5128_Ch10_pp530-561 1/13/06 3:50 PM Page 530
Chapter
10
Parametric,
Vector, and Polar
Functions
I
n 1935, air traffic control was conducted with a
system of teletype machines, wall-sized blackboards,
The Pythagorean Theorem
Geometry 9.2
By now, you know the Pythagorean Theorem and how to use it
for basic problems.
The Converse of the Pythagorean Theorem states that:
If the lengths of the sides of
5128_Ch09_pp472-529.qxd 1/13/06 3:44 PM Page 472
Chapter
9
Infinite Series
crucial to the
analysis of the world is p.
One mathematical constant The p-series
1
1
1
1
p2
1
22
32
42
52
6
approximates t
Name_ Period _
Segments and Angles
Geometry 3.1
All constructions done today will be with Compass and Straight-Edge ONLY.
Duplicating a segment is easy.
To duplicate the segment below: Draw a light, s
Pythagorean Introduction
Geometry 8.0
We will dive right into using the Pythagorean Theorem.
We have been using the Pythagorean Theorem for several area applications and applications involving circles
Geometry EOC
EOC Review:
Focus Areas:
Trigonometric Ratios
Area and Volume including Changes in Area/Volume
Geometric Probability
Proofs and Deductive Reasoning including Conditionals
Properties of Po
Circle Review
Geometry 6.1
Name each part of the circle below:
1. Name two chords.
2. Name one diameter.
3. Name three radii.
A
4. Name one tangent.
5. Name two semicircles.
6. Name nine minor arcs.
7
Lecture 22
Geometric Applications
Volumes
Geometric Applications - Volumes
Solids of Revolution
Geometric Applications - Volumes
Solids of Revolution
Taking the limit, we get
n
volume(V) =
lim
n
r(xi
Lecture 21
Geometric Applications
Areas Between Curves
Geometric Applications - Areas Between Curves
Reminder
We can in fact measure the area between y = f (x) and y = g(x):
n
area(A) =
lim
(f (xi )
Lecture 16
Antiderivatives/ The Indefinite Integral
Antiderivatives/ The Indefinite Integral
Problem
Let F(x) be differentiable over some interval
(Indefinite) integration: Given f (x), we want to fin
Lecture 17
Antiderivatives/ The Indefinite Integral (cont.)
The Definite Integral
Preliminaries
(Defining) The Definite Integral
Antiderivatives/ The Indefinite Integral (cont.)
Example 1
Example 2
Th
Lecture 10
(Derivatives of) Compositions of Functions, A.K.A. the Chain Rule
(Derivatives of) Logarithmic Functions
(Derivatives of) Compositions of Functions, A.K.A. the
Chain Rule
Question.
What is
Lecture 18
Example (cont.)
The Definite Integral - In General
Example (cont.)
Divide [0, 1] into n subintervals (of equal length),
i
n1
,
],
.
.
.
,
[
[0, 1n ], . . . , [ i1
n n
n , 1]:
1
n
1
n
sn =
Transformations
Geometry 14.1
A transformation is a change in coordinates plotted on the plane.
We will learn about four types of transformations on the plane:
Translations, Reflections, Rotations, an
Inductive Reasoning
Geometry 2.1
Inductive Reasoning:
Observing Patterns to make generalizations is induction.
Example:
Every crow I have seen is black, therefore I generalize that all crows are
black
EOC Geometry Sample Items Goal 1
North Carolina Testing Program
1.
The angle of elevation from point G on the ground to the top of a flagpole is 20. The
height of the flagpole is 60 feet.
60 feet
G
20
Similarity
Geometry 11.2
What makes two polygons similar?
Corresponding angles must be _.
Corresponding sides must be _.
Basically: Same shape, different size.
Are both definitions necessary?
Try to d
Area
Geometry 8.1
Rectangles, Parallelograms, Triangles, Trapezoids, Kites:
1. What formula can you use for the area of any parallelogram (including
rectangles, squares, and rhombuses)?
Diagram why th
Building Blocks of Geometry
Geometry 1.1
Points:
Have no size, only represent a location.
.
Named with a capital letter.
Lines:
Have no thickness, but infinite length.
.
.
Named using any two points w
MAA American Mathematics Competitions
2016
AMC
8
TEACHERS MANUAL
A guide for administering a successful and rewarding AMC 8 Competition
COMPETITION DATE: TUESDAY, NOVEMBER 15, 2016
Mathematical Associ
EOC Mathematics Geometry Sample Items Goal 3
North Carolina Testing Program
1.
Point J (p, q) is a vertex of
quadrilateral JKLM. What are the
coordinates of J after JKLM is
rotated 180 about the origi
Solid Geometry
Geometry 10.1
Vocabulary and More Drawing!
A polyhedron is a geometric solid made up of polygon faces which meet
at straight-line edges that come together at vertices.
Like polygons, po
Geometry 4.0
Triangle Basics
First: Some basics you should already know.
1. What is the sum of the measures of the angles in a triangle?
Write the proof (Hint: it involves creating a parallel line.)
2
EOC Geometry Sample Items Goal 2
North Carolina Testing Program
1.
2.
The conditional statement all 45
angles are acute angles is true.
Based on this conditional statement,
which of the following can
Geometric Proof
Geometry 13.1
Geometry begins with Undefined Terms: Point, Line, and Plane.
Using these three terms, all other geometric terms can be defined.
Practice: Try to define these three terms
Lecture 12
Application: Related Rates
Maximum and Minimum Values
Application: Related Rates
Idea:
Certain quantities are related to each other. Some of the quantities
vary at a certain rate. How can w
Lecture 8
Higher Derivatives
Calculating Derivatives
Polynomials
Exponentials
Products
Quotients
Other possible issue: the graph of f might also have a vertical tangent.
Example 1
Higher Derivatives
S
Lecture 19
Properties of the Definite Integral
The Fundamental Theorem of Calculus
Properties of the Definite Integral
Proposition
I
I
I
I
I
Rb
a 1dx = b a,
Rb
Rb
a k f (x)dx = k a f (x)dx, where k is
2016 AMC 10A
February 2nd, 2016
1
What is the value of
11! 10!
?
9!
(B) 100
(C) 110
(A) 99
2
(B) 2
(E) 5
(C) $87.50
(D) $90.00
(E) $92.50
The remainder can be defined for all real numbers x and y with
AP Calculus BC
2000 Scoring Commentary
The materials included in these files are intended for non-commercial use by AP
teachers for course and exam preparation; permission for any other use must be so