Exam 3
·
(will have questions
Sumulative
Roughly
-
·
importantdeformation
48%
of questions
-
-
new
content
140
pts
models
4:I question (Differential
Equation, notpopulation
Chapter
18: I questions
Chapter
-Chapter 11:
8
questions
(5:88 PM-7:45PM)
Time:I have & YS minutes
assemedations: I have 28 minutes
With 1.5xtime
max
-
-
·
topic)
18 pte, total
out
total, each worth
questions
of
I
7:
questions
Chapter
I question
8:
Chapter
-
·
I
I4
-
·
on
Exam 1& Exam
topics, reston
prior
over
Location:
Science
Center, Room 2.383
Learning
What
to
bring:
Pencil (s) and her
-
·
-
pen
lead. If regular pencil, bring
extra
sharpence
of
being
to
be
safe!
Bring spaces backups
C
alculator
Scientific
allowsch
Strictly
I graphingcalculators
·
·
mechanical
pencil,
content
am
Grams
Intatts.
(f(x)g(x)dy f(x)g(x) jf(x)g(x)dy
-
=
Tip take
get(is hr(x)
Auign
f(x) tosomethingwhere file &easy to
/ f(x)g(x) dy
If bounds exist, f(x)g(x) [f(x)g(x)]a apply hands to
is
-
->
-
7.2
With
cas &
·
Fagonometrics
olf
-
Integrate
one
sin:
them (cos
of
or
the
sin) to
power of
is
an
odd #,
to
be
split
ITX
Lie
wesin), use cos(x)+siny)=I (rearranged) &
concept
if sos,- cases,
integrate url a-sub (u sin() or cas(x))
a-ouh & ears
are even, apply one of the
identities, then
If both
following
same
=
-
sin?(x) = (1-ex(2x)
ex(x) I(1 cas(2x))
Useful identity: sin(x)cas(x) x
·
=
·
-
·
+
=
=
With tan & see
Same idea
as above,
-
-
-
of
see
pause
even
is
of tan power odd,
Useful things:
is
-
·
·
·
use
identity
82,
set
use
process
remove see (x) tan
Itan (x) dy In Ise(x)) 1
sec(x) dy ha) sec(x) tan(x))
=
(x):1+tavi(x)
(x), use
+
=
+
2
+
Other
useful identities:
-
-
-
(B) I[sin (A-B) ein (A B)]
Sin (A) ein (B) I/ess (A-B) -car(A +1)]
Cos (A) crs(B) I[cas (A-B) ear(A+B)]
Sin (A)
exe
+
=
=
+
=
+
from in less,
a
u
tan(), d=c(x)
=
sec(x), di-sec(y) tan (X)
=
7.3
·
Figemetic
.
Substitution
---
·
Ex,
xx,
Na,
=
x:a
a
x=
like
Treat
*
7.4
Integration
of rational
functions by
Partial
fractions
·
·
a cas (A)
ein(), work out,
a.ex(F)
tanlt), work out,
->
x a
->
ex(f), work out
u-sub,
use
x
-
a
tan (t)
instead
of
u, use
power denominator
power,
If numerator
use
trig
identities
long division
power denominator
power, do following:
If numerator
out
denominator (is Yx(x (3x+3))
If possible, factor
-
2
Once
-
1.
implified, apply
2
at
2.
at
one
3.
of
+2)
following forms
the
ab)2...
numerator (A, AxtB, A...) by other
denominators
by highest
power
of case 3 used, only multiply
teller
Set
to
solve
numerator
(A, B, ...)
