10.5 The Comparison, Ratio, and Root Tests
Example 1
Use the comparison test to determine whether the following
series converge or diverge.
n
1
3
c) 3
a)
b)
k
n =1 3n + n + 1
k =1 2 + 5
n =3
n2
Example 2
Use the limit comparison test to determine whethe
10.6 Alternating Series; Conditional Convergence
Example 1
Use the alternating series test to determine whether the
following series converge or diverge
a ) (1)
k =1
k +1
1
k
b) (1)
k =1
k +1
2k
4k 2 3
c) (1)
k =1
Approximating Sums of Alternating Series
10.7 Maclaurin and Taylor Polynomials
f ( x) f ( x0 ) + f '( x0 )( x x0 )
p ( x) = f ( x0 ) + f '( x0 )( x x0 )
f ( x0 ) = p( x0 ), and f '( x0 ) = p '( x0 )
Local Quadratic Approximations of f at x = x0
f ( x ) p ( x ) = c0 + c1 x + c2 x 2 , at x = x0
sa
10.8 Maclaurin and Taylor Series; Power Series
x 2 k +1
For example Maclaurin series for sinx is (1)
( 2k + 1)!
k =0
k
Power Series in x x0
Example 1
Find the interval of convergence and radius of convergence of
the following power series.
xk
a) k
k =0 5
10.9 Binomial Series
Example 1 Find binomial series for
1
1
a)
b)
(1 + x) 2
1+ x
Example 2
xk
a) Show that the power series of f ( x ) = ln (1 x ) is
k =1 k
With interval of convergence (-1, 1).
b) Using part (a) find the sum of the series :
1
k 5k
k =1
10.10 Differentiating and Integrating Power Series
Example 1
x 2 k +1
x3 x5 x 7
sin x = (1)
= x + + , R =
3! 5! 7!
( 2k + 1)!
k =0
k
d
( 2k + 1) x 2 k = 1 3x 2 + 5 x 4 7 x 6 +
[sin x ] = (1)k
dx
3!
5!
7!
( 2k + 1)!
k =0
x2k
x2 x4 x6
= (1) k
= 1 + + , R
Chapter6 Integration
6.1 An Overview of Area
6.1.1 THE AREA PROBLEM f is continuous and nonnegative
on [a,b ] . Find the area under the curve and x-axis.
Method 1: Rectangle Method
Divide the interval [a , b ] into n subintervals of length b a
n
Example 1
7.2
Indefinite Integral: Integral curves
6.2.1 Definition: A function F is an antiderivative of a function f
on an interval I if F '( x ) = f ( x ) for all x in the interval.
x4 x4
x4
x4
3
Example 1
,
+ 3,
6,
+ c are antiderivatives off (x ) = x
44
4
4
6
6.3
Integration by Substitution
F is an antiderivative of f
d
[F ( g ( x)] = F ' ( g ( x) g ' ( x)
dx
F ' ( g ( x) g ' ( x)dx = F ( g ( x) + c
f ( g (x ) g '(x )dx = F ( g (x ) + c
u = g ( x) du = g ' ( x)dx
f (u )du = F (u ) + c
Example 1 Evaluate
a )
10.4
Convergence Tests
Example 1
a)
n =1
n
n +1
1
c)
n =1 n
b)
n =1
1
n2
en
d)
n =1 n
Example 2 Find the sum of the series
1 1
n
n1
2
n=1 3
Example 3
Determine whether the following series converge or diverge.
1
a)
n =1 n
b)
n =12
4
n
Example 4
D
10.2 Monotone Sequences
Testing for Monotonicity
Example 1 Show that the sequence is a strictly increasing
n
n + 1 n=1
Monotonic By Derivative f(x)
For example
1111
3, 4,17, , , , ,.
2345
Example 2
Show the sequence
4n
is eventually decreasing
n ! n=1
8.6 Matches Requiring Special Substitutions
(A) u = x
1
n
Example 1
1
a)
0
(B)
x
1+ x
3
dx
in which n is the LCD of the exponents.
Evaluate
dx
b)
2+2 x
c)
1 + e x dx
Rational Function of sin x and cos x
Some examples are
sin x
sin x + 3cos 2 x
,
,
2
cos
CHEM101:
General Chemistry I
Textbook: Zumdahl and Zumdahl
Instructor: Dr. Wolfgang Frner
Office: Buildg. 4, No. 147-3
Tel.: 3553
e-mail: [email protected]
Quizzes: in recitation classes, you MUST write a quiz in the section you are registered, or
put y
Objectives: early history, laws for calculations, atoms, molecules
Early history
3000 years ago:
iron was produced from mined minerals (weapons), in Egypt
embalming fluid were in use
2000 years ago:
4 fundamental substances (elements) were postulated by t
Objectives: Atomic masses, mole, stoichiometry, molar masses, maybe % composition of
a compound
Atomic masses: e.g. for electrons me is detected by deflection and oil-droplet exp.
Definition: (1961): 126C has exactly a mass of 12 amu (atomic mass unit)
At
Objectives: Reactions in solution, electrolytes, reaction equations, composition of
solutions
Chemical reactions in aqueous solutions
The water molecule has a dipole moment, is a polar molecule, so it attracts charged
ions.
Figure 2 (figures are at the en
Gases
Objectives: Pressure, gas laws, molar weight, density
gases are the most easy state of matter to study (liquids and solids are much more
difficult.
Reason: just molecules in random motion
Thus gases always fill up the container in which they are and
Objectives: electromagnetic radiation, particle-wave dualism, atomic structure
That light consists of electromagnetic waves was long time known.
But about 1900 new observations about light and matter brought new explanations and
new insights.
These new th
Objectives: Bonding, electronegativity, dipole moments
Bonding
Types of Bonds
Ionic bonding: Coulomb's law
+ distance r - : Q1 = Z1, Q2 = Z2 (only charge numbers, + or -)
energy of the bond between ions:
E = 2.31 10 -19 J nm
Q1 Q 2
r
In the number the ele
Objectives: Hybridization of atomic orbitals, AO-s, Localized electrons (LE) model
Hybridization
We have seen, that e.g. in H2 bonding comes about by mixing the two unpaired 1s electrons
from different hydrogen atoms:
H
H
+
1sA
1sB
overlap or mixing
The
6.5
Definite Integral
A=
n
lim
max x k 0 k =1
*
f (x k )x k ABC
Definition.
A function f is said tot be Riemann integrable on [a, b]
If the limit
n
lim
f
max x k 0 k =1
*
( x k ) x k
b
= f (x )dx
a
Exists and does not depend on the choice of the
*
partiti
6.6
The Fundamental Theorem of Calculus
b
A = f ( x )dx
a
A '( x ) = f ( x )
A (a ) = 0, A (b ) = A
A ( x ) is antiderivative of f ( x ),
T hen F ( x ) = A ( x ) + c
F (b ) F (a ) = [ A (b ) + c ] [ A (a ) + c ] = A (b ) A (a ) = A 0 = A
b
f (x )dx = F (
6.8
Evaluating Definite Integrals by Substitution
Example 1
1
dx
a)
0 3x + 1
/8
b)
4 2
3 / 8
1
d)
sin x dx
2x
sin 8 x cos5 x dx
0
cot 2 d
0
c)
ln 2
2
1
e)
dx
x 4x
1
xe
x 2
dx