MATH 161 (TIR)
Strong
Auditorium
Exponential
# 1: Intro &
Sep
•
15
Placement
:
<
score
Sep
Functions
(R)
exam
50%
to 141
dropped
•
Recitations start week after
•
Do Web Work homework
12022 ⑨
( Sep 19 week)
exam
o
r
Notation
/ Definitions :
Set of
•
integers
2-
:
Set of rationals
⊕ ( fractions of
Set of real numbers
IR
•
integers)
:
•
QUITE
=
:
-
{¥
:
•
floes
Get
f-
-
let
For
Rt
=
is
2k
=
Rules
•
§;
.
.
.
{Meira ;
}
.
.
.
}
EXPONENTIAL FUNCTIONS
1.1
•
;
is
a
an
exponents
b
be
a
power
"
"
function
"
polynomial
exponential function
"
"
of
a
/
"
:
fired positive
constant
positive integer exponent
n
bm-b.b.i.hn
times
Then
-
and
EI
more
24
:
define
we
2-
generally
=
M
,
(bⁿ )
Cbm )
We
So
-
→
We
:
.
bm
=
-
=
positive integers
:|
understand
b
in
the
"n
bm
gaps
what does 21T
,
n
b
pill
n
blm + m
=
define
then
we
are
n
then (bn )
-
b
:
-
1^-6
&
m
-
=
16
=
Notice if
-
tb
'
b-
=
Fb
Tbm
"
•
=
for
to
all
define
Erb)m
=
rational
b
"
r
=
Mn
for irrationals
mean?
D
IT ≈
3,14159
This tells
us
:
.
.
.
3 < IT
23
2 4
:
3 I <
.
17.23.2
3. 14 Lt
< 3.15
2T
8 < 2'
@
Similarly
<
211
2
"
2
"
2
'
<
29
216
<
< 2" <
23.2
2%15
23.14159
< 21T
8. 8249616
Calculator
For
positive real
any
b' +8
•
All
br Y
of
.
:
number b. and real numbers
BY
) ( be)Y
+
b÷y
=
exponential
+7
functions for b
1-
:
"
a
at
r
=
0
:
:*
y
=
@b)
> I
defined by the following
has
e
tangent line slope
ge.is
-
=
-
+7
•
be
14160
.
8. 8249778
=
SUMMARIZE
To
-
23
2 8. 8250228
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:
<
=
•
=
Y
Ri
bag
aol.br
have the
following shape
^
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er
.
:
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#2 Inverses boss
:
,
,
domain /
Sep
range
A
•
•
input
A
•
function
F-
can
function has
a
have 1-
only
6 2022
output
domain of inputs
So
,
we
Graphing be
may
f's domain D= /
have
for each
specified
± :*
→
:
a c-
domain
D:
"
IR Ir
such
on
⊕
-1-0}
That
R
\
goes be
Graphing glad
!
( never
'
④
=
on
;
Dogon
domain
=
reaches
0)
l0iᵗ•y¥@g
/
Dgcsd
?
x
•
specified
Unless
which
is
E± :
to
the
Q
what is
:
A
Dfw
→
D goes
is
fuel ?
g (2)
:
(6) =
9rad
what
:
A
:
±
=
is
Range
points
the
EO ;
C2
=
+
defined
that
fbd
may
a)
output
.
6]
,
Cg GED
of
range
O
2
B-2
the
be
can
GI
-
g
Q
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it
to)
range of
glse)
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:
A
C2 ,
=
set
has domain
function fbe)
TEE
=
① let
Q
=
Range (f) is all
Range ( posed
:
a
,
largest possible
let fool
→
otherwise
increasing
is
range of
(1*2)
}
=
_z&
10 ;
+
D)
on
range
G&¥
domain C2 ;
,
+ a)
¥
;¥÷
.
The
-
•
if
Define
it
:
.
never
EI
:
Q:
A
"
A function f is 1- 1 ( one to
attains the same output more than
-
on
since
fC2)
is
there
There
:
is
a
D)
;
domain for which
9
[0 ;
:
•
)
for>
=
if
no
✓
line
horizontal
is
l l ?
