Multivariable Calculus Exam Questions with Verified Answers Latest Update 2024 ( 100% Pass)
Distance between two points (a,b,c) and (x,y,z) - Answers d = sqrt((x-a)^2 + (y-b)^2 + (z-c)^2)
cross sections vs level curves - Answers 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 - Answers 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 - Answers 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 - Answers 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? - Answers horizontal plane
Level surfaces - Answers 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 - Answers z = x^2/a^2 + y^2/b^2
hyperbolic paraboloid - Answers z = -x^2/a^2 + y^2/b^2
Ellipsoid - Answers x^2/a^2 + y^2/b^2+ z^2/c^2 = 1
hyperboloid of one sheet - Answers x^2/a^2 + y^2/b^2- z^2/c^2 = 1
hyperboloid of two sheets - Answers x^2/a^2 + y^2/b^2- z^2/c^2 = -1
cone - Answers x^2/a^2 + y^2/b^2 - z^2/c^2 = 0
Plane - Answers ax + by + cz = d
cylindrical surface - Answers x^2 + y^2 = a^2
parabolic cylinder - Answers y = ax^2
, limit - Answers 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) - Answers 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 - Answers vector representing displacement i.e. change from a point P to point Q.
unit vector - Answers has magnitude of 1 unit
how to add, subtract, scalar multiply vectorys - Answers add, subtract and scalar multiple by
components
magnitude - Answers ||*v*|| = sqrt(v1^2 + v2^2 + v3^2)
||c*v*|| = |c|||*v*||
algebraic properties of vectors: commutativity, associativity, distributivity, identities - Answers 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*
dot product - Answers *v* • *w* = ||*v*||||*w*||cos(x) where x is the angle between the vectors. =
v1w1 + v2w2 + v3w3
magnitude using dot product - Answers *v* • *v* = ||*v*||^2
angle between two vectors (x) - Answers cos(x) = *v* • *w*/||*v*||||*w*||
perpendicularity using dot product - Answers if *v* • *w* = 0 then *v* and *w* are perpendicular
algebra of dot product: commutative, associative, distributive - Answers comm: *v* • *w* = *w* • *v*
assoc: *v* • (c*w*) = c(*v* • *w*) = (c*v*) • *w*
Distance between two points (a,b,c) and (x,y,z) - Answers d = sqrt((x-a)^2 + (y-b)^2 + (z-c)^2)
cross sections vs level curves - Answers 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 - Answers 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 - Answers 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 - Answers 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? - Answers horizontal plane
Level surfaces - Answers 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 - Answers z = x^2/a^2 + y^2/b^2
hyperbolic paraboloid - Answers z = -x^2/a^2 + y^2/b^2
Ellipsoid - Answers x^2/a^2 + y^2/b^2+ z^2/c^2 = 1
hyperboloid of one sheet - Answers x^2/a^2 + y^2/b^2- z^2/c^2 = 1
hyperboloid of two sheets - Answers x^2/a^2 + y^2/b^2- z^2/c^2 = -1
cone - Answers x^2/a^2 + y^2/b^2 - z^2/c^2 = 0
Plane - Answers ax + by + cz = d
cylindrical surface - Answers x^2 + y^2 = a^2
parabolic cylinder - Answers y = ax^2
, limit - Answers 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) - Answers 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 - Answers vector representing displacement i.e. change from a point P to point Q.
unit vector - Answers has magnitude of 1 unit
how to add, subtract, scalar multiply vectorys - Answers add, subtract and scalar multiple by
components
magnitude - Answers ||*v*|| = sqrt(v1^2 + v2^2 + v3^2)
||c*v*|| = |c|||*v*||
algebraic properties of vectors: commutativity, associativity, distributivity, identities - Answers 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*
dot product - Answers *v* • *w* = ||*v*||||*w*||cos(x) where x is the angle between the vectors. =
v1w1 + v2w2 + v3w3
magnitude using dot product - Answers *v* • *v* = ||*v*||^2
angle between two vectors (x) - Answers cos(x) = *v* • *w*/||*v*||||*w*||
perpendicularity using dot product - Answers if *v* • *w* = 0 then *v* and *w* are perpendicular
algebra of dot product: commutative, associative, distributive - Answers comm: *v* • *w* = *w* • *v*
assoc: *v* • (c*w*) = c(*v* • *w*) = (c*v*) • *w*
