Calculus Review Section by: javier
Calculus Review Section by: javier
Chapter 1 limits and Indeterminants
Chapter 1: Limits Indeterminants
Chapter 1: Limits Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , Limits
∞ ∞
Chapter 1: Limits Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , Limits PIM, Graph, EM, ReWrite,
∞ ∞
Chapter 1: Limits Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , ∞ ∞ Limits PIM, Graph, EM, ReWrite, L’Hopital,
Chapter 1: Limits Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , ∞ ∞ Limits PIM, Graph, EM, ReWrite, L’Hopital, Taylor Series Example: lim xx
In
0° ×ln× hiya
x→0+
e×k¥ ekxx
¥y
=
h¥= F kina
.
!,%
=
=
E
Kai
ftp.xlux
=
@
xz
=
line
not
=
Eµ ;
'T
=e° 2
=
'
Z
×
°
c
,
Chapter 1: Limits
Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , ∞ ∞ Limits PIM, Graph, EM, ReWrite, L’Hopital, Taylor Series
Example (fibonacci golden ratio): suppose an+1 = an + an−1 is the fibonacci sequence determine the limit an+1 } 13,2434 lim An ={ 1,112,315,8 n→∞ an -
-
.
kf÷='t±÷÷
,
THE ,¥FI¥¥
=
1
,
2,1-5,1-67
,
-
bn= g.
.
ba
Itt
{ Htylt'T { lttntt
't 's
" '
's 't ,
'T
't
} .
'
1.6/1-6251%61
kntltnttl
.
set
An
An
=
,
An
.
,
A
=
'
'
3
Agy
Any
+
,
+
An
E[
=
1
+
Anat
±
It
=
Eilt ,
,
.A÷IL L
AT
AT
AT
ksm
'
ti
-
o
=£±F54T÷ 2
L=1±t
" '
l±a¥d*+£÷ ) ,=±#9dTaE
Chapter 1: Limits Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , ∞ ∞ Limits PIM, Graph, EM, ReWrite, L’Hopital, Taylor Series Example: lim x2 · ln(x) x→0+
Chapter 1: Limits Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , ∞ ∞ Limits PIM, Graph, EM, ReWrite, L’Hopital, Taylor Series ! " 1 x Example: lim 1 + x→∞ x
Chapter 2 Integration
Chapter 2: Integration by # u-sub $ √ 8 + xdx
u
=
due
8tr×
tzxtdx
HE
.
jam €m£¥tx .
A
f
=
=
§
ru
-
.
.
du
Cut ) du
µ8u"Ddu }u" 2- 2.8 }u" "
=
=
=
2
2
-
.
.
§(
strxjhtc 2.85 ( strxjh .
+
c
Chapter 2: Integration by # parts ln xdx uv
=
xlnx
=
=
=
u
=1nx
}:
-
xlnx
xlnx
dkdx
du= txdx heck
frd9
fttxd
-
-
-
fdx Xtc
d#(
xkx
-
)
xtc
ftaxjfluxtxdadnxjddxatdd,
=
lnxtxtx
=
=
Inx
-
1
to
(c)
Chapter 2: Integration by # parts sec3 xdx UV
=
-
§c3
secxdx
=
=
dkseixdx
sec 's
du= secxtaaxax
V=taux
zfseaxax
fvda
seoxtaux
=
U
=
=
secxtanx
+1
"
ltanxtsecx
1
+
(
ftanxsecxtahxdx
fseaxax.tk#ax+lnlmxtsecx1)fsec3xdx=secxtaux-/tan2xsecxdxfsec3xdx=secxtanx-/@ec2x-hsecxdxfsec3xdx=secxfanx Jedxdx Jsecxdx +
-
2
2
¢ec3xd×=
fseaxdx
=
sextant fsecxdx sextant
fsecx
.FI?ftseJldx2fseaxax.saxtauxtfqIaxtEIetxY#M tanxtsecx duisedxtsecxtanxdx -
Zfseihdlseexfanxtfdy
Chapter 2: Integration by # trig sub 1 dx 3 + 4x2
Chapter 2: Integration ' ×2KTDHH
⇐×E×,y)=(÷t±atx÷tt÷z ) (
by PFD #
x2 (x
=
1 dx + 1)(x + 2)
duftxaxt
If
1
=
AK )l×tD( xtz )
+
×2(xtD(xtD
B(xtDAdD+cP(xtD+DX4XtD
iii;t÷ljisnf÷ItIIknHk .IE#EtsfIIfkFixIFFI*.s+u
t*d×t§Mt}f±µ×
1=6
3+3+2
At
2=4+6+6-1 2=12 -
9
Tz=A
A
th
.**
Chapter 2: Integration
tnyrixxi
improper integrals # ∞ 2 sin (x) dx x2 1 µIgQdx£⇐edx
=
finite
Chapter 3 Applications
Chapter 3 Applications Work lifting chains, empty water tanks, pull spring s etc,
Chapter 3 Applications Center of mass lamina , wires, etc.
Chapter 3 Applications differential equations 2 +1 dy = yx+1 dx
f ÷f÷ -
,
arotauy
=
ken )
tc
Chapter 3 Applications differential equations: suppose f = cos(x) + i sin(x) solve df icosx = if f! dx =
save
:
ftp.fidx hf=
e
i×
e
f=ei× ei×=
cosxksinx
sinx
-
+
ifsinxt 'D ')=i(
Tass
.
jdgf×=
-
id¥×= -
isinx
(
×=
if
cos
eosxtisinx
iddfe are EE ¥
-
×
)
Chapter 3 Applications Fluid force
Chapter 3 Applications Fluid force
free
do
Chapter 3 Applications Fluid force
Chapter 1 limits and Indeterminants
Chapter 1: Limits Indeterminants
Chapter 1: Limits Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , Limits
∞ ∞
Chapter 1: Limits Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , Limits PIM, Graph, EM, ReWrite,
∞ ∞
Chapter 1: Limits Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , ∞ ∞ Limits PIM, Graph, EM, ReWrite, L’Hopital,
Chapter 1: Limits Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , ∞ ∞ Limits PIM, Graph, EM, ReWrite, L’Hopital, Taylor Series Example: lim xx
In
0° ×ln× hiya
x→0+
e×k¥ ekxx
¥y
=
h¥= F kina
.
