MATH 101 Study Guide – The Ultimate Survival Pack for First-Year Calculus

math 101


Semester 2




January

2024

, Week 1 -

definite integral



Sf(x) ax ↑ Area between the curve & the X-axis from (a , b)
Y




- a 7 * b
~
L




X
taking
and

2 ->
small

adding
summation
sections

up,
& defining functions using



?
definite integrals




dX
TAKE THE INTEGRAL
TO FIND FUNCTION
Ex Sketch &
simplify Alx) =
So tat x0
I
[xdx
Y L bounds




unt >
- A(X) =
tasheatt
=
2x2
AX =
Pa t
O *




Approximating Integrals
①
Right Riemann Sum >
Definite Riemann Integral one definition of definite integral is
-




1 f(x)ax =
Ax(fx ,
+ fX2 + ..
) fX4
&
as limit of Riemann Sum



+ (xi) ·

A
-
x4
f
x
Cf(xdx = m x
-x
!
O
-
1 X




Summation Notation Geometric sum

-
↳
EX . 4 + 6 + 8 + 10 + 12 OR 2 + 2 + 2 + 2 + 2 S =
1 + p + r2 +... + un

S =
rn -


1 ↓ ↓ ↳
Si
5
r
-
1
& S =
:
can

① ② ③
add constant factor constant commutative addition
S S S



"Sc = 10
= G(i
i 1
=
+ 2)
= 2



i= 1

, Week -
2
Integration Using Symmetry

Even Enx Odd En X



f(x) =
f) -


X) Symmetric f(x) = -

f(X) symmetrical
even around y-axis around origin




S
f(x) f) X) ↓
↳
= -
·




↳
Sef(x(ax =
2(of(x)dX
(af(x)ax =
2(f(x)dx Sf(xax = 0




odd Basic Integral Properties

f(x)
=
-

f(x) 0Sakf(x)dx = kSaf(x)aX
↳ ② S (f(x) +
g(x))dx
=
SaPf(X(ax +
*
Sa g(x)ax

Saf(x(aX =
O ②
Sf(x)dX =
- Sif(x)dX
①
Saf(x) =
Saf(x)dX + Sif(x1dX



Fundamental Theorem of Calculus I apDlY elmitS




Saf(t)d + A'(X) 11mD h) A(x)
=
A(x) = A(x +
-




n= 0 n




curve approcess em f(x)
=
=


FTC 1

A(x) =
Saf(t)dt
Y
a xynth
me13 A(X) =
Saf(t)dt where est) is continuous


A'(x) =
f(x) ↓

A'(X) =
f(x)



Fundamental Theorem of Calculus #


Assume f cont on La ,
bJ , let F(X) be fux statement




F(X) =
f(x) (a , b) (such that F's called antiderivative of f)
FTC 2 Then for a = X = b


Integrals & derivatives
Saflt)at F(X) F(a)
23 Sf(t)dt F(a)
= -



= F(x) -


*

inverses of each other




-
All off
Indefinite Integrals possible antiderivatives

* NO BOUNDARIESA

↑ (x) is fax statement
> (f(x)ax =
F(x) +
2
↑
F'(x) =
f(x) then
abitrary constant




using linearity of integrals
↓
P(X) =
anxn +
an - ,X + ...
+

ax + do ant0

-
Sp(x) =

an