MATH 21C Discussion Handout
Midterm Review, Power Series and Applications
October 11
(Updated October 13, 2016)
1
Useful facts
A power series about x = a is a series of the form
X
cn x n ,
n=0
where the coefficients cn is a real number for all n 0.
Radi

ac
LECTURE 20 SUMMARY 3/3/2014
Let y0 be a xed constant. The intersection of the plane y = y0 with the surface
z = f(x, y) is a curve in the plane y = y0 , namely, z = f(x, y0 ). The slope for the
tangent to the curve z = f(x, y0 ) at (x, y0 ) is given by

0.05ex
Summary 3-7-2014, Lecture 21
A function f (x, y, z) is dierentiable at (x0 , y0 , z0 ) if x f, y f, z f exist and are continuous at
(x0 , y0 , z0 ).
Suppose f (x, y), x = x (t) , y = y (t) are dierentiable functions. Then
df (x (t) , y (t)
= (x

LECTURE 22 SUMMARY 3/12/2014
The linearization for f(x, y) at (x0 , y0 ) is the just the tangent plane at (x0 , y0 ):
z = f(x0 , y0 ) + x f(x0 , y0 )(x x0 ) + y f(x0 , y0 )(x x0 )
The tangent line to the level curve g(x, y) = c (c is a constant) at (x0 ,

SUMMARY 22, 3/10, 2014
The rate of change for f(x, y) in the direction ^ (a unit vector) is dened as
v
D^ f(x, y) = lim
v
t0
f(< x, y > +t^) f(x, y)
v
.
t
If f(x, y) is differentiable at (x, y), then
D^ f(x, y) =< x f, y f > ^
v
v
The gradient vector of f

HOMEWORK 8, DUE 2/27/2014, MATH21C
Quiz 4 (on Thursday, 2/27/2014) will be based on HW 6 and 7, Sections 12.1
through 12.5 in the book
Suggested Readings: Section 12.5, 13.1
Problems with
will be discussed by the TA (time permitting).
Feel free to discuss

HOMEWORK 9, DUE 3/6/2014, MATH21C
Suggested Readings: Section 12.6, 14.1, 14.2, 14.3
Problems with
will be discussed by the TA (time permitting).
Feel free to discuss the problems with your fellow students and the TA.
Section 12.6, 2, 4 ,5, 6, 12, 33, 38,

HOMEWORK 10, DUE 3/13/2014, MATH21C
Quiz 5 (on Thursday, 3/13/2014) will be based on HW 8 and 9, Sections 12.6, 13.1,
14.1, 14.2 in the book
Suggested Readings: Section 14.3, 14.4, 14.5, 14.6 (just the tangent plane part+linearization),
14.7, 14.8
Problem

LECTURE 7 SUMMARY JAN 22, 2014
Given a function f(x), we want to know how to generate the power series for it.
Elementary methods: Substituting new variables in the power series. Example:
1
= (x3 )n . Differentiating,
n=0
1+x3
(1)n x2n+1
1
1+x2 dx = n=1

LECTURE 16, 2/14, 2014
v
The magnitude for the cross product u is the area for the parallelogram spanned
by u and v, i.e.,
u v = u v sin ,
where is the angle between u and v. The direction is perpendicular to both u and
v and is determined by the right

ac
LECTURE 20 SUMMARY 3/3/2014
We say that limxx0 ,yy0 f(x, y) = L if for all paths
(t)
r
leading to (t0 ) =< x0 , y0 >, we have limtt0 f(t) = L.
r
r
Methods of calculating limits: (1) Direct substitution = if the function f(x, y) is dened and has no sing

LECTURE 19 SUMMARY 2/28/2014
Consider a function f Rn R for n = 2, 3. The set of points in Rn for which f is
dened is called the domain; the set of all possible values for f is called the range.
Example: ln(1 x2 y2 ) the domain is the interior of the unit

LECTURE 18 SUMMARY 2/26/2014
The distance from a point P to a plane is the shortest line segment connecting P to
a point in the plane. One elementary way to get it is as follows: (1) Determine the
line through P in a direction parallel to a normal vector

Section
FQ2016 MAT21C
Student ID
return to
EC2
1. [12.2.54] Vectors are drawn from the center of a regular n-sided polygon in the plane to the vertices of
the polygon. Show that the sum of the vectors is zero. (Hint: What happens to the sum if you rotate

Math 21C
Quiz 7
TA: Ernest Woei
February 24, 2011
Last name:
First name:
1 (5 points): Find a parametrization for the line in which the planes
x + y + z = 1 and x + y = 2
intersect.
In order to obtain a parametrization for the line in which the planes int

MATH 21C Discussion Handout
Convergence of Sequences and Series
September 27
(Updated October 2, 2016)
1
Useful facts
A sequence cfw_an converges to L (written lim an = L) if for every > 0, there is a positive
n
integer N such that n > N implies |an L|

Math 265 SI Exam 1 Review 2/ 18/2015
Find the length and direction (when dened) of u x v when u = 2i and v = -3j.
F iii . (at lamb/HMO
" ducted: ~z
Z O O
O 3 0
@ Find the area of the triangle determined by the points P, Q, and R. Then, nd a unit vector
pe

SUMMARY JAN 8, 2014
A monotone sequence either diverges to (or ) or converges. If it is bounded,
then it converges. (Monotone Convergence Theorem)
Three basic ways of showing monotonicity: (1) The associated function is monotone. (Derivative test). (2) Ra

LECTURE 13 SUMMARY 2/7/2014
Graphs in R3 :
(1) x = constant represents a plane parallel to the plane determined by the y, z
axes, called the y-z plane.
(2) likewise, y = constant represents a plane parallel x-z plane; z = constant represents a plane paral

12.1 THREE-DIMENSIONAL COORDINATE SYSTEMS
2. The line through the point (l,0, 0) parallel to the y-axis
4. The line through the point (1,0, 0) parallel to the z-axis
6. The circle x2 + y2 = 4 in the plane 2 = 2
12. The circle )c2 + z2 = 3 in the xz-plane

LECTURE 14 SUMMARY 2/10/2014
vectors vs. points: a vector is an action and its a dynamic object; a point is a static
object. We will represent a vector in coordinates by < x1 , y1 , z1 >, which means that
we move x1 units in the x direction, y2 units in t

LECTURE 15 SUMMARY 2/12/2014
Applications of dot product: (1) Find the angle between two vectors, u and v, via
the formula
uv
.
u v
(2) Detect orthogonality: two non-zero vectors are orthogonal (i.e. the angle be
tween them is 90o if and only if u v

LECTURE 17 SUMMARY 2/24/2014
Given a vector v =< a, b, c > and a point (x0 , y0 , z0 ), the line going through the point
and parallel to v can be represented in one of three forms:
vector equation: =< x0 , y0 , z0 > +t
r
v
x = x + at
0
parametric equation