius of this circle can be anything positive, however small
boundary point: in 2 dimensions, a point (x0,y0) such that every disc with that point as its center contains both points inside and outside R. the boundary point does not need to be part of R.



in 3 dimensions this definition is modified to make R a sphere rather than a circle

interior of a region: set of interior points in a region
boundary of a region: set of boundary points in a region
open region: a region that only has interior points
closed region: a region that contains only boundary points

special cases:
a set with a half interval (a,b] is not open, nor is it closed
the empty set and the set containing the entire plane are both open and closed

Answer 2

bounded region: a region contained within a disc of finite radius
unbounded region: a region not contained within a disc of finite radius

ex: f(x) = sqrt(y-x^2)
the range of this function is all real #'s greater than 0, since a square root cannot be negative (only considering real #'s at this time)

so this function is basically the parabola y = x^2. since the function contains all the boundary points along y = x^2 but cannot be contained within a circle of finite radius it is considered closed and unbounded

Answer 3

\({\bf{Curves~and~Contours:}}\)



level curve: the set of points where f(x,y) = c
if you are asked to find the level curve at a certain c value just set the function equal to c and solve. the resulting function is your level curve.
graph or surface: the set of all points (x,y,(f(x,y)) in the domain of f
level surface: the set of points where f(x,y,z) = c, basically the same thing as a level curve but in three dimensions

Answer 4

Source material is section 14.1 of Thomas' Calculus, Early Transcendentals, 14th edition by Hass, Heil, Weir, et. al.