112
Chapter 7. The Riemann Integral
Exercise 7.2.4. (a) () Assume there exists a sequence of partitions (Pn )
satisfying
lim [U (f, Pn ) L(f, Pn )] = 0.
n!1
Given > 0, choose PN from this sequence so that U (f, PN ) L(f, PN ) < .
Then Theorem 7.2.8 implie

348
Chapter 20
Electric Charge, Force, and Field
Section 20.5 Matter in Electric Fields
48.
32. In his famous 1909 experiment that demonstrated quantiza-
tion of electric charge, R. A. Millikan suspended small oil
drops in an electric field. With field st

20.4 Fields of Charge Distributions
339
!
Here the Eis are the fields of the point charges qi located at distances ri from the point
where were evaluating the fieldcalled, appropriately, the field point. The r^is are unit
vectors pointing from each point

Number Sense Tricks:
1. Subtracting Reverses
a. 3 digit or 4 digit (work in pairs): subtract first and last digit/pair, then multiply by 100 and subtract difference
i. 634-436 = 198 (6-4 = 2, 100x2 2 = 198)
ii. 1495-9514 = -8019 (95-14=81, 81x100-81 = 801

Whats the Best Prep
Method for YOU?
Major Prep Methods and
How to Decide
By Allen Cheng
www.PrepScholar.com
TABLE OF CONTENTS:
Intro.3
1: In-Person Tutoring.4
2: Classes.7
3: Self-Study.10
4: Online Prep Program.13
5: Online Prep + Tutoring.16
6: And Now

Cape
Town
Honolulu
2005
MATHCOUNTS CHAPTER
SPRINT ROUND
1. We are given the following chart:
Cape
Town London
Honolulu
6609
Bangkok
5989
London
To nd the distance between Honolulu
and Cape Town, go to the row labeled
Honolulu and l

2008
Chapter Competition 1. What is the average student headcount for the spring terms of the 1- Students
0203, 03-04 and 0405 academic years? Express your answer
to the nearest whole number.
Total Student Headcount (2082-2003 to 2005-2006)
Fall Ter

. In the integer 45,075,123, the 2 represents the value 20. By what
factor would the value represented by the 5 in the thousands place
have to be multiplied to equal the value represented by the 5 in the
millions place?
Twenty-seven increased by twice a n

20.2 Coulombs Law 333
Quantities of Charge
All electrons carry the same charge, and all protons carry the same charge. The protons
charge has exactly the same magnitude as the electrons, but with opposite sign. Given that
electrons and protons differ subs

PART THREE
SUMMARY
Thermodynamics
Thermodynamics is the study of heat, temperature, and related
phenomenaand their relation to the all-important concept of energy.
Thermodynamics provides a macroscopic description in terms of parameters like temperature a

67
4.4. Continuous Functions on Compact Sets
Exercise 4.4.3. Because compactness is preserved by continuous functions, the
set f (K) is compact. By Exercise 3.3.1, y1 = sup f (K) exists and y1 2 f (K).
Because y1 2 f (K), there must exist (at least one po

7.6. Lebesgues Criterion for Riemann Integrability
127
(c) The cos(1/x) term in the formula for g 0 (x) oscillates between +1 and 1
as x ! 0. Because the other term in this formula converges to zero, the net
eect is that g 0 (x) attains every value betwee

109
6.6. Taylor Series
Applying LHospitals rule we can write
8/x5
4/x2
=
lim
2
2 .
x!0 2e1/x /x3
x!0 e1/x
g 00 (0) = lim
One more application of LHospitals rule lets us conclude
8/x3
4
= lim 1/x2 = 0.
2
x!0 2e1/x /x3
x!0 e
g 00 (0) = lim
2
In general, whe

73
4.6. Sets of Discontinuity
Exercise 4.6.4. This argument is very similar in spirit to the proof of the
Monotone Convergence Theorem.
Given c 2 R, lets prove that limx!c f (x) exists for an increasing function
f . Our first task is to produce a candidat

61
4.3. Combinations of Continuous Functions
4.3
Combinations of Continuous Functions
p
p
Exercise 4.3.1. (a) Let > 0. Note that |g(x) c| = | 3 p
x 0| = | 3 x| where
c = 0. Now if we set = 3 , then |x 0| < = 3 implies | 3 x| < . This shows
g(x) is continu

7.3. Integrating Functions with Discontinuities
115
Focusing on the intervals that make up [a, b]\O (the good points), we partition
these so that all the resulting subintervals have length less than . This puts us
into a situation like the one in Theorem

76
Chapter 5. The Derivative
(b) To avoid confusion with the notation in Theorem 5.2.4, lets set h(x) =
1/x. By the Chain Rule,
0
1
g 0 (x)
= (h g)0 (x) =
.
g(x)
[g(x)]2
Then using the product rule (Theorem 5.2.4 (iii), we have
0
f
(x) = [f (x)(h g)(x)]0

64
Chapter 4. Functional Limits and Continuity
However,
|f (x0 ) f (y)| c|x0 y|,
must also be true, and because 0 < c < 1 we conclude that x0 = y.
In summary, if f is a contraction on R, then f has a unique fixed point, and
every sequence of iterates conv

70
Chapter 4. Functional Limits and Continuity
such that |f (x) f (y)| < whenever |x y| < . Given this , we use the fact
that (xn ) is a Cauchy sequence to say that there exists an N 2 N such that
|xn yn | < whenever m, n N . Combining the last two statem

Name '- " . '
. Date '-
MathCounts Competitions
' (2008) ,._' l. The ndings from a recent analysis of rental housing costs 1. 3
are given in the table. How much more expensive is the
cheapest monthly rental cost for a 3 Bedroom in the District
of Columbia