-
^
Horizontal line test CHCT)
only
set
C- D; 03
and
f•
one
1- 1
NOT
is
one
fl -27=4
=
Er
.
to
domain
y= x2
)
"
-
y
=
x2
.
food is
intercept the graph
:
y
=
1-1 if and
more
than
once
••
Let f
be
suppose f
domain R and
is
A-
f- Ig)
defined
l
-
l
.
on
a
Then
range D
,
domain
D and range R and
function fy=fGe) if and only
there exists
such that
a
1
m
.
if
Note If f:
exist)
1-1
is not
f
,
not invertible
is
f-
'
doesn't
-1
1
NANNINI f- Get ≠ (pln) )
EI suppose g is 1-1 and
1127 ; g- ' (3) ;
Find
:
ga )
:
ga )
2
=
g-
1
G)
1
=
1
g- (3) =3
g
^
5
157--7
g
g- (7)
^
Note To find formula for y= fB)
3
=
→
'
go )=7
g- (7)
g-
D
god =3;
=L ;
a
:
solve
equation
the result
→
The
Graphing
reflecting y=fbo
•
:
:
let for>
-2/-8
plot
a
-
=
l
l
-
O
O
1
1
21
8
fbe) for
r
graph
-
f
-1
r
of y
in terms
of
(g)
letters g and
f
'
=
-
'
②
g- for
:
y
I
g- f- Get
r →
-
is
about Y R
Find a formula and
r?
obtained by
-
r
-
→
=
is
Interchange
→
EI
y
)
Get given
-
!/
l
O
81
f- Get
&
-2
-
l
O
,
z
{
f-
'
graph
(a)
=
'
f- (2)
Tae =r"3
Using algebra
y=r3→ gits
:
interchange xg
-
R
|y=r1=f-/_
÷
reflect
/
r
Logarithms
let
R
thru
→
,
•
"
b > 1
bgbcx)
y
for>
.
=
2^-3
:
:
is
-
the
bgb
be)
inverse
called
is
"
log
function of
"
base b
r
be
n
¥-::t÷=
:
Cb
⊕
flat
=
bk
has
domain
1
=
begs
(f)
C- • i
=
CO ; a)
(a) has domain Lf -7
range
⊕ f- be?
(f)
D)
±
range
=
=
CO ;
C- D; a)
a)
>
1)
•
Basic
formulas
:
K
f-
•
[ If
)
1
for
For
for)
b
=
)
b b9bk
-
-
=
)
f- (FCK )
1
"
;
x
1
f- (a)
a
f
se
r
bgb ( b4
(a) toga 11 )
(b) bag 6 (1)
:
bg , (9)
(d)
logs (B)
=
whenever
=R
he
c-
domain (f)
IR
40--1
60=1
g-
3-
=
9
log , 197--2
2
112
bge, (2)
?
logo ( %)
bgbtr)
Ir
bg form
"
=
(f)
y
=
Lf )
c-
since
?
=
whenever
RE domain
>0
0
=
(c)
(e)
0
=
--R
bgb (a)
=
whenever
=
EI
•
Hse) )
f- G-
→
of
equation
6
=
BY
=
'
=
a
Mr
exponential
of
form
equation
.
f-
bagel:)
=
-1
# f.
Natural
logarithm
Trigonometry
Sep
8
2022
⑧
#4
Sep
trigonometry (continued )
Inverse Trig Functions
:
13 2022
⊕
.
Basic 4 identities Cant )
•
:
.
(3)
:
(A)
:
sin Cat
la
cos
b)
±
Sina
=
b)
cosa
=
Double angle formulas
•
-
Sin
20
cos
20
25in Q
=
cost
=
20052 ⊖
=
→
can
cosine
use
in terms of
cos 20
cos
=
It
cos
⊖
cos
LQ
l
double
costs
2
=
sinb
Gina
sinb
⊕
=
28in 2-0
1-
angle
to write
-1
Loos 8-
lcoso-n-lteg.IT
-
Similarly :/
F
:
Loo} A
20
-
Cosa
⊖
gin
_
±
:
-
-
costs
gin2⊖=1-÷2
-
sin
20 and
cos 20
Graphing big functions
Ez
:
Graph
of
y
:
sink
=
and
y
=
esek
on
the
same
set
axes
;÷i÷
y
-1
-
0
-
!
