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AP Calculus AB: Chapters P and One

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Donald Phan

on 27 March 2015

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Transcript of AP Calculus AB: Chapters P and One

Chapters P and One
Chapter P - Finding Intercepts

Example!
Find any intercepts for the following equation: y=(x-3)/(x-4)
To find the y-intercept, set all x values to 0
x = 0, y = (0-3)/(0-4)
y = 3/4
(0, 3/4) y-intercept

To find the x-intercept, set all y values to 0
y = 0, 0 = (x-3)/(x-4)
0 = x-3
x = 3
(3, 0) x-intercept
Chapter P - Testing for Symmetry
Tests for Symmetry Rules
1. the graph of an equation in x and y is
symmetric with respect to the y-axis
when
replacing x by -x
yields an equivalent equation
2. the graph of an equation in x and y is
symmetric with respect to the x-axis

when
replacing

y by -y
yields an equivalent equation
3. the graph of an equation in x and y is
symmetric with respect to the origin
when replacing
x by -x and y by -y
yields an equivalent equation
Using Intercepts and Symmetry to Sketch a Graph
If you know the intercepts and the symmetry of an equation, you can use this information to sketch a graph.
Example!
Test for symmetry with respect to each axis and to the origin
Jack Kiriazes and Donald Phan
DP
DP
y² = x² - 5
with respect to the y-axis
y² = (-x)² - 5
y² = x² - 5
with respect to the x-axis
(-y)² = x² - 5
y² = x² - 5
with respect to the origin
(-y)² = (-x)² - 5
y² = x² - 5

y² = x² - 5 is symmetric with respect to
both axes and the origin

DP
Example!
y= -½x + 3
y-intercept: y = -½(0) + 3 = 3
(0, 3)
x-intercept: -½x + 3 = 0
-½x = -3
x = 6
(6, 0)
Symmetry: None
Finding Points of Intersection
Given two equations, make sure that
they are
both set equal to y
.
Once they are both equal to y,
set both
equations equal to each other.
Solve for x
, and
plug back in
to equation
to get y value.
Example
Given: 5x + 3y = -1
x - y = -5
5x + 3y = -1
y = 1/3(-5x -1)
x - y = -5
y = x + 5

1/3(-5x -1) = x + 5
-5x - 1 = 3x + 15
-16 = 8x
-2 = x


When x = -2, y = 3
Point of Intersection is: (-2, 3)
Finding the Slope of a Line
Slope Formula
Example!
Find the slope of the line passing through the following points
(-7, 8) and (-1, 6)
(6-8)
(-1-(-7))
_________
=
____
-2
6
-1
3
____
Slope =
Finding an Equation of a Line
Point: (-3, 0)
Slope: m = -2/3
Point-Slope Form
y - 0 = -2/3 (x - (-3))
y = -2/3 (x + 3)
Slope-Intercept Form
DP
DP
DP
DP
Rate of Change
The purchase of a new machine is $12,500, and its value will decrease by $850 a year. Use this information to write a linear equation that gives the value V of the machine t years after it is purchased. Find its value at the end of 3 years.
Slope = -850
V(t) = -850t + 12,500
V(3) = -850(3) + 12,500 = $9950

The value of the machine after 3 years will be $9950.

DP
Evaluating a Function
DP
Finding Domain and Range of a Function
The Domain of a function is the set of all real numbers for which the equation is defined. (the x-values)
The Range of a function is the set of all real numbers that the function can equal. (the y-values)
DP
Find the Domain and Range of:
Domain: (-infinity, infinity)
Range: (-infinity, 0]
DP
Example!
Important Graphs
to Know!
Transformations!
Composite Functions
Even and Odd Functions
The function y = f(x) is
even
when
f(-x) = f(x)
The function y = f(x) is
odd
when
f(-x) = -f(x)
Example!
f(x) = 1 + cos(x)
f(-x) = 1 + cos(-x)
= 1 + cos(x)
this function is even

DP
DP
DP
DP
DP
DP
Chapter 1- The Tangent Line Problem
Chapter 1- The Area Problem
As point Q approaches point P, the slope of the secant approaches the slope of the tangent. Thus, the slope of the tangent is the limit of the slopes of the secant lines.
Slope of the secant line
Pre-Calculus
Spoiler Alert!
Pre-Calculus
One can estimate the are under the curve by putting rectangles underneath it, and then finding the area of the rectangles. The more rectangles that are used, the better the estimation becomes. The final goal is to determine the limit of the sum of the areas of the rectangles as the number of rectangles increases without bounds.
JK
JK
Finding the Limit Numerically and Graphically
Numerically
Graphically
JK
Evaluating Limits with an Indeterminate
By Factoring Example!
By Rationalizing Example!
JK
More Fun with Limits
Piecewise Limit
No Limit
JK
Finding the Limit of Polynomials
Just Plug It In
JK
Limits of Trig Functions
Squeeze Theorem
Continuity
Graphs That Aren't Continuous
Removable
Removable
Non-Removable
JK
Greatest Integer Function
Intermediate Value Theorem
Infinite Limits
As f(x) approaches 2 the function shoots up from the left and down from the right. Thus the limits are - infinity from the left and + infinity form the right.
JK
JK
Continuity Example
f(x)= x^3+3x^2+1
@ (2,21)
1) f(2)= 21
2) lim x^3+3x^+1 = 21
x approaches 2
3)
lim x^3+3x^+1 = f(2)
x approaches 2
Continuous
JK
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