Ap Calculus notes

1 . Limits and continuity

Rin C0S+ I
sin = Dim tan b
eim
·
· - ·
-
-
=
0
* > *
8 O *

·
eim m + + b
=
ma + b
*+ a

and tanx any interval

secy are cont . Over not containing x =

I
sex and coty are cont .
Over any intertal not containing =

IVT
:
Let fixe be continuous over laibs such that frarefibe -o then by IT ACE Ca,belfies so

if fixe if fix l
lim exists
lim 1 lin f(x)
Nim
-
=

+
-ab
-

Continuity at a point : lim free -flas

* *A

Sandwich theorem :
for usyoffixsuits if lim was im vixt
=
d
* A
f(x) l
* A
then Rim =

* a

types of discontinuity :
jump discont : ates a
limatis Flim * a
free

removable discontinuity : at =a lim exists but frar is not defined

removable discontinuity at
lina fixi
=
non !
a

classes of continuous functions :

·
free sind giye-coss are cont

·
If f and gare cont on an
open interval 1, ftg , -g fog , , Ig are continuous

over I

·
Polynomials and monomials are continuous

left -

and right continuous

left cont: lim free:fras right cons : lim (1x) = fla

a
+
& -a

·
a function is cons . on Ca , by it it is cons on (a ,b) right cons as y =
a and left at = b

e
=

lim
(1
+ Critical points
e
&
Extreme value theorem fr=0 fl does not exist endpoints
:

,
or can

be extremes
2 . Differentiation

Definition of derivative :

free im fi
U o

n -
1 y

+ y UV v v'u
y
=

nx
=
= =

yy'
n
.
y' nuun y
=

u
y
= =

2
~

derivatives of
trigonometric functions :

y'
y'

y =
sin + =
cosy u
=

286x =
-

SC

y's
y'

y Cos Sin y= sex secxtany

= -
=
+

Y'
sex
y

& =
anx =

y
=
CS x
=

excoty

derivatives of inverse trigonometric functions :

un - I

y =
sin' mus 4 = 1 - 42
y =
tancur
i =

un

y COS" ru 4 Cour -
y u
=
=
-
=

u -,

y = Secu - = ↑

- I
141-1
y = CSC rur
y' =
ur
-

- ~

right and left hand derivatives :

right hand :
at x =
a Rim -Fran exists
U - 8
+

left hand : at + = a lim frathe-flas

->
exists
8 -

* H

non differentiable functions :

·
not continuous
·
Vertical tangent lim Fraue Rim f(x) =
+0

n +0 +
&-

·
corner point Rin free e Rim fry) absolute Valued
&- I
-
 mean value theorem :

if a function is diff on an intervall then the average rate of change between 2 points =

instantaneous rate of change of function at some point in the interval

if fiscont . on Tabs and difs on Cabe there exissel point cera ,

be

such that free fibe-frad

b -
a

Rolle's theorem
:

Let fr be a cont function over Saibs & fre is difs over ra,b)/fraiffbe
by Rolles theorem 3

<(a , brifr 21 = G

linearitation :
(exe-fras + far (x-as linearitation of fatx =

Slope , tangent

Newtons method - =
a -
fran -n =
+n -
+1
:

Frag fy n
-
1
=
40 -
frx0
T
↑ =0 &

Other derivatives :

1 y

-logau
ex
1x y
-

y y e
=
= =
= =

*
na
g
un e
y' y

y In u =

y
= =
=

fixo
I
a al

&
y' '

Ina -

y
=
=

- =
b flar

f(a) =
b
3 . Integrals

definition of
a definite integral :

(a free
Dim in der

Properties of definite integral

1 . Sa fidx = o

C .
/am fre de =
ma fix de

3 .
S fraudx --1 fix de

s .
If memil on Sa , b) and Mamas on Zaibs then mibas fre axs Mib as
- .
If fr o , on Ja,bs then free des o

Sudx
M
=
C
n + 1

1
substitution method :
Su'u das Sudu U
n +
+C
=-

integrals of trigonometric functions :