equal
for capital
-
Multiply
each
·
-
S
-
7.5
Strategy for
Integration
to
denominators:
Apply
back into
fractorsI then
find
↳
integrals
7.8
Type I:Infinite Integrals
If I f(x)dx, di Satf(x)dx
·
Improper
Integrals
d (F(x)Ja-i (F(t) -F(a)], scle
olf S f(x) dx, Sef(x)dx [F(x)?- [F(b) FCH)], solve
(him 3,-0)
exists, convergent, otherwise
divergent
of limit
(--,%), split
integral 21 (where I a real #
olf hands
For S, Idx, convergent
ifp>I, divergentif
p11
-
-
->
C
-
-
-
-
-
is
-
are
is
-
Type II: Discontinuous dategrals
·
-
-
-
-
solve
(a,b] Ja f(x)dx,-ttSef(x) dy, then
save
(a,b)S!f(x)dx, i Sf(x) dx, then
-
of limit
exists, convergent, otherwise
divergent(limit -a -0)
=
If discontinuous 21, where ab, splitintegral & solve
diverge
If I partdiverges, entire thing
=
= 8 when
=
Comparison:f(x) g(x)
Sa g (x) dxalso converges
If Sf(x) dx converges, then
ofS,g(x)dx diverges, then Saf(x) dx also diverges
·
-
x
⑧
a
C
E
-
When
the Length. When
8. I
·
eld) dx
f(x), I:Sa
fly), I Say): dy
am2
8.2
·
S=2πSavar
of
Surface of
Area
-
2
de
or
opposite of
dedx
used the
*Ydy & dx or by depends the variable
b (ie f(x) dax &dx, f(y) d*de&dy)
function
original
or
Revolution
in
on
->
+
y y.x f(x) f(x).
9.1
=
=
=
Modelling
/Differential
Example: Show
Equations
Left(pull
y
I-y
=
y)
x
(ddea isto
find
integral
yetis
that
a
dennative):
(l-ce) (cet)-(lect) fce
=
(l-xet)
2
d
bez math
=
g(x)f(y)he)-e ,
=
=
·
Separable
-
Equations
-
& solutions
solution
of y
E
y
9.3
dy
rotation
axis
war is
-
*/g)
or
Right (plug
=
(y2-1)
y
in
&
simplify)
getareamaet er
↓
[(Azette) 2cette)
H+
+
=
(1
-
xet)
-
2
(h(y)dy (g(x)dx a(y) b(x) +
-
=
=
=
condition given
for (if
starting
Solve
dy/ 9(x)/h(y) forem
u/dx
Sometimes
question
given
=
18.1
leaves
x
·
Parametric
-
-
·
=
as
that
outside variable
determines
x &
& substitute for t
Parametric Cartesian: Rearrange
lire: x=ecast & y=rsint,010_2K
-
Equations
(Freatt
y g(t)
-
·
Defined
by
f(t)&
=
·
x
x
rcos(af) & y rsin(b8)-same
=
=
h rexst&y k cint,88- 2T
+
=
=
+
Does counter-dockwise,
Cycloid:
X
switch
ein &
r(A-sint) &
=
y
radius & origin
and/or range
domain
as above w/ different
for circle of radius 2(h,k)
wide
for clockwise
a
cas
r(1-cost)
=
of
for circle
y)
18.2
dy
·
Calculus ol
Parametics
-
Lowes
=
edt, y=Caylex) (or
daivative
of
deld
ity
dx/dt
--dy/dy (ifa point t, plug t
Slope of largent
f(x) 0
lensanity: d/dx2: ift, then 1; if- then 1 (check between
Fangents: Vertical of dydt=8,78; Harizental & deldt+0,dy/dt=0
Area: A:Soy.xdy (I, g(t) f(t)dt, f(t)&y g(t))
in
DNE)
=
·
·
x
the
·
·
18.3
Surface area:
*
At
S=2πJa" nor (*/dt)"dt(war opposite of
2:S."
is
-
ImportantPolac
(r,f)
⑦
Example
traits
came as
(r,pIn)
·
(-r,f)
In, fIn)
·
·
same
18.4.thea:
Areas
Polar Leschmal
1-2, 54)
original
12, S4)
t-as- e
.
are
or
1- 2,4u)
y rsint- smt)=y/r
or
y
=
-3
(2, *(y)
I
!
(2, -4(u)
1-2, -34)
e
=
A=SdO (Symmetry: change
Length:I S?
=
·
12, 94a)
original
aginal apped
_rgz)
tan 8:*y
as
drection
(2, 4)
original
12. AF2π)
D
-
of rotation)
axis
e is
/retation)
in
Lengths
=
Polar format: (s,A)(+8=counter-clockwise, -8:clockwise) Itin
rachane only)
to
from center endpoint
of line
length
·
Polar
berdinates
length:
=
e
2
(**/dA)2 dA
+
a
seb to
a
line
of symmetry, multiply by 2)
New Exam
Content
-(Heu)
11. I
Sequence
·
Sequences
Rearrange & use
of an Ants
of
·
or
is
or
an
--
farn?I,
An>Anx
have ratified
requence is masters
bothconditions
findinin
g
diverges
sequence
decrease
the
·
-
If
of
·
11.2
ifhis An=1. If limit DNE
I'hopitals
ifneeded
for II, sequence increases. If
converges
-
both
aist,
sequence
sequence
bounded.
