MATH 12 CLASS 4 NOTES, SEP 28 2011
Contents
1. Lines in R3
2. Intersections of lines in R3
3. The equation of a plane
4. Various problems with planes
4.1. Intersection of planes with planes or lines
4.2. The angle between two planes
4.3. The distance of a
MATH 12 CLASS 5 NOTES, SEP 30 2011
Contents
1. Vector-valued functions
2. Dierentiating and integrating vector-valued functions
3. Velocity and Acceleration
1
3
4
Over the past two weeks we have developed the basic language of vectors and
geometry in dime
MATH 12 CLASS 5 NOTES, SEP 30 2011
Contents
1. Arc length of curves
1
1. Arc length of curves
Now that we know how to take derivatives of vector-valued functions, we briey
describe how to solve a natural and common geometric problem. Suppose we have a
cur
MATH 12 CLASS 7 NOTES, OCT 4 2011
Contents
1. Plotting functions of several variables
2. Limits of functions of several variables
3. Computer systems for 3D graphing
1
3
5
1. Plotting functions of several variables
We now begin to seriously study multivar
MATH 12 CLASS 8 NOTES, OCT 7 2011
Contents
1. Partial Derivatives
2. Higher partial derivatives
3. Tangent planes
1
3
3
Now that we have some idea of how to dene a limit for multivariable functions, and
how the behavior of limits of multivariable function
MATH 12 CLASS 9 NOTES, OCT 10 2011
Contents
1. Tangent planes
2. Denition of dierentiability
3. Dierentials
1
3
4
1. Tangent planes
Recall that the derivative of a single variable function can be interpreted as the
slope of the tangent line to the graph o
MATH 12 CLASS 10 NOTES, OCT 12 2011
Contents
1. Directional derivatives
2. The gradient
1
2
Quick links to denitions/theorems
Directional derivative denition
Gradient denition
Computing the directional derivative using the gradient
1. Directional deriv
MATH 12 CLASS 11 NOTES, OCT 14 2011
Contents
1. The direction of maximum increase
Suppose we have a function f (x, y ) and a point (a, b) we are interested in. If we
think about the surface z = f (x, y ) and the point above (a, b), then there should be
a