The Fundamental Theorem of Calculus
This theorem bridges the antiderivative concept with the area problem. Indeed, let f (x) be a
function defined and continuous on [a, b]. Consider the function
F(x) =
f (t)dt
defined on [a, b]. Then we have
=
Since
f (t)
Partial derivatives
In two dimensions, when we have a function y(x), we can readily define dy/dx as the
slope of the curve y(x). Here we'll use two concrete examples to illustrate partial
derivatives: first we'll look at a curve y(x) that is also a functi
Analytical derivatives
But what if we 'know' the formula for the function x(t)? I have put 'know' in quotation
marks, because for anything in physics, the only things that we know are the
measurements. There are only a finite number of these, so we just
Trigonometric functions
Sine and cos functions are important, especially in circular motion, simple harmonic
motion, components of forces and other cases involving components of vectors.
Fortunately, the derivatives here are simple. Let's work them out, u
Population vs Sample
The population includes all objects of interest whereas the sample is only a portion of the
population. Parameters are associated with populations and statistics with samples. Parameters
are usually denoted using Greek letters (mu, si
Statistics: Data Description
Definitions
Statistic
Characteristic or measure obtained from a sample
Parameter
Characteristic or measure obtained from a population
Mean
Sum of all the values divided by the number of values. This can either be a population
Stats: Counting Techniques
Fundamental Theorems
Arithmetic
Every integer greater than one is either prime or can be expressed as an unique product of
prime numbers
Algebra
Every polynomial function on one variable of degree n > 0 has at least one real or
Stats: Hypothesis Testing
Definitions
Null Hypothesis ( H0 )
Statement of zero or no change. If the original claim includes equality (<=, =, or >=), it is the null
hypothesis. If the original claim does not include equality (<, not equal, >) then the null
Statistics: Frequency Distributions & Graphs
Definitions
Raw Data
Data collected in original form.
Frequency
The number of times a certain value or class of values occurs.
Frequency Distribution
The organization of raw data in table form with classes and
Stats: Probability
Definitions
Probability Experiment
Process which leads to well-defined results call outcomes
Outcome
The result of a single trial of a probability experiment
Sample Space
Set of all possible outcomes of a probability experiment
Event
On
Stats: Estimation
Definitions
Confidence Interval
An interval estimate with a specific level of confidence
Confidence Level
The percent of the time the true mean will lie in the interval estimate given.
Consistent Estimator
An estimator which gets closer
The Fundamental Theorem of Calculus
This theorem bridges the antiderivative concept with the area problem. Indeed, let f (x) be a
function defined and continuous on [a, b]. Consider the function
F(x) =
f (t)dt
defined on [a, b]. Then we have
=
Since
f (t)
Integration: How do the results of a variable rate add up?
Let's leave displacement time graphs for a moment, because my favourite example
of an integrator is a bucket. A bucket integrates the flow of water from a tap above
it. In our example, someone is
Analytical derivatives
But what if we 'know' the formula for the function x(t)? I have put 'know' in quotation
marks, because for anything in physics, the only things that we know are the
measurements. There are only a finite number of these, so we just
Partial derivatives
In two dimensions, when we have a function y(x), we can readily define dy/dx as the
slope of the curve y(x). Here we'll use two concrete examples to illustrate partial
derivatives: first we'll look at a curve y(x) that is also a functi
Trigonometric functions
Sine and cos functions are important, especially in circular motion, simple harmonic
motion, components of forces and other cases involving components of vectors.
Fortunately, the derivatives here are simple. Let's work them out, u
Calculus Notes
Definition of an anti-derivative-A function f is an antiderivative of f on an interval I if F(x) =f(x)
for all x in I.
Theorem 5.1 Representation of Antiderivatives- If F is an antiderivative of f on an interval I then
G is an antiderivativ
Negative exponents-for any real number a and any integer n where a does not equal 0 a to the
negative equals one over a to the n and 1 over a to the negative n equals a to the n
Multiplying powers-For any real number a and integers m and n, a to the m tim
Order of operation:simplify the expressions inside grouping symbols, such as parentheses,
brarckets, braces, and fraction bars-evaluate all powers, do all multiplications and divisions from
left to right, then all additions and subtraction from left to ri
Definition of a Polynomial in One Variable- A polynomial of degree n in one variable x is an
expression of the form a0xn An-2x +a,n where the coefficients a0,aa1,a2 an represent real
numbers, a0 is not zero, and n represents a nonnegative integer
Definiti
EXERCISE 2.3 Data
Presentation
Objectives
After completing this exercise, you should be able to
1. Explain the difference between discrete and continuous variables and give examples.
2. Use one given data set to construct a line graph.
3. Use another give
Shi 1
Simulations
Makes
Misses
Above/Below
26
1.
Simulation 1
20
19
Below
Simulation 2
27
12
Above
Simulation 3
14
25
Below
Simulation 4
12
27
Below
Simulation 5
24
15
Below
Since we are using a six sided die and Shaq had a shooting percentage of 53%, it
Zero product property-for any real numbers a and b if ab=0 then either a=0 or b=0, or both
Quadratic Formula- The solutions of a quadratic equation of the form ax2+bx where a=/= 0, are
given by the following formula
Quadratic Formula
Sum and Product of ro
1. Find the sum of the measures of the interior angles of a(n):
a. Octagon
b. 16-gon
2. Find the value of x:
a.
b.
c.
3. The measures of the interior angles of a quadrilateral are x,
2x, 3x, and 4x. What are the degree measures of the
angles?
1. The sum o
Chapter 6 Review: Station 1:
1.
2.
3.
4.
5.
Station 2:
1. The perimeter of a rectangle is 154 feet. The ratio of the length to the width is
10:1. Find the length and width.
2. Solve the proportion:
y
3
=
a.
20 10
b.
4
2
=
a 3 5
c.
2p +5 9p
=
3
1
Station 3
Quarterly 2 Review Answers:
1. a. x = 30
m6 = 50o
b. x= 3
9. x = 40o
c. x = 95
2.
1
2
3.
1
1
a. slopes and - parallel
2
2
10. a. equilateral
b. isosceles
2
2
and - - neither
3
5
b. slopes
c. slopes
4
1
and - - perpendicular
1
4
c. scalene
11. a. right tr
Station 1:
1. _
Transitive Property
2. _
Converse of a statement
8. _
Straight angle
A. 2 lines that intersect to form right
angles.
B. AB=AB
C. 2 coplanar lines that do not
intersect.
D. 2 noncoplanar lines that do not
intersect.
E. 2 angles whose sum is
Geometry Quarterly 1 Answers:
Station 1:
1. G
2. I
3. K
4. A
5. F
6. B
7. N
8. T
9. D
10. L
11. M
12. C
13. H
14. R
15. J
16. S
17. Q
18. O
19. E
20. P
Station 2:
9. plane BAC
10. AC or EF
11. AE or AC
Station 3:
1. 7 and 4 , 3 and 6
2. line HD perpendicu