Textbook Notes
(363,556)

Canada
(158,423)

Western University
(15,371)

Calculus
(4)

Calculus 1000A/B
(4)

Alan Mac Isaac
(1)

Chapter 1

# Calc 1000a Notes All of Chapter 1 (concise)

Unlock Document

Western University

Calculus

Calculus 1000A/B

Alan Mac Isaac

Fall

Description

Sec. 1. 1- 1. HONE
LOG s
otare
1.5
x exponent
or power
a a o
1 a
Base.
a
Q: where is 3) ER saueeze
a
a a
op
a.
a)
le
lecture
Logs inverse of
a y
a TS1
04 a 41
what
No
Papert
tt
time
re
f re
Q. E
4 2
era
strand
e m
Q.E.D
Se
Sec. 1. 1- 1. HONE LOG s otare 1.5 x exponent or power a a o 1 a Base. a Q: where is 3) ER saueeze a a a op a. a) le lecture Logs inverse of a y a TS1 04 a 41 what No Papert tt time re f re Q. E 4 2 era strand e m Q.E.D Seen a
IMPOT:
smaller
Ex
mean
TRIGS
APPEND. D
HAL GH
SCHOOL
Angles measures
THINGS
reES
360
RADIAN
2 RADIANS
th
le is
277 RADIANS
for acute anges via
Sin 6
20 For a tuse ok even negative
Cos G
ties
3) cos
sind
ssinx
Lect
THE
Y sinf2x) sin x cos x
1- sin x
COS
cos x (1- x)
Product
can Be re-written
as
costk
tan (x)+ 1.
COS
X
was Y.Cos
Sin
eaux. Cas
sinx. Sun
tan tan
1- tan X- tan.
GRAPHS:
IMPORTANT TO
sin cos, tan cot HOME
RECALL
cot
se
VERSE TRTGS
make
SIn
0, H1
ta
1- 1
en a IMPOT: smaller Ex mean TRIGS APPEND. D HAL GH SCHOOL Angles measures THINGS reES 360 RADIAN 2 RADIANS th le is 277 RADIANS for acute anges via Sin 6 20 For a tuse ok even negative Cos G ties 3) cos sind ssinx Lect THE Y sinf2x) sin x cos x 1- sin x COS cos x (1- x) Product can Be re-written as costk tan (x)+ 1. COS X was Y.Cos Sin eaux. Cas sinx. Sun tan tan 1- tan X- tan. GRAPHS: IMPORTANT TO sin cos, tan cot HOME RECALL cot se VERSE TRTGS make SIn 0, H1 ta 1- 1stope at
stabe ai
Xeo is 1
TUPICA
the eook)
Sin at
See mamy
PROBLE
Let
ED
HOME Read Sec. 1.1
1. 6. AND
APPENDIX D
Quiz 1
lecture
Sec. 2. 2. at HoME
VWstivation:
Intuition
X Sin
cannot
Det lim L if one com make f b) argitrarig elose to L
cien
oon be NOT deigned at
Note
Ex. UNO LIMIT
at
lim
lim f
33
ded limits
One-S
Here tends to 1
X 7
what
tend to o
amit at x o? Det
t one com m
from the
taring
S
te Limia
lim t
Q: C Guess T
Ex
tam tanx
e Cam
Bitran
X-e a
m tan x 7
at
small
tan x
too t leant one there
the
Vertica t asymtote at t a
Q: lim Sec x
sect
m sec X
stope at stabe ai Xeo is 1 TUPICA the eook) Sin at See mamy PROBLE Let ED HOME Read Sec. 1.1 1. 6. AND APPENDIX D Quiz 1 lecture Sec. 2. 2. at HoME VWstivation: Intuition X Sin cannot Det lim L if one com make f b) argitrarig elose to L cien oon be NOT deigned at Note Ex. UNO LIMIT at lim lim f 33 ded limits One-S Here tends to 1 X 7 what tend to o amit at x o? Det t one com m from the taring S te Limia lim t Q: C Guess T Ex tam tanx e Cam Bitran X-e a m tan x 7 at small tan x too t leant one there the Vertica t asymtote at t a Q: lim Sec x sect m sec XLeat
RRE
le multipli
tive
tuition 7
DIFFICULT
sect
cost
X. S
LAWS 0F LIMITS
The
Let
limfo) F lim gGK G and c is a constant
Th
en
C. F
X-7
Coro/
TNTTS OF GOOD FUNCTIONS.
some real number
tim
LAWS
hn X
X a
Exampl
linn 1001 x