I
'
i
'
¥
.
¥
,
l
:*
.
÷¥;
¥
:*
EI
Graph
:
y=cosr ; y=
seek
i
i
i
-
-
-
!
-
-
÷
t
1
1
!
-
-
-
+
.
A
!
y=
-1
I
i.
i.
!
'
y.com
ER
y
:
NOTE
Recall
:
:
=
tank
.
tank
faux
=
is
a
-
(tank ± KITXKG 8)
periodic
=
fan
a)
tint
cos 8
y
'¥;÷:i¥÷i÷
y
I
tank
i
i
I
l
'
1
=
l
i
f
I
I
'
1
:
i
:
'
1
.
i
-
-
i. *
-
!!
END OF BSE MATERIAL
:
!
-
-
-
-
Solving trig
•
equations :
Solve for Q
⊕
Lte)
EO ;
in
such that
sin ⊖
=
D
For
< 0
sin ⊖
Q in
→
III. II
I
quadrants
"
:÷¥÷s
sin
∅
=
[
{
⊖
⊖
④ Solve
∅
IT
=
=
+
Lt
for
∅
-
t
such that
%
=
¥
=
'
∅
=
in
[0
20ft
( look at unit circle)
¥
21T
,
-
)
cost
=
O
D
cost (Lcost
te
[ te
-
1)
=
t % ; 3% }
{
"
bi
0
[
cost -0
Least
-
I
-0
¥
.
?⃝
•
Inverse
trig functions
→
s"÷¥
.
( see
section
g- sink
this
:
-
→
.
Restrict domain to interval
that
function
where
1.
is
( pails
5)
NOT 1- 1
NLT)
sink
y=
is
:
-1712
÷
:
I
b-
f-
'
(a)
=
Siri
Domain :[-1
;
Range:[ %
;
g-
-
=
"
x
13
%]
singed oil
domain
[%
"
i
÷t*
"
;
"
%]
""
"
&
I
, a
K
l
-
l
and invert
CE
Similarly
:
cos R
NOT
is
choice of domain
Natural
y=
cos r
,
[0,
→
IT
restriction
•
f-
c- "
K
=
Domain
Range
:
:
a
cos
-1
4-
-
-
-
C- *
on
for
,
a)
ability for
invert
]
÷
-
1- 1
-
.
¥
.
Y
x
E- 1 ; 13
[0 ;
IT
]
y
I
=
cos
-1k
Tag
y= tank
:
Natural restriction :(
%
→
12 ;
"
_;#:÷
f-
Gc)
=
=
tank
)
on
"
D= C- 12;
↑
/
a)
I
i.
'
y=
12
tan -1 (a)
arctan
R
R
i .
# F. Inverse Trig Functions (cont )
Composition
Ex
:
Find
①
→
i.
sin
that
so
where
≤ 0 ≤
Sin
^
-
cos
FAI
:
sin
cos
^
-
(sink
-
cos
^
"
=
+113
)
is in
'
GED
=
R
.
be
:
c-
whenever
r
tan
-1
since
→
Its
loosp.si?j-.!?
(positive
±
sneaky sneaky
"
whenever
x
EM2 ; 12 ]
can
-
_
defined
are
x
=
tan ,
,
Er :@cos -1 (cos %)
"
-4¥ )
f- Cf
=R
(sin -1 be) )
,
sin
=
trig functions
sin
① cos
-
(-53-2) %
"
f- 4- Ge))
Inverse
⊖eIM
→
whenever these
•
0=-4
sin
¥
(F)
"
'
:
④
.
(Ia ) Is
sin
Find
Recall
152022
=
① Find
•
(Eg )
^
-
Want ⊖
e.
