The Earth-Atmosphere System
Atmospheric Composition
Table 1. Composition of the Atmosphere (Clean dry air near sea level)
Gas Constituent
Chemical Formula Volume Mixing Ratio Sources
Nitrogen
N2
0.78083
Biological
Oxygen
O2
0.20948
Biological
Argon
Ar
0.0
9. INDEFINITE INTEGRALS
9.1. Antiderivative
9.1.1. Definition
Let us consider a function f : IR IR: x f ( x).
The function F is an antiderivative of f if F ( x) f ( x) , or dF ( x) f ( x) dx .
Example
Let f : x x 4 .
A few antiderivatives of f are:
x5
F1
5. DIFFERENTIATION
5.1.
The derivative in a point
5.1.1. A rate of change
When a train travels 108 km in a time interval [t1,t2] = [0,8;2,3].of (=90 min = 1,5 h) we
s 108
km / h 72 km / h. .
conclude that its average velocity was v
t 1,5
s2
s
(km)
s1
t
t
3. LIMITS
The concept limit allows us to investigate the behaviour of a function in the close
neighbourhood of a point. The functional value in this point is not important here.
Example 1
(x 1)(x 2)
f(x)
x 1
met
dom f = IR \ 1
3
The function f is not def
7. INFINITE SEQUENCES AND SERIES
7.1. Infinite sequences in IR
7.1.1. Definition
An infinite sequence of real numbers is a mapping of IN ( or IN0 ) in IR
f : IN 0 IR : n f (n) un
Every n corresponds with a real number un. The natural order 1 , 2 , 3 , . o
11. APPLICATIONS OF DEFINITE INTEGRALS
11.1. Surface of a plane area
11.1.1. Surface limited by y = f(x), the x-axis and the lines x = a and x = b
Let us repeat 10.2.4. :
If the function f is continuous on a, b and f 0 for every x a, b then the value of t
4. TRANSCENDENTAL FUNCTIONS
4.1.
Trigonometric functions
The trigonometric functions are important because they are periodic, or repeating, and
therefore model many naturally occurring periodic processes, e.g. a mass attached to a
string, the pendulum, an
8. COMPLEX NUMBERS
8.1. Definition and operations
Imagination was called upon at least three times in constucting real numbers: first the set
of all integers was constructed from the counting numbers, then the set of rational
numbers was constructed from
2. CONTINUITY
2.1. Continuity at a point
Definition: a function f is continuous in a point a if the graph of the functions does not
show a jump in a.
If the function is not continuous in a, it is defined as discontinuous in a.
Examples
y
y
y
f(a)
f(a)
a
a
Name: _ Class: _
Integrating f (ax + b) and Integration by Substitution
Question 1: Find the antiderivative of each function with respect to x:
a)
b)
F ( x) sin( 2 x 3)
F ( x) (3 x 5) 6
Question 2: Integrate with respect to x:
a)
b)
4
(2 x 3) dx
Question
Question 3: What does the second derivative of a function tell you about a function?
The first derivative of a function tells us the rate at which a function changes. If the first
derivative is positive over some interval, then the values on the function
Conics Summary
Academic Skills Advice
Conics are curves that are produced when a double cone is intersected by a plane. The 3
main types of conics are:
Parabola
Ellipse (including the circle which is a special case)
Hyperbola
General Equations of Conics:
Academic Skills Advice
Sketching Curves Summary
When sketching a curve its useful to have an idea of the general shape.
= 2
= 2
= 3 +
= 3 +
Then there are 2 things to think about:
the and intercepts (where does it cross the axes)
the turning point
IB Math HL Integration Practice
Name: _
Integrate each of the following.
1) (a)
(b)
4 x dx
(c)
x
4 x 2 dx
(d)
x
4 x dx
2) (a)
3xe
x 2 5
4
x 2 dx
(b)
dx
3xe
x 5
dx
3) (a)
(b)
sin dx
2 x
3
4) (a)
2
x 6 x 13
3x dx
sin dx
3 x
3
(b)
x 3
x 2 6 x 13 dx
(c)
x
2
Integration by Substitution
Dr. Philippe B. Laval
Kennesaw State University
August 21, 2008
Abstract
This handout contains material on a very important integration method
called integration by substitution. Substitution is to integrals what the
chain rule
Integration: Summary
Philippe B. Laval
KSU
September 14, 2005
Abstract
This handout summarizes the various integration techniques. It also
give alternatives for finding definite integrals when an antiderivative cannot be found.
1
Strategy for Integration