!,%
=
=
E
Kai
ftp.xlux
=
@
xz
=
line
not
=
Eµ ;
'T
=e° 2
=
'
Z
×
°
c
,
Chapter 1: Limits
Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , ∞ ∞ Limits PIM, Graph, EM, ReWrite, L’Hopital, Taylor Series
Example (fibonacci golden ratio): suppose an+1 = an + an−1 is the fibonacci sequence determine the limit an+1 } 13,2434 lim An ={ 1,112,315,8 n→∞ an -
-
.
kf÷='t±÷÷
,
THE ,¥FI¥¥
=
1
,
2,1-5,1-67
,
-
bn= g.
.
ba
Itt
{ Htylt'T { lttntt
't 's
" '
's 't ,
'T
't
} .
'
1.6/1-6251%61
kntltnttl
.
set
An
An
=
,
An
.
,
A
=
'
'
3
Agy
Any
+
,
+
An
E[
=
1
+
Anat
±
It
=
Eilt ,
,
.A÷IL L
AT
AT
AT
ksm
'
ti
-
o
=£±F54T÷ 2
L=1±t
" '
l±a¥d*+£÷ ) ,=±#9dTaE
Chapter 1: Limits Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , ∞ ∞ Limits PIM, Graph, EM, ReWrite, L’Hopital, Taylor Series Example: lim x2 · ln(x) x→0+
Chapter 1: Limits Indeterminants ∞ − ∞, ∞0 , 00 , ∞ · 0, 1∞ , 00 , ∞ ∞ Limits PIM, Graph, EM, ReWrite, L’Hopital, Taylor Series ! " 1 x Example: lim 1 + x→∞ x
Chapter 2 Integration
Chapter 2: Integration by # u-sub $ √ 8 + xdx
u
=
due
8tr×
tzxtdx
HE
.
jam €m£¥tx .
A
f
=
=
§
ru
-
.
.
du
Cut ) du
µ8u"Ddu }u" 2- 2.8 }u" "
=
=
=
2
2
-
.
.
§(
strxjhtc 2.85 ( strxjh .
+
c
Chapter 2: Integration by # parts ln xdx uv
=
xlnx
=
=
=
u
=1nx
}:
-
xlnx
xlnx
dkdx
du= txdx heck
frd9
fttxd
-
-
-
fdx Xtc
d#(
xkx
-
)
xtc
ftaxjfluxtxdadnxjddxatdd,
=
lnxtxtx
=
=
Inx
-
1
to
(c)
Chapter 2: Integration by # parts sec3 xdx UV
=
-
§c3
secxdx
=
=
dkseixdx
sec 's
du= secxtaaxax
V=taux
zfseaxax
fvda
seoxtaux
=
U
=
=
secxtanx
+1
"
ltanxtsecx
1
+
(
ftanxsecxtahxdx
fseaxax.tk#ax+lnlmxtsecx1)fsec3xdx=secxtaux-/tan2xsecxdxfsec3xdx=secxtanx-/@ec2x-hsecxdxfsec3xdx=secxfanx Jedxdx Jsecxdx +
-
2
2
¢ec3xd×=
fseaxdx
=
sextant fsecxdx sextant
fsecx
.FI?ftseJldx2fseaxax.saxtauxtfqIaxtEIetxY#M tanxtsecx duisedxtsecxtanxdx -
Zfseihdlseexfanxtfdy
Chapter 2: Integration by # trig sub 1 dx 3 + 4x2
Chapter 2: Integration ' ×2KTDHH
⇐×E×,y)=(÷t±atx÷tt÷z ) (
by PFD #
x2 (x
=
1 dx + 1)(x + 2)
duftxaxt
If
1
=
AK )l×tD( xtz )
+
×2(xtD(xtD
B(xtDAdD+cP(xtD+DX4XtD
iii;t÷ljisnf÷ItIIknHk .IE#EtsfIIfkFixIFFI*.s+u
t*d×t§Mt}f±µ×
1=6
3+3+2
At
2=4+6+6-1 2=12 -
9
Tz=A
A
th
.**
Chapter 2: Integration
tnyrixxi
improper integrals # ∞ 2 sin (x) dx x2 1 µIgQdx£⇐edx
=
finite
Chapter 3 Applications
Chapter 3 Applications Work lifting chains, empty water tanks, pull spring s etc,
Chapter 3 Applications Center of mass lamina , wires, etc.
Chapter 3 Applications differential equations 2 +1 dy = yx+1 dx
f ÷f÷ -
,
arotauy
=
ken )
tc
Chapter 3 Applications differential equations: suppose f = cos(x) + i sin(x) solve df icosx = if f! dx =
save
:
ftp.fidx hf=
e
i×
e
f=ei× ei×=
cosxksinx
sinx
-
+
ifsinxt 'D ')=i(
Tass
.
jdgf×=
-
id¥×= -
isinx
(
×=
if
cos
eosxtisinx
iddfe are EE ¥
-
×
)
Chapter 3 Applications Fluid force
Chapter 3 Applications Fluid force
free
do
Chapter 3 Applications Fluid force