is
Larger-upper
monotonic& bounded, then
sequence
is
Tactic
series: EAI" (if Irk1,
·
of converges, sum=Fr
Firergence test: of tim 8,
converges
series
diverges
converges, otherwise
series
A
Serie
bound, smaller-lover bound
-
·
series
·
Telescoping series:(A-B)+(c-D) (B
+
-
-
inconclusive)
is
diverges (if
8, text
series
E) (D-F)
+
...
terms
(use not
eliminated
-
an
term)
surving
((s) (s)) (((4) 26)) (s)ha())..4)
Example: E
0
beaume:(n13)+hn(4) In (n+1), limit to
-, diverges
th(n+2) ((n
-
=>
+
+
-
-
+
+
=
-
-
Frend: (a-B) (a.-Bn)
Sum of series-in
of sequence
-
-
·
an 8 (only seiner
is convergent, in
if convergen
If series
of series:
Propertier
Elan an
Can bn) iE an ba
=
·
·
-
n
11.3
Integral
test&
Estimates
of
sums
·
1
=
->
-
Conditions:f(x) +,
=
f(x)
sentinuous on [1,8)
an is
then
convergent
=
-,
If ,f(x) dxcamerges,
If Sf(x) dx diverges, then an divergent
test. In converges ifp>I, diverges
P-series
if p11
be estimated
Remainder Estimate: of convergentviaintegral
text,
wl firstIS terms: n= 1.58
test, appx
Example: In converges integral
"exor of appe
-
I
n
1
=
-
is
·
erior can
via
/
11
*
In
dy
ii)
exes
Rn=
->
sam
hidy-e-(+)-
4 Ya Rist/s
+
(0.8005, how
many
terms
needed?
In,. (In 0.0005) ->
=
n
2000
=
-
-
0.8656
=
11.4
Ean & Son
·
Comparison
Faste
If
-
w/ positiveterms
series
And by
If
-
An bu
for all
for
An is
givenS an
*
-
of
is
i
()" "On
·
Alternating
converges,
diverges,
compassion
Ian & Shn
an also
then
an also
then
converges
diverges
wl+ terms, then:
series
San &Sba diverge/converge
both
an/bu=1, where 38, then
P-series & conditions
met,
*
11.5
If
&Shn
all n Ibn
Limit
Test:
Comparison
·
i
where
go
to
LCE
bn78,
laxbn for all
Note: olf above fails, due to
(-1)"series divergen
If ASTsatisfied, IRn)=1S-snlbuel
Basically, heros, then s-but
a
Dibat8,
series converges
*
Series
·
-
11.6
conditionally emergentif an converges BUT Slant diverges
Ean & Staal
San absolutely
convergenti fboth
converge
an
then
an
convergent,
convergent
olf
absolutely
Ratioest: When in 1-2 and:
absolutely
11, series
convergent
2, series diverge
I1, inconclusive, need another
test
San
·
Absolute
-
is
is
Convergence
& root,
·
ratictests
is
is
·
-
-
-
=
·
RootTest: When him an1:1
-
-
-
11,
2,
I1,
=
and:
absolutely
convergent
series
series
diverge
test
inconclusive, need another
11.7
for
Strategy
I
esting series
1)
I
P-Series:Eni,p>1=con, p11:Div
2) Erometric:S ar"or Sar",
this an
test: Find
3)Divergence
4)comparison
Irk1: can, dir= Irk1
an
ital, ifnot8 then
divergent
test:Looks like
or geometric,
P-series
5) Alternating
test:Here we
series
6)Ratio test:
we known
compare
series
alternating
Factorials, cords of p& geometric
7) Rost:Ontha
(Lastresort):
8)
test(f(x)
Cubegral
Saf(x)dx Dir, SAn=Div,
=
11.8
Power Series
else if
=
+,
f(x)
=
-,
S!fle)dx=con,
Power Series:E.In(x-al centered elabout a
variable used to
Treat
find when
x as untouched
-
-
(x-ak1 = -1<x-ac1 =
a
·
-
8,R
-2,
-
0
=
0d:(
-
(x-cK1,
con
convergent
is
sorice
1(X-1 a
x
a-1&
=
a + 1.