Sein x de--Coskx
V
+ Scosux de =

sex
+a

Sseevade- + SCSC vide -

-
V Vo

(seauxtanxx da -sexte escuscoturdos-sx+

Vo E
 Stanx de =

(de-inlost
=
In seit

Scott af
= dx =
insint

Sseex de =
In isecyttanti +C

Sescy de =

in lesex +coty +

integrals of inverse trigonometric functions :

SFax
u
- sin u + -cosu+ 1412

Si de -tan'u+c-cot'U+C

S 42 -
1
der setuxstute I

S un af =

sin + Ita
-
a -
u

S
de tan"
es is

S ur ax
= se + 1
#
uz -
a

4 Applications and properties of integrals

mean value theorem for definite integrals :

the average rate of a diff function on a closed interval is attained at least once

a
e era , be/far a
freude fice

First fundamental law of calculus :

Ify = fitat = fr

1f
Greg fre as u u'res +rurse
=

y
=

44&

y =

g fitat =
-

Säfstedt Sa
fitat

Trapezoidal Sum

h A trapezoid
ba (b, +
ba
=
=

=
(f(1) + 2912) + 2f(3) ...
+ fin
 The second fundamental law of Calculus

if f is a continuous function on a closed interval La i b) then Safredy-Fibe-fra

where fredsf'd Fres is the

antiderivative of free

Sf(x) dy =
F(x +

area of a function under the curve :

Salfreude
volume of solids of Revolution

disk method = Sa freid de

S free
"
washer method :
-gredy

shell method :

27 hiserres de

solids with known Cross-section :

Sa Dresde

cross section of a sphere is a circle with radius ures-

the average value of a continuous function on a closed interval:

far - a( free dx

5 .
Other Important Remarks

Rules for a finite sum

k
n

E Ca =
ca = nu
.. i = i =

Summeation formular
I

⑭ ne
U

21 =
n
nun +18 E -2
I -nun + 2122n +12
2 i
i 1 =
G
=

Trigonometry : sin'x+cosy -
,
1
Ittanke Sechs ,
It coty =sche

Sumrule : Sindatbe -

sinacosb + cosasinb

Cosratbrs cosacosb-sina sind

 double-angle formula : Sin (2x) = 2 sinXCOS2
cosidas = 1-2 sind =
10sa -
1

=cos(ge-sin(e
inverse trigonometric functions :

domain range
Sin C C-E E , E1 is ,

Sire"C [1 , 1 [ - , * odd

COS & 21t or ,

2 -

1
, 18
cas" & 5-1 , 18 Co, 1
tan &-

E -
IT
I
R
tan"d R (
y y ,
*
odd

See" (eu -

cosy sin" y +Cos" x

=

E
sa" (4) =
Sin, sex+csa =

I
Cot "11 = tan" ( tan"x+coty = gresso
T
-
gro

inverse functions :
fifixe symmetric about yee, fre has to

Be 1 -
1

Exponential functions :

Exponential function laws exponents

:
of

b
1 eab =
pa +

e= in domain (R) 2 .
reab =
gab

&
b
range (0 tor ea -

, , 3 .
=

y =
ey
sex
&4
you b
e

4 e
-

y
=

. =

Continuous
compounding
rate

A =
pert-time effective rate :
R = e-1
Initial amount o

money

Logarithmic Functions :

natural logarithmic function y S dt o

=
:
Inf =

y = in
= sin y
=

y logay
properties of the natural logarithm :

In1 =
0 In e = 1 IMN MIN ( same properties
apply for 109 a +
In IM-IN Innen
 special angles

L in degrees Jo · 30° us 600 900120° 135° 150° 1800

L in radians O
T
T I -
T

O i 1
Sind

COSL
-E -
I O
Es
-
/
/
O -
-
tan a Es
5
/ B
-V -
-

/
Sin T = G

Cosrevent =
1 Lexample
:

2, Un, GT

Cosrodite= -1 /T, 31, Ste

cos(oddi =
o

=
sin
(odd

+1 : Sa F
Il T

COSL =X assume cos (x) =

a

a
=
a + 2VIT
-

a + 24π

sinn =x assume sin (xe =

a

L =
a + Lit

-A +2

tandsf assume tan"(x) =

a

& a + VI
=
 etfre Ein fre
ein ,

eine fugiese freim gra

* * C - od