④
functions
of
Sep
quadrant ④
c-
E- 1 ; 13
similar
is in
}
x
range of
cos
05^605 E)
-
=
-1
Er :
tan ( sin -1 ( t ) )
Evaluate
ñ
D
Let
sin
sin
Draw
( t)
-1
✗
1-3
=
triangle
a
a
(A)
and
0C
=
<
✗
:
2
I
tana
=
,
tonkin
Simplify
Er :
tan (sin -1
G)
÷
cos
Ira
Han
-
'
-
'
G))
a)
D
let
cos
tan tr
-
Han
Note
:
cos
"
a)
-72
Han
=
<
-
'
tana
a
=
,
tan
e)
-1
x
70
2*12
(t ) )
=
-
=
=
tana
LE
R
Vertical and
•
-
Let
c
Horizontal
70
shifting
)
g- for
→
:
.
+
is
e
shift
vertical
a
of
the
graph
g- flies
.
-
Ex
:
*÷
{y
y
Er
f- be c)
=
:
f- be +c)
=
y
y
=
=
Irl
!x
-
graph
y
=
fed
to the
{
r
-
r
,
r
so
31
.
¥
r≥ o
,
right
left
-
=
=
.
-
shifts
-
*
y
be
ya
+
31
t¥ ¥¥
"
.
•
-
Horizontal and vertical
Let
c
>1
:
y
y
=
=
stretching /compressing
cfbe)
{ flat
-
-
stretch
compress
:
vertically
vertically
.
R2
-
2
g
yn
?÷÷÷•-÷!i
Er
:
y
1
-
-
-
=
Sinn
y =3 sink
i
,
'
-1T
I
I.
-
let
c
> 1
:
y
y
Er
:
y
=
sin
Notice
:
floes
=
f-
=
(2k)
sin
Eck)
→
→
horizontal
-
G. E)
=
sin IT
=
*
¥,
3¥
-
-
stretch
¥
14
-
3
compression
0
"
¥
-
.
-3¥
¥
-
.
.
Ex :
a)
sin
is
IT
\
i
g-
-
sin
({ a)
'
I
-21T
1
!
•
Reflecting
•
-
y
-
y
=
=
-
317
:
f be)
→
f- C- a)
reflect y
=
for) about
→
_
Er :
-:
y
=
x2
-
f
re
-
axis
y-axis
¥-
R
y
=
1
-
x2
Er
"
y=x3 +1
:
t
:#
•
-
Composition
f- g) (a)
Er
◦
:
f- be)=
(f
→
(
g
ER
→
:
f- (a)
(f
o
22
◦
f) to
f)
◦
gtk)
,
g)
(a)
x2 +1
the>
cg g) be)
G- go f)
◦
flagged
=
o
=
functions :
of
=
=
=
=
=
-
_
sink
flglx))
glflx ))
=
=
a)
f- Gin
gli )
glad 32+2
)
f- Cftr)
f- beat 1)
sink
=
=
Sin Gi )
=
,
=
glglse) )
=
flglftr) )
9132+2)
=
=
flg
(x2
( 3N
(x2
=
=
+
+
+
1)
2
+1
3 (3×+2)+2
f-(3×2+5)
572 1- 1
D)
=
Note
For
domain
in
Er
:
:
let
& g)d)
◦
g
g
,
feed
domain
:
IR
,
be)
flogged
=
must be
to
in
be
defined
domain
gbex
domain
The domain of
G-
:
◦
→
f.
x >0
g) Get
=
fcgcai))
is
[0; D)
r
must be
Consider fcx)
•
x2
-
2+2
-
/
1.999/3.99970
193.97T
1.99
As
→
3.9970
We say
to 4
•
2.001
2
4.00030001
4
for) approaches
,
limit of flies
the
2
4. 0301
2.01
approaches
R
A-
need
2.↑É
for
•
202022 TO
Sep
#6 : Limits
as
approaches
r
2 is
equal
.
( x2
him
We write :
_
✗ +
2)
=
4
✗→ 2
Define
•
values
of
:
hiya f- be)
fbe)
are
close to
sufficiently
•
WARNING
:
high
flat
the
EI
:
means
L
close to
a
The behavior of
.
him
Consider
k
→
2
as
we
we
can
by restricting
like
flies
near
a
=a
is
r=a
RI
22
_
4
D
Notice
7k¥
is
guarantee that the
a
.
