Honors Calculus II
Quiz 5
1
centered at a = 3 . What is the
1! x
radius of convergence and interval of convergence of the series. Does the series
1
converge to
on this interval?
1! x
1.
Determine a power series for the function
2.
Determine a power series
CALCULUS 133: TECHNIQUES OF INTEGRATION
The purpose of these techniques is the following: you are given the problem of
nding an antiderivative of a complicated function, and these techniques allow you
to reduce it to nding an antiderivative of a simpler f
Analysis of Hoover Dam
The dam was built in the years 1931 to 1936. Its height is 726 feet. The length of its
crest is 1,244 feet. A straight line across the arc at the top is 860 feet. In determining the
net force of the water against the dam in the forw
Higher Dimensional Integration
Jed Keesling
So far we have only considered integration of a function on the real line. It is easy
enough to generalize this to higher dimensions. However, you need to be alerted to some
caveats. This will be a topic that wi
Net Force on a Dam
Consider a dam holding back an incompressible fluid such as water. Let us suppose that
the xyplane is parallel to the ground and the zcoordinate perpendicular to the ground.
The purpose of this exercise is to show that the total net for
Circumference of the Earth
The Greeks knew that the earth was spherical and derived estimates of its
circumference. Eudoxus of Cnidos (408-355 BC) was supposed to have estimated the
circumference of the earth to be 400,000 stades (circa 40,000 miles). The
Centroids
Let A be a geometric figure in Euclidean space. The Euclidean space may be R 2 or R 3 .
It could in fact be R n . The figure may be any dimension. The coordinates of the
centroid or center of mass of the figure are given by the following formula
Cavalieris Determination of the Volume
of the Sphere
Bonaventura Cavalieri (1598-1647) was a contemporary of Galileo who
considered him the greatest geometer since Archimedes. One of his powerful tools has
been called Cavalieris Principle which states tha
Cavalieris Determination of the Area of an Ellipse
The principle is the following. Suppose that two planar figures have the same
height and at the same level the cross-sectional lengths are in the same ratio r. Then the
areas are in the same ratio r.
Cava
The Catenary and Parabolic Cables
The Catenary
y ! axis
Assume that one has a cable which
hangs under its own weight with only
tangential forces acting along the length of
"
T0
the cable. The cable is assumed to have
uniform density, w0 , along its length
STRIKING RESULTS WITH BOUNCING BALLS
Andr Heck, Ton Ellermeijer, Ewa Kdzierska
ABSTRACT
In a laboratory activity students study the behaviour of a bouncing ball. With the help of a high-speed camera they
can study the motion in detail. Computer modelling
Limits and Continuity
Definitions of Limits
For all ! > 0 there is a ! > 0 such that for all x with x ! a < " and
lim f ( x ) = L .
x !a
with x ! a , f ( x) ! L < " .
lim an = L
For all ! > 0 there is an N such that for all n ! N , an ! L < " .
lim f ( x
Newton Iteration
Consider solution to the equation z 2 = 2 on the real line. The solution is clearly
z = 2 . Now consider approximating the solutions of this equation using Newtons
Method. We first set up the function to be set to zero, f ( x ) = x 2 ! 2
Honors Calculus II
Quiz 4
1.
2.
For what values of N will it be true that
For what values of N will it be true that
!
!
! ! ! !
!
! (! ! )
! !
!
!
!
! ! ! !
!
< !"?
!
! (! ! )
! !
!
!
< !"?
Honors Calculus II
Quiz 4
1.
Determine the radius of convergence and the interval of convergence for each of
the following power series.
(a)
( x ! 1)n
#n
n =1
(b)
"n
"
!
n =1
!
(c)
xn
2
xn
" nn
n =1
Honors Calculus II
Quiz 3
1.
Show that the following series converge. Explain to the best of your ability.
!
(a)
"x
n
for x < 1
n=0
!
(b)
n =1
2
(!1)n +1
#n
n =1
"
(c)
1
"n
Honors Calculus II
Quiz 2
!
1.
Show that
"x
n=0
n
=
1
for x < 1 .
1# x
"
2.
Show that ln(1 + x ) = # (!1)n
n=0
"
3.
Show that arctan( x ) = # (!1)n
n=0
4.
x n +1
for x < 1 .
n +1
x 2 n +1
for x < 1 .
2n + 1
1
Show that .9999! = 1 . Show that .3333! = .
3
Honors Calculus II
Quiz 2
!
1.
Show that
"x
n=0
n
=
1
for x < 1 .
1# x
"
2.
Show that ln(1 + x ) = # (!1)n
n=0
"
3.
Show that arctan( x ) = # (!1)n
n=0
4.
x n +1
for x < 1 .
n +1
x 2 n +1
for x < 1 .
2n + 1
1
Show that .9999! = 1 . Show that .3333! = .
3
Honors Calculus II
Quiz 1
1.
Show that the following series converges.
!
"x
n=0
2.
n
=
1
for x < 1
1# x
A ball bounces with a coefficient of restitution 0 < e < 1 . If the ball is released
from a height of 5 feet, how long will it bounce before coming to
Honors Calculus II
Quiz 1
1.
Determine the power series for the following functions centered at the given
point.
(a) exp( x ) at a = 0
(b) sin( x ) at a = 0
(c) cos( x ) at a = 0
(d) 1 + x at a = 0
Taylor Polynomials and Power Series
Taylor Polynomials and Taylor Series.
Suppose that f has N derivatives at a. Then let
f !(a )
f ( N ) (a )
2
PN ( x ) = f (a ) + f !(a )( x " a ) +
( x " a) + ! +
( x " a )N
2
N!
This is the N th Taylor polynomial for f
Numerical Solution of Ordinary Dierential Equations
James Keesling
1
Picard Iteration
The Picard method is a way of approximating solutions of ordinary dierential equations.
Originally it was a way of proving the existence of solutions. It is only through
Bifurcation Diagram for the Quadratic Family
1
x ! axis
f ( x ) = ! x ! (1 " x )
0
3.5
4
! axis
The diagram is the bifurcation diagram for the quadratic family, f ( x ) = ! x ! (1 " x ) .
7
! ! 4 and 0 ! x ! 1 . The diagram represents the
2
long-term beh
Archimedes Quadrature of the Parabola
Consider a parabola given by the formula f ( x ) = a ! x 2 . Let b < c be given.
Archimedes determined the area of figure bounded by the line joining the points
(b, a ! b 2 ) and (c, a ! c 2 ) . By modern methods we w
Series
!
The harmonic series is given by the following formula
"
n =1
!
"
n =1
1
n
1
n
. Its limit is
= ! . There are several clear proofs of this property. One that is especially simple is
that if the series does not go to infinity, then the series is bo
Self-Similar Sets
James Keesling
Outline
Brief History of Self-Similar Sets
Iterated Function Systems
Hausdorff dimension
The Code Space
Shift-Invariant vs Sub-Self-Similar
The Boundaries of Tiles
Summary
Brief History
Helge von Koch 190
Romberg Integration
James Keesling
1
The Trapezoidal Rule for Estimating the Integral
A common way of estimating an integral is to use the Trapezoidal Rule. Let f (x) be
b
the given function over the interval [a, b]. We want to estimate the integral a f (
MAC 3473 Honors Calculus 2
Keesling
Review Test #4
Test Date
4/22/08
1.
Consider a circle of radius R rolling along the xaxis with the speed of the center
of the circle being R. Give a parametric representation of the motion of the point.
Suppose that the