Multivariable Calculus – Questions/Answers
Distance between two points (a,b,c) and (x,y,z) ✔️Ans - d = sqrt((x-a)^2 +
(y-b)^2 + (z-c)^2)
cross sections vs level curves ✔️Ans - cross sections: vertical slices of the
graph of f(x,y) formed using vertical planes x=c (f(c,y)=z) or y=c (f(x,c)=z).
Level curves: horizontal slices of f(x,y) using horizontal planes z=c (f(x,y) =c).
Both used to plot graphs of functions with multiple variables.
contour diagrams ✔️Ans - a family of graphs of the equation f(x,y) = c
plotted in the xy-plane, for set values of c, usually labeled by the values. Values
of c usually plotted in equal increments. (like topographical map)
m and n ✔️Ans - m= slope in positive x direction dz/dx (holding y constant)
n = slope in positive y direction dz/dy (holding x constant)
equation for plane passing through the point (x0, y0, z0) with slope m in +x
direction and n in +y direction ✔️Ans - z-z0 = m(x-x0) + n(y-y0)
Point-slope: f(x,y) = z = z0 + m(x-x0) + n(y-y0)
Slope-int: f(x,y) = z = c + mx + my. c = z0- m(x0)-n(y0)
What does f(x,y) = c (constant) look like? ✔️Ans - horizontal plane
Level surfaces ✔️Ans - Used for visualizing a function of three variables w =
f(x,y,z). For various constants w = c, plot the surface whose graph is f(x,y,z) =
c. Creates a 3D contour diagram
elliptical paraboloid ✔️Ans - z = x^2/a^2 + y^2/b^2
hyperbolic paraboloid ✔️Ans - z = -x^2/a^2 + y^2/b^2
Ellipsoid ✔️Ans - x^2/a^2 + y^2/b^2+ z^2/c^2 = 1
hyperboloid of one sheet ✔️Ans - x^2/a^2 + y^2/b^2- z^2/c^2 = 1
hyperboloid of two sheets ✔️Ans - x^2/a^2 + y^2/b^2- z^2/c^2 = -1
, cone ✔️Ans - x^2/a^2 + y^2/b^2 - z^2/c^2 = 0
Plane ✔️Ans - ax + by + cz = d
cylindrical surface ✔️Ans - x^2 + y^2 = a^2
parabolic cylinder ✔️Ans - y = ax^2
limit ✔️Ans - lim (x,y) -> (a,b) f(x,y) = L , if f(x,y) can be made as close to L as
we please whenever the distance from point (x,y) to the point (a,b) is
sufficiently small, but not zero
Finding the limit of a function along a line, or another function (ex. y = mx, y =
x^2) ✔️Ans - Plug function into each y in the larger function and evaluate
the limit at the point. If the limits approaching a point along multiple different
functions differ the limit does not exist.
displacement vector ✔️Ans - vector representing displacement i.e. change
from a point P to point Q.
unit vector ✔️Ans - has magnitude of 1 unit
how to add, subtract, scalar multiply vectorys ✔️Ans - add, subtract and
scalar multiple by components
magnitude ✔️Ans - ||*v*|| = sqrt(v1^2 + v2^2 + v3^2)
||c*v*|| = |c|||*v*||
algebraic properties of vectors: commutativity, associativity, distributivity,
identities ✔️Ans - comm: *v* + *w* = *w* + *v*
assoc:(*u* + *v*) + *w* = *u* + (*v* + *w*)
a(b*v*) = (ab)*v*
dist: (a + b)*v* = a*v* + b*v*
a(*v* + *w*) = a*v* + a*w*
ident:1*v* =*v*
*v* + *0* = *v*
0*v* = o
*w* - *v* = *w* + (-1)*v*
Distance between two points (a,b,c) and (x,y,z) ✔️Ans - d = sqrt((x-a)^2 +
(y-b)^2 + (z-c)^2)
cross sections vs level curves ✔️Ans - cross sections: vertical slices of the
graph of f(x,y) formed using vertical planes x=c (f(c,y)=z) or y=c (f(x,c)=z).
Level curves: horizontal slices of f(x,y) using horizontal planes z=c (f(x,y) =c).
Both used to plot graphs of functions with multiple variables.
contour diagrams ✔️Ans - a family of graphs of the equation f(x,y) = c
plotted in the xy-plane, for set values of c, usually labeled by the values. Values
of c usually plotted in equal increments. (like topographical map)
m and n ✔️Ans - m= slope in positive x direction dz/dx (holding y constant)
n = slope in positive y direction dz/dy (holding x constant)
equation for plane passing through the point (x0, y0, z0) with slope m in +x
direction and n in +y direction ✔️Ans - z-z0 = m(x-x0) + n(y-y0)
Point-slope: f(x,y) = z = z0 + m(x-x0) + n(y-y0)
Slope-int: f(x,y) = z = c + mx + my. c = z0- m(x0)-n(y0)
What does f(x,y) = c (constant) look like? ✔️Ans - horizontal plane
Level surfaces ✔️Ans - Used for visualizing a function of three variables w =
f(x,y,z). For various constants w = c, plot the surface whose graph is f(x,y,z) =
c. Creates a 3D contour diagram
elliptical paraboloid ✔️Ans - z = x^2/a^2 + y^2/b^2
hyperbolic paraboloid ✔️Ans - z = -x^2/a^2 + y^2/b^2
Ellipsoid ✔️Ans - x^2/a^2 + y^2/b^2+ z^2/c^2 = 1
hyperboloid of one sheet ✔️Ans - x^2/a^2 + y^2/b^2- z^2/c^2 = 1
hyperboloid of two sheets ✔️Ans - x^2/a^2 + y^2/b^2- z^2/c^2 = -1
, cone ✔️Ans - x^2/a^2 + y^2/b^2 - z^2/c^2 = 0
Plane ✔️Ans - ax + by + cz = d
cylindrical surface ✔️Ans - x^2 + y^2 = a^2
parabolic cylinder ✔️Ans - y = ax^2
limit ✔️Ans - lim (x,y) -> (a,b) f(x,y) = L , if f(x,y) can be made as close to L as
we please whenever the distance from point (x,y) to the point (a,b) is
sufficiently small, but not zero
Finding the limit of a function along a line, or another function (ex. y = mx, y =
x^2) ✔️Ans - Plug function into each y in the larger function and evaluate
the limit at the point. If the limits approaching a point along multiple different
functions differ the limit does not exist.
displacement vector ✔️Ans - vector representing displacement i.e. change
from a point P to point Q.
unit vector ✔️Ans - has magnitude of 1 unit
how to add, subtract, scalar multiply vectorys ✔️Ans - add, subtract and
scalar multiple by components
magnitude ✔️Ans - ||*v*|| = sqrt(v1^2 + v2^2 + v3^2)
||c*v*|| = |c|||*v*||
algebraic properties of vectors: commutativity, associativity, distributivity,
identities ✔️Ans - comm: *v* + *w* = *w* + *v*
assoc:(*u* + *v*) + *w* = *u* + (*v* + *w*)
a(b*v*) = (ab)*v*
dist: (a + b)*v* = a*v* + b*v*
a(*v* + *w*) = a*v* + a*w*
ident:1*v* =*v*
*v* + *0* = *v*
0*v* = o
*w* - *v* = *w* + (-1)*v*
