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ing
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rve S
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ontal
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5.2
ympt
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Lets examine the function
x2 + 2
y= 2
x +1
Lets examine the function
x2 + 2
y= 2
x +1
X
y=
x2 + 2
x2 + 1
0
2
1
1.5
2
1.2
5
1.038462
1
er
apt
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ntial & ions
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thmic
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ogar ns
8.3 L nctio
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Laws of Exponents
ay = x
log b a y = log b x
y log b a = log b x
log b a log b a
log b x
y=
log b a
8y = 5
l
TER
AP Two
CH
ives
rivat
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heet
orks ey
W
wer K
Ans
#1
y = 3x ! 5x + 2x ! 7x + 2
4
3
2
y' = 12x ! 15x + 4x ! 7
3
2
#2
2 1
y = 3x + 2x ! + 2
x x
2
!1
!2
y = 3x + 2x ! 2x + x
2
!2
y' = 6x + 2 + 2x !
er
apt
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ntial & ions
pone Funct
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thmic
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es of ns
rivativ nctio
.4 De mic Fu
8
garith
Lo
Laws of Exponents
d
1
( ln x ) =
dx
x
Let y = ln x
Then e y = x
Differentiating this implic
er
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s for
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Proc a C
5.5
wing
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ple
Exam
#5 page 240
y=
1. Domain:
2. Intercepts:
3. Asymptotes:
Vertical asymptote:
Horizontal asymptotes:
R
TEve
AP Fi
CH
C
ing
ketch
rve S
u
rtical
5.1 Ve
otes
ympt
As
In this chapter we will look at further aspects of curves ~
vertical asymptotes, horizontal asymptotes, concavity, and
points of inection
February 03, 2014
Limits and Rates of Change
1.2 The Limit of a Function
Concepts like Terminal Velocity, population
curves etc will all use limits.
February 03, 2014
We begin by investigating the beh
Chapter One
Limits and Rates of Change
1.1 Linear Functions & the
Tangent Problem
Gavin Denham
Tuesday, January 28, 2014 2:27:17 PM MT
A linear function is a function f of the form
f ( x ) = mx + b, m
R
TEur
AP Fo
CH
ues
e Val
xtrem
E
g or s
reasin nction
.1 Inc g Fu
4
easin
Decr
When discussing the concept of a function in a left to right manner
increasing or decreasing in nature, we need to ask
er
apt
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ntial & ions
pone Funct
Ex
thmic
ogari
L
rowth
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onen cay
5 Exp & De
8.
In many sciences, certain quantities grow or decay at a rate that
is proportional to their size.
In man
TER
AP Two
CH
ives
rivat
De
.1 The
2
ative
Deriv
In previous sections we saw that limits of the form
f (a + h) f (a)
lim
h0
h
occur as slopes of tangents, computing velocities and
other functions tha
R
TEur
AP Fo
CH
ues
e Val
xtrem
E
and
ximumlues
.2 Ma um Va
4
Minim
As discussed in previous lessons, a maximum or a minimum occurs
when the rst derivative of a function is made equal to zero.
As dis
TER
APONE
CH
es of
Rat nge
its & Cha
Lim
its to
ng lim s
i
4 Us
1.
gent
d tan
n
In section 1.1 we found the tangent line to the parabola y=x2 at
the point (1,1) by computing its slope as the limit of
Math 31 Final Exam Prep #1
1. Determine the following limits if they exist
3+ t " 3
t
3+ t " 3 3+ t + 3
lim
t !0
t
3+ t + 3
lim
t !0
x 2 " 16
lim
x! 4
x"4
x " 2x " 8
x!2 x 2 " 7x + 12
( 2 )2 " 2 ( 2 )