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 a triangle satisfy the Pythagorean Theorm,
then the tri
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, large table maps, and movable markers
representing airp
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 the value of p. The error, or remainder, of such an appr
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, straight line. Set your compass to the
length of the ori
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.
On a sheet of paper, sketch the following TIC-TAC-TOE
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 Polygons and Circles
I will be trying my best to incorpor
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. Name one major arc.
8. Draw a circle concentric to ci
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 )2 xi =
max(xi )0 i=1
Z b
a
r(x)2 dx
Example 1
Exampl
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 ) g(xi )xi =
n
max(xi )0 i=1
Z b
a
(f (x) g(x)dx
That is,
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 find a function F(x)
such that F 0 (x) = f (x).
Definition
Lecture 17
Antiderivatives/ The Indefinite Integral (cont.)
The Definite Integral
Preliminaries
(Defining) The Definite Integral
Antiderivatives/ The Indefinite Integral (cont.)
Example 1
Example 2
The Definite Integral
Preliminaries: notation
It allows t
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 (f u)0 (x)?
Idea:
Theorem (The Chain Rule)
Let y(x) = (
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 = 0 +
1
n2
+
1
n
22
1 (i1)2
n2 + . . . + n n2 + . . . +
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, and Dilations.
Translations simply move the coordinates o
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. Inferences made by inductive reasoning are not necess
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
x
Which equation could find the distance from point G
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 draw two polygons with congruent angles that are NOT sim
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 this formula will work even for the tilted shapes.
h
h
b
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 with a line symbol above them.
Planes:
Have no thickness
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 Association of America
PO Box 471 Annapolis Junction, MD 2070
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 origin?
+PQR is the image produced after
reflecting +PQR acr
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, polyheda are named with prefixes we have already used.
Oc
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. In an isosceles triangle, the base angles will always
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 be
concluded from the additional
statement the measure
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 using point, line, and plane.
Ray
Circle
Angle
Polygon
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 we find the rate of change of the other
quantities?
Exam
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
Since f 0 (x) =
derivative:
df
dx (x)
is a function of x
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 any real number,
Rb
Rb
Rb
a (f (x) + g(x)dx = a f (x)d
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 y 6= 0 by
x
rem(x, y) = x y
y
j k
where xy denotes t
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 sought
from the Advanced Placement Program. Teachers may