Name MATH 221B Multivariate Calculus Spring 2004 Exam #4 Instructions: You can work on the problems in any order. Please use just one side of each page and clearly number the problems. You do not need to write answers on the question sheet. This exam is a
Math 280, Toews
Spring, 2013
Concept Review for Chapter 13
A vector valued function is a function r(t) = (x(t), y(t), z(t).
As t changes, r(t) traces out a path in R3 . The set of points on this path is called a curve. Any curve
can be parameterized in
Name V Math 280
Calcuius 11
Spring, 2013
Quiz 4
(1) Sketch the curve < t2 + it > for t E {ma-1,1]. Compute the tangent vecter at t I l ané add it to the sketch.
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Math 280, Toews
Spring, 2013
Concept Review for Exam 2
What is a vector valued function?
What is the dierence between a curve and a path?
Fact: every curve can be parameterized in innitely many ways
Know how to do calculus on vector valued functions:
Math 280, Toews
Spring, 2013
Concept Review for Exam 3
Double integrals
The double integral of z = f (x, y) over a region R is the signed volume of the region between the
graph of f and the region R.
R
1dA can be interpreted as the area of R.
If f (x,
Math 280, Toews
Spring, 2013
Concept Review for Exam 1
Geometric idea of a vector (i.e. has a tail and a head)
Algebraic idea of a vector (a 3-tuple of real numbers)
How to reconcile geometric and algebraic viewpoints: dene vectors to be equivalent if
Study Questions for Exam 2
1. Let a be a positive constant, and consider the parabola y = ax2 . Show that the curvature of this
parabola is maximal at x = 0.
2. Write down the equation of the tangent line to the path c(t) = (t, t2 , log t) at the point P
1. Write down the equations of two planes (in 3-D) such that the acute angle formed by the planes is 30
degrees.
2. Draw a left-handed coordinate system.
3. In what sense does the algebraic equation x2 + y 2 + z 2 = 4 describe a sphere?
4. Let P = (1, 1,
Study Questions for Exam 3
1. Evaluate
R
e3x+4y dA for the region R = [0, 1] [0, 2].
2. Evaluate the integral of f (x, y) = (x + y)1 over the region bounded by y = x, y = 1, y = e, and x = 0.
3. Let W be the region bounded by z = 1 y 2 , y = x2 , and the
Name MATH 221B Multivariate Calculus Spring 2004 Exam #5 Instructions: You can work on the problems in any order. Please use just one side of each page and clearly number the problems. You do not need to write answers on the question sheet. This exam is a
MATH 221B
Name Multivariate Calculus
Spring 2004
Exam #2
Instructions: You can work on the problems in any order. Please use just one side of each page and clearly number the problems. You do not need to write answers on the question sheet. This exam is a