Chain rule for functions of 2, 3 variables (Sect. 14.4)
Review: Chain rule for f : D R R.
Chain rule for change of coordinates in a line.
Functions of two variables, f : D R2 R.
Chain rule for functions dened on a curve in a plane.
Chain rule for change o
Limits and continuity for f : Rn R (Sect. 14.2)
The limit of functions f : Rn R.
Example: Computing a limit by the denition.
Properties of limits of functions.
Examples: Computing limits of simple functions.
Continuous functions f : Rn R.
Computing limits
MA 234-0 Exam 2 Spring 2014
1. (25 pts) Find the work done by the force eld F = :1: sin yi + yj on a particle that moves on the parabola
y = x2 from (1,1) to (2,4).
0
{:1 XI
: SUN? r M (7) 11 : ga anX2 g 2x3 >011
{:4 y:\
: z+l\z:[$"/z)+w
.2 Z #I 3
/_.
=\i
Integration and projectile motion (Sect. 13.2)
Integration of vector functions.
Application: Projectile motion.
Equations of a projectile motion.
Range, Height, Flight Time.
Integration of vector functions
Denition
An antiderivative of a vector function v
Partial derivatives and dierentiability (Sect. 14.3)
Partial derivatives and continuity.
Dierentiable functions f : D R2 R.
Dierentiability and continuity.
A primer on dierential equations.
Partial derivatives and continuity
Recall: The following result h
Partial derivatives and dierentiability (Sect. 14.3)
Partial derivatives of f : D R2 R.
Geometrical meaning of partial derivatives.
The derivative of a function is a new function.
Higher-order partial derivatives.
The Mixed Derivative Theorem.
Examples of
Scalar functions of several variables (Sect. 14.1)
Functions of several variables.
z
On open, closed sets.
Functions of two variables:
f(x,y)
Graph of the function.
Level curves, contour curves.
y
Functions of three variables.
x
Graphs, level surfaces.
Sc
Vector functions (Sect. 13.1)
Denition of vector functions: r : R R3 .
Limits and continuity of vector functions.
Derivatives and motion.
Dierentiation rules.
Motion on a sphere.
Denition of vector functions: r : R R3
z
Denition
A vector function is funct
The length of a curve in space (Sect. 13.3)
The length of a curve in space.
The length function.
Parametrizations of a curve.
The length parametrization of a curve.
The length of a curve in space
z
r(t)
Denition
The length or arc length of a curve in
the
Math 234 Quiz 4 SOLUTIONS: Section 67 Spring 2014 (May 6)
1) (5 points) Compute the line integral given below, where C is the part of the curve y2 = x3 from
the points (1, 1) to (4, 8):
y x dx + x x dy
C
SOLUTION: There many different ways to parametrize
Math 234 Quiz 2: Section 67 Spring 2014 (April 15)
Name: Solutions
1) (5 points) Set up integrals to nd the mass and centroid of the plane lamina bounded by the
curves y = 0, x = 1, x = 1, and y = exp x2 in the xy plane, with a density of (x, y) = |xy|.
Y
Math 234 Quiz 1: Section 67 Spring 2014 (April 8)
Name: Solutions
1) (5 points) Evaluate the following iterated integral:
sin(x)
y dy dx
0
0
SOLUTION: Take the anti-derivative with respect to y:
sin(x)
sin(x) 2
y
y dy dx =
2
0
0
0
=
0
1
2
y=sin(x)
dx
y