0,0
=
R varies, a
=
21,5)
-
R 8 &d:x a
=
San
on
+
Check bounds w/ratic testwhen
When time ...
·
contin
calculated
from insquality
div, ()
If son, 21, if
ene
to fit
form.Plugpart intheredefined
abe batter
·
11.9
Representations-
of functions
When
as
Power Series
-
11.18.
Taylor
sides)
decinatives
of both
=
-
&
Maclarin
has R78, then:
power series
f(x) nCn(x- a)* (pull
-
·
Series
+
the
same
f(x) s*(x-al"
seria:
=
a
atabout
at centered
Taylor
Remainder
when
series
of Taylor
Ca is
E
=
a
both
side
a mightchange I needs to
be checked
original,
as
Madaurin series: f(x)
-
·
R stays
Taylor
-
(pall integral of
Sf(x) 1 Sen
=
a
where
# to
plug
where
x
f "(a)
with derivative
of
is
f (a)
nth derivative
of
is
f(a)
8
=
Series:I
study this
get it atall
give up,
do not
I
f(a)
in)
on
your
own
because
Formulas You Are Expected To Memorize
Trigonometry:
1
csc x
1
csc x =
sin x
sin2 x + cos2 x = 1
1
sec x
1
sec x =
cos x
tan2 x + 1 = sec2 x
sin x =
cos x =
sin(2x) = 2 sin x cos x
cos(2x) = cos2 x
sin2 x =
1
(1
2
cos2 x =
cos(2x))
1
sin x
=
cot x
cos x
1
cos x
cot x =
=
tan x
sin x
1 + cot2 x = csc2 x
tan x =
sin2 x
1
(1 + cos(2x))
2
Di↵erentiation:
d
(c) = 0
dx
d x
(e ) = ex
dx
d
(sin x) = cos x
dx
d
(csc x) = csc x cot x
dx
d
1
(sin 1 x) = p
dx
1 x2
d
1
p
(csc 1 x) =
dx
|x| x2 1
Constant Multiple Rule:
Sum and Di↵erence Rule:
Product Rules:
Quotient Rule:
Chain Rule:
Integrals:
Z
k dx = kx + C
Z
Z
Z
Z
Z
Z
Z
Z
ex dx = ex + C
Z
sin x dx =
cos x + C
cot x dx =
ln | csc x| + C
Z
Z
2
sec x dx = tan x + C
csc x cot x dx =
d n
(x ) = nxn 1
dx
d x
(a ) = ln a · ax
dx
d
(cos x) = sin x
dx
d
(sec x) = sec x tan x
dx
d
1
(cos 1 x) = p
dx
1 x2
d
1
p
(sec 1 x) =
dx
|x| x2 1
d
1
(ln x) =
dx
x
d
(tan x) = sec2 x
dx
d
(cot x) = csc2 x
dx
d
1
(tan 1 x) = 2
dx
x +1
d
1
(cot 1 x) =
2
dx
x +1
d
cf (x) = cf 0 (x)
dx
d
f (x) ± g(x) = f 0 (x) ± g 0 (x)
dx
d
f (x)g(x) = f 0 (x)g(x) + g 0 (x)f (x) or (uv)0 = u0 v v 0 u
dx ✓
◆
⇣ u ⌘0
d f (x)
f 0 (x)g(x) g 0 (x)f (x)
u0 v v 0 u
=
or
=
2
dx g(x)
v
v2
g(x)
✓
◆
d
f g(x) = f 0 g(x) g 0 (x)
dx
xn dx =
xn+1
+ C (n 6=
n+1
1)
1
dx = ln |x| + C
x
Z
cos x dx = sin x + C
sec x dx = ln | sec x + tan x| + C
Z
Z
sec x tan x dx = sec x + C
tan x dx = ln | sec x| + C
csc x dx =
csc2 x dx =
ln | csc x + cot x| + C
cot x + C
csc x + C
1
1
dx = tan
2
2
x +a
a
1
⇣x⌘
a
+C
Z
p
1
a2
x2
dx = sin
1
⇣x⌘
a
+C
Z
1
p
x x2
1
dx = sec
2
a
a
1
✓
|x|
a
◆
+C