Pre- Calculus & Basic Calculus.docx

Pre - Calculus 1.1 Arithmetic sequences (Partial Sum) s n=

n ( 2 s1 +( n−1 ) d ) 2

n a b (n−r ) r

n−r

1.7 Conics Section

n s n = ( s 1+ s n ) 2 1.7.1 Distance Formula

1.2 Arithmetic sequences

y 2− y 2 ¿ ¿ ¿2 (x 1−x 2)2 +¿ √¿

s n=a1+ ( n−1 ) d 1.3 Geometric sequences s n=a1 r

s n= s ∞=

n−1

a1 (1−r ) n

1−r

1.7.2 Standard Equation (Circle) n≠1

(x−h)2 +( y −k )2=r 2

a1 1−r

1.7.3 General Equation (Circle) 2

1.4 Mathematical Induction

1.7.4 Midpoint Formula (Circle)

A. Prove n = 1 B. Assume n = k

h=

x 1+ x2 2

k=

y1 + y1 2

C. Prove n = k+1

1.5 Binomial Theorem n ( a+b ) = n an + n a n−1 b+ n an−2 b2 0 1 2

() ()

nCr=

2

x + y + Dx+ Ey + F=0

()

n! r !(n−r )!

1.6 Finding the term that contains ar in the expansion (a+b)n

1.7.5 Standard Equation (Parabola) 2

( y−k) =4 p(x −h) opens to the right ( y−k)2=−4 p( x−h) opens to the left (x−h)2=4 p( y−k ) opens upward (x−h)2=−4 p ( y−k ) opens upward

P= distance of vertex to focus or vertex to directrix 4p = Latus rectum

1.7.6 Standard and General Equation (Ellipses)

COS

1

SEC

1

TAN

0

x2 y2 + =1 a2 b2 a = x-axis

1 √2

1 2

2 √3

√2

2

1 √3

1

√3

√3

1

1 √3

√3 2

0

b = y- axis COT

f 2=p 2−q 2 f = foci p = major axis

+¿ sin ¿ CSC

q= minor axis

1.7.7 Standard and General Equation (Hyperbola) 2

2

2

2

x y − 2 =1 opens to the left and right 2 a b

+¿ tan ¿ cot

x y − 2 =1 opens upward and downward 2 a b 1.8 Trigonometry 1.8.1 Special Triangles 0

SIN

CSC

°

0

30

°

°

°

45

60

1 2

1 √2

√3

2

√2

2 2 √3

°

90

1

1

ALL (+)

+¿ cos ¿ SEC

0

Radian = Degree (

π ¿ 180

2.2.3 Reciprocal Identities cotθ=

1 tanθ

tanθ=

1 cotθ

1 A= r 2 θ , θ must be in radians 2

sinθ=

1 cscθ

2.0 Linear Speed/ Angular Velocity

cscθ=

1 sinθ

cosθ=

1 secθ

secθ=

1 cosθ

Degree = Radian (

180 ¿ π

1.9 Area of a Circular Sector

V=

s t

V =rw 2.1 Angular Momentum θ W = , θ must be in radians t 2.2 Basic Trigonometric Identities 2.2.1 Pythagorean Identities 2

2

sin θ+cos θ=1 2

2

2

2

1+tan θ=sec θ

1+cot θ=csc θ

2.2.4 Half - Angle Formulas

√ √

sin

A 1−cosA =± 2 2

cos

A 1+cosA =± 2 2

2.2.5 Double - Angle Formula sin 2 A=2 sinAcosA

cos 2 A=cos2 A−sin 2 A 2

¿ 2 cos A−1 2

2.2.2 Quotient Identities sinθ tanθ= cosθ cosθ cotθ= sinθ

¿ 1−2 sin A tan 2 A=

2 tan A 1−tan A

2.2.6 Sum and Difference Formulas sin ( A + B )=sinA cosB+ cosA sinB

sin ( A−B ) =sinA cosB−cosA sinB

tan ( A+ B ) =

tanA +tanB 1−tanA tanB

cos ( A+ B ) =cosA cosB−sinA sinB

cos ( A−B )=cosA cosB+ sinA sinB

tan ( A−B )=

tanA+ tanB 1−tanA tanB

Basic Calculus 1.1 Solving Limits 1.1.1 Table Method lim ( x+ 1) x→ 0

¿1

As “x” approaches “0” from the positