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CH
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ing
ketch
rve S
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ontal
Horiz otes
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
100
1.000100
1000
1.000001
Lets examine the function
x2
er
apt
Ch ht
E ig
Lo
ntial & ions
pone Funct
Ex
thmic
gari
ithmic
ogar ns
8.3 L nctio
Fu
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
log 8 5 = y
log b 5
log 8 5 =
log b 8
if we let b = e
lo
er
apt
Ch ht
E ig
Lo
ntial & ions
pone Funct
Ex
thmic
gari
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 implicitly
y dy
e
=1
dx
e y e y
dy 1 1
= y =
dx e
x
Examples
er
apt
Ch ve
Fi
Cu
hing
ketc
r ve S
s for
edureur ve
Proc a C
5.5
wing
Dra
ple
Exam
#5 page 240
y=
1. Domain:
2. Intercepts:
3. Asymptotes:
Vertical asymptote:
Horizontal asymptotes:
4. Interval of Increase or decrease:
5. Local Max and
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. We use them, together with intervals of
increase or d
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 behavior of the function
This tells us that the graph is u
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 and b are constants
This equation has a
slope of m &
a
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 ourselves how does
this relate to what we have learned
er
apt
Ch ht
E ig
ntial & ions
pone Funct
Ex
thmic
ogari
L
rowth
tial G
onen cay
5 Exp & De
8.
In many sciences, certain quantities grow or decay at a rate that
is proportional to their size.
In many sciences, certain quantities grow or decay at a rate
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 that change
In previous sections we saw that limits of th
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 discussed in previous lessons, a maximum or a minimum occu
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 slopes of
secant lines
In section 1.1 we found the t
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 ) " 8
a) lim 2
x!2 2
( ) " 7 ( 2 ) + 12
2
lim
=
lim
b)
x