Slide 1
Section 6.1
Slide 2
Basic Geometric Figures
Reality
Slide 3
Note that the rectangle or "slice" is
perpendicular to the axis of revolution.
Slide 4
Slide 5
2x2 ,
y
y
(a)
the x - axis
(b)
0,
the line x = 2
Slide 6
V
b
a
OR
2
IR
2
dx
x
2
Slide 7
2x2
MATRICES AND LINEAR ALGEBRA: A QUICK INTRODUCTION
1. An Introduction to Matrices
1.1. Denition of a matrix.
Denition 1.1. An m n matrix is a rectangular array of numbers with m rows and n
columns.
Example 1.2. A 2 1 matrix is a column vector and a 1 2 mat
Ritz/Lac, (AIL/def
11.1 and 11.2 CPA
This first CPA will be completed together in class (at least mostly) and you will submit it in Blackboard no
later than midnight tonight. All future CPAs must be submitted by 9:30 am. before the day we will cover
the s
Calculus III
Section 11.5
1. Whatisavector-valuedfunction? )
g ()epwht meimloLc-g lmy? (are, 00MP0mcn+g 4? (i 4/19
95% camplnml vmcs (Nil/L. (L$(<A" lo 9 <Inglt- hilt/NH
VrJ'thLJ-(I- be
2. Two Sing: points in R3 determineaunique line. Alternatively,one
K
Calculus III CPA
ggion 12.5 The ghgin Rule
Write the basic chain rule from Calculus | where you have one independent variable and one dependent
variable. (7( I J l 1
A1 (MN F (go-3)- 5 <f)o(%1:cfw_v
Write the chain rule where you have one independent var
Calculus Ill CPA
Section 11.7 - Motion in Space
1. Let the vector-valued function r(t) : (x(t). y(t), z(t) describe the position of a moving object
at times t 2 0.
What is meant by the trajectory of the object?
m w M aw les
How do we find the instantaneou
Section 11.9 Curvature and Normal Vectors
1. Imagine driving a car along a winding mountain road. There are two ways to change the velocity
of the car (that is, to accelerate).
a. You can change the SEC/Cd ofthe car, or
b. You can change the (A! fact (U /
Calculus III CPA
Section 12.3 Limits and Continuig
1. Write the Limit Laws for Functions of Two Variables
Let L and M be reai numbers and supposed that iim [(x, y): L and
(xy)->Ca.bJ
(x,y1)in(lab) 90:, y): M. Assume cisa constant, and m and n are integers
Calculus Ill CPA
Section 12.1
Intuitively, a Q [530 1 = is a flat surface with infinite extent in all directions. Three
. c
[SQ/V Co H lugs: points cfw_notaiimthesameneideteminea W) L4 [2 cfw_1 plane
in 1R3. A plane in 1R3is also uniquely determined by on
Calculus 3, Exam #3
Spring 2017
Leave this box blank. Your instructor will fill it in. (2 pt each)
13.1, 13.2 CPA 13.3, 13.4 CPA
13.6 CPA 13.5, 13.7 CPA Ch 3 Review CPA
Show your work extremely neatly and circle your answers. Solutions without correct s
Calculus III
11.3 and 11.4 CPA
You must post as a single pdf using the appropriate link in Blackboard. Hard copies, email submissions,
submissions containing multiple files, and submissions using any other format will not earn credit.
Your paper must be c
Calculus II Chapter 6 Practice Sheet
Name _
1. A spring has a natural length of 2 ft, and a force of 15 lb is required to hold it compressed at a length of 18 in. How
much work is done in stretching this spring from its natural length to a length of 3 ft?
Calculus II, 8.2, Series and Convergence
TV s01
An infinite series is a sum of the form
a
n
= a1 + a 2 + . + a n + .
n =1
The nth partial sum S n is the sum of the first n terms of the series. S n = a1 + a 2 + . + a n
If the sequence of partial sums conve
Slide 1
Calculus II
Section 6.2
Slide 2
y
4x
(a)
(b)
the y axis
(c)
y
the x - axis
x2 ,
x=7
Slide 3
V
b
a
2 ( rad )( height ) dy
Note that the rectangle
or "slice" is parallel to
the axis of revolution.
0
Slide 4
1)
2)
x2 ,
y
4x
y
y
x , y 0, x 4
x 2 , abo
Section 6.6 Work
The work done by a variable force, F(x), moving an object along a straight line from x=a to x=b is
b
W = F ( x) dx .
a
To evaluate work you need to find a force function.
* For springs use Hookes Law: F = kd
* For filling and emptying tan
Calculus II Section 7.7 Improper Integrals
Recall that the definition of the definite integral
b
a
f ( x) dx requires that [a,b] be finite. Also, the Fundamental
Theorem of Calculus requires that f be continuous on [a,b]. Now we are going to learn to eval
2/3/2010
Formula
Integration by Parts
n
Calculus II
Section 7.1
3 Basic Formats for Integration
by Parts
u dv = uv v du
Basic Guidelines
n
n
n
n
n
Let dv be the most complicated portion of
the integral that can be integrated with basic
integral formulas.
2/3/2010
sin
Trigonometric Integrals
n
n
m
n
n
Use the odd exponent to rewrite if
you have one. If m & n are both
even then use a half-angle identity
to reduce powers.
rewrite as (_
angle identity
n
n
n
x dx
2 m/2
)
dx
and use a half-
pull off one and re
Calculus II Section 8.1 Sequences
A sequence is a function whose domain is the set of natural numbers.
Write the first 5 terms of the sequence a n
Compare: f ( x )
x1
and
an
Write the first 5 terms of the sequence a n
Simplify:
Simplify:
(n
2)!
n!
( 2 n 2
Section 8.3, The integral test p-series and harmonic series
In this section we restrict our attention to Positive term series and convergence tests
Theorem 9
The Integral Test
If f is positive, continuous, and decreasing for x 1 and a n
f ( n ) , then
a n