Exam (elaborations)
Multivariable Calculus – Questions/Answers
- Course
- Institution
Multivariable Calculus – Questions/Answers
[Show more]
Seller
Follow
Studycafe
Reviews received
Content preview
Multivariable Calculus – Questions/AnswersDistance 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*
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller Studycafe. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $9.99. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
72042 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now