The value
of f at
.
limit
as
L
=
undefined
at
a
=
2
is
not
what determines
what determines
floe)=:÷i
go.gg/o.2goog
✗
t.90.de#
1.
him
✗→
2.0001 ↓
0
.
24499
approaches
for) approaches -4
2
"
0.25
=
&
(?)
is
i÷÷
I
,
-
i
,
2
,
.
EI
=
!
↓
÷
:
i
-
Ask
z
:
for )
4Eur
0.239
2. I
Guess
→
:
Consider
:
goes
41¥
=
-
3
when
✗
when
2=2
+2
D
gbe)
is
tim
✗
→
2
got
the
gcse)
=3
same
as
¥
(
=
is
fbd except god
assessing goes
irrelevant
to
¥72
with
god
a
.
is
defined
near
,
.
not at
2)
•
WARNING
limit
EI
Cannot simply make
:
chart
a
of
values
evaluate
to
a
.
hi?
:
Sin
◦
( Ee )
D
is
Sin
/
r
Sin
undefined
at
a
( Ee )
FAI
/
sin
:
tiny
does not
sin
exist
.
É=
0.01
O
=
since
(1001-1)=0
as
does not
Value
(100001-1)--0
r
approaches
approach
a
0
,
Sirte
specific
FEI
Consider
flu
0.00001
sin
)
:
Yn
=
•¥÷
•
{
1
a
,
0
,
→
limit :D N
.
.
≥ 0
a
co
::: ::: : : :: : ::
+7
As
a
→
0
and
from left
from the
right
flat
,
right disagrees
.
→
1
One-sided limits
•
Define
-
•
t
→
FAIT
•
In
{
f- be
) =L
means
n→
{
for) =L
a
example
fbe )
f 6)
consider
him
:
means
him
:
bigger
→
"
=L
Ot
=
0
=
l
at
Hyo
,
=D
→
him
for>
flat
=
•
_
f-
=D
;
a
from the
"
from the
finna
Ifat
approaches
right
.
f- be) =L
=L
flat
for>
D. N E
=
.
¥-4
can
arbitrarily large by restricting
%g
-
a
D. NE
but because the value of
say
.
a →o
Technically ¥%fbe)
we
.
him
say king for>
R → a-
as
:
•
We
limit
left
previous
✗
Ed
+
to
f-
a-
) him
✗→ a
•
:
Um
+7
:
a
made
close
( similar to
.
him
✗
be
→
at
for)
=
to 0
,
king poses
•
are
=
-
*
)
defined similarly
.
EI
foe >
:
Fe
=
•
for>
alight
470
•
is
:
y
both
Er
=
If
:
infinite
EI
z
f- be)
-
Define
D
=
we
he
hate
so
for both of
either
,
=
-
say
and
a
k=a
is
a
hiya
Ptsd
.
Wm
or
✗
vertical aft
of
→
at
f- G)
K
.
¥2
Y=
vertical
asymptote
at
a
=
0
:
R
I
tank
→
him
-
✗
→
has
tank
vertical
≥
•
;
asymptotes
!+
tank
at
=
✗
-
=
so
In
+
KIT 1k£
EI
y= Mr
°?u•
1%+4
-
R=0
→
is
vertical
a
asymptote
y=fM
:
consider
{
R
when
2
when
2=0
when
02×23
☒
D
f(07=2
>
¥50 for
f.
=
0
_
lim
→
d-
%o
f- be)
fbe)
(3) =L
him
,
=
0
lim
*→ +
-
O
flat
=3
for)=
laimgfbe)=
1
D. N E
-
pose)=
N
It
✗
•
.
K
EI
.
when
x
a
<O
≥ 3
Evaluate
Er :
lgqz.tl?-
and
←
him
x
→
Z
zt
-
R
D
-
-
As
gets
x
close
2
to
From the left Csec 2)
So,
&
2
222
when
so
gets
at
,
-
close to 4
> 0
r
and
k
close to 2
d- R
→
2%
therefore
gets
re →
-
From the
(similarly to
LARGE
KI
lim
:
"
2-
right
r
2-
"
=
2.
✗→
since
to
x
2- x 20
Ck > 2)
< 2)
as
:
him
✗
→
Lt
2%
=
-
•
÷y
→
LARGE