002
1010
X- 1
Faa)
heoREm
Example,
X-2
at 2.
lim
at x
laws
tim (x-2) 3-1 at o
na
emb-2)
X-2.
Theorem
X) ration
the domain then
and a
ld
Exa
f rat
ona funct
C-2 (2-3)
man
Example C CANCELING 1
X 3
1 -s true.
ntu.
02 3
X-9
Example what
nd No
BAD POINT
EFFT
lim
dit danger
1.0
m 8
tive
mum
Now
ver
ma
posit/
lim
and we have a vertical asymptoze
rahh
Leat RRE le multipli tive tuition 7 DIFFICULT sect cost X. S LAWS 0F LIMITS The Let limfo) F lim gGK G and c is a constant Th en C. F X-7 Coro/ TNTTS OF GOOD FUNCTIONS. some real number tim LAWS hn X X a Exampl linn 1001 x 002 1010 X- 1 Faa) heoREm Example, X-2 at 2. lim at x laws tim (x-2) 3-1 at o na emb-2) X-2. Theorem X) ration the domain then and a ld Exa f rat ona funct C-2 (2-3) man Example C CANCELING 1 X 3 1 -s true. ntu. 02 3 X-9 Example what nd No BAD POINT EFFT lim dit danger 1.0 m 8 tive mum Now ver ma posit/ lim and we have a vertical asymptoze rahhExa
eAD point ont cance
posiere
X 2.
X-3
5 X 6
XE3
only at
lecture 6
ta ton of
2-2-2. 3
Contra
2.17
lath
last
X 4.2
X-3
xample
4-
Because AROUND O
I When a n
Jump ouurs
Exam/2
33
be counse
S is
em does not exist.
Very approxim
sketch
ate Show
lehavior at
Lecture
If f(x) around a x-a-whate
Theorem
Com
So in
squeeze Theorem C very IMPORT.1 JVf F6) g s
x around a
-a whatever
AND
lim h 6
de funct
xample
e Ca
Here tend to 1
We
lim H
in Hit) does not exist
ds too
We cannod do it as em as
x near
ds to ouR
X-20
IS eu po
Sa
Example
One
this
no
limi
to use
00
lim sin.
0
Show
X10t
Exa eAD point ont cance posiere X 2. X-3 5 X 6 XE3 only at lecture 6 ta ton of 2-2-2. 3 Contra 2.17 lath last X 4.2 X-3 xample 4- Because AROUND O I When a n Jump ouurs Exam/2 33 be counse S is em does not exist. Very approxim sketch ate Show lehavior at Lecture If f(x) around a x-a-whate Theorem Com So in squeeze Theorem C very IMPORT.1 JVf F6) g s x around a -a whatever AND lim h 6 de funct xample e Ca Here tend to 1 We lim H in Hit) does not exist ds too We cannod do it as em as x near ds to ouR X-20 IS eu po Sa Example One this no limi to use 00 lim sin. 0 Show X10tContinue
We define ugood
ne ugood
Zef continuous at z
Discon
La ST if it cont. at eve
cont.
from start
to Fri
without ever dona.
ch point of
s are.
do
graphs
C Re
Sin x
COSX
COS x
nonntals
Ex
Q: it cont. at
X- 1
tio
)we brow that for poenom. Gm exist and
ever
cont at
Th
Pola no
Rem
at evi
"We
me way for one fenofons instead
proceed
the
Let and are ann at x
Conti
anal a
re cont at
easy see P.12
r0
cont at
Rational
ctions
Point
re.
main,
tanxet.
are cont
sec x, CSC x
at 8 Point of the aomain
ever
Lecture talking Gout
tamm
lets talk agout Bad' nts
1) Remo
when it can ee removed
THREE
THPES
defining fla) emfl)
CONT
nuvent here more
XE.2
X-3
removable at X3.2
X 2
and f
3 th
ore
IIMPORT
Exam
lim e
Sun X.
because whateva
lim sinx
is going Ho
xe R
r a
Theorem
cont at as
le 9: Prove
that e
cent at
reo
fram
cont at
f-e
Q E.D
where
Example
contin
ery where domain
nt. every where
i
all R.
defined
les where is arcces, le +L
contimous
Fram
e 1.
SO
ever where
in
rts d
cont
main
2) d
in e
0m
NEVER
y defined
e SO
Continue We define ugood ne ugood Zef continuous at z Discon La ST if it cont. at eve cont. from start to Fri without ever dona. ch point of s are. do graphs C Re Sin x COSX COS x nonntals Ex Q: it cont. at X- 1 tio )we brow that for poenom. Gm exist and ever cont at Th Pola no Rem at evi "We me way for one fenofons instead proceed the Let and are ann at x Conti anal a re cont at easy see P.12 r0 cont at Rational ctions Point re. main, tanxet. are cont sec x, CSC x at 8 Point of the aomain ever Lecture talking Gout tamm lets talk agout Bad' nts 1) Remo when it can ee removed THREE THPES defining fla) emfl) CONT nuvent here more XE.2 X-3 removable at X3.2 X 2 and f 3 th ore IIMPORT Exam lim e Sun X. because whateva lim sinx is going Ho xe R r a Theorem cont at as le 9: Prove that e cent at reo fram cont at f-e Q E.D where Example contin ery where domain nt. every where i all R. defined les where is arcces, le +L contimous Fram e 1. SO ever where in rts d cont main 2) d in e 0m NEVER y defined e SOIntermediate v
75m.
be van nous on
La
6]
fa) between fa) and fe)
then
there
exists such value.
X, Y c
that
s that
RouGHLy Speaking
thm
La, Bi takes
fa)
ox ts h
that
sh
PCx
roots
1-1, z
P (2) -19
th
ere exists CE 2.3
sit, pla -o root
ecture Example f
112
10 for some
all real x
funct.
then
PROD u C
also
value bet
10
takes ev
O and
SinX
Theorem:
trig
Rad length
PHI S
See
ride By sin x
Sec
XeTo
assumed
ee
Sinx
en when
hm
Sin
Sinux
Gm
COS X
Ore
urme
COS
1-Sino X
COS X
tim
tan yr
Example.
Limili at
lect
ure MD
amfs)-4 if f is defined for all
a a-some.
2) tG) is a
close to
targ
f6
-L f is defined for all urea, a-some
to L
arB. cl
fx
e above
en
L a a horiaon tat asymptoze
rizon
asymptoks:
Ex
Large,
Intermediate v 75m. be van nous on La 6] fa) between fa) and fe) then there exists such value. X, Y c that s that RouGHLy Speaking thm La, Bi takes fa) ox ts h that sh PCx roots 1-1, z P (2) -19 th ere exists CE 2.3 sit, pla -o root ecture Example f 112 10 for some all real x funct. then PROD u C also value bet 10 takes ev O and SinX Theorem: trig Rad length PHI S See ride By sin x Sec XeTo assumed ee Sinx en when hm Sin Sinux Gm COS X Ore urme COS 1-Sino X COS X tim tan yr Example. Limili at lect ure MD amfs)-4 if f is defined for all a a-some. 2) tG) is a close to targ f6 -L f is defined for all urea, a-some to L arB. cl fx e above en L a a horiaon tat asymptoze rizon asymptoks: Ex Large,Recal that
lect.
tional neum6e
er, s.
Polynomials at
2X
Elim /2x
that
tine 2x
thod
7 o 7
Rational Fanchons a
(-2.
2.2
nt
ana PA fiamme
tote
rarh
Ex
lim x
to sketch the graps of such fi
tion
ok it using
not
ne-side limit
3 41
2.3
X+3
h the stetch
ed Anne ont what heorens as
X 13
X +3
Other Amm. at
Ex
5-2 x
X-3
BAD
BAD
at
d tra
a a
Ex
Ex
a
a K1
N
the exams
not
ZNE
a i
Ex
tim Sin
DNE
cos x DNE
DNE.
d of Sec. 2.6
skip
def
t the
Sec. 2.1 and 2.8
Derivatives
t line to
Def
t P
that
araP
inst touc

More
Less
Related notes for Calculus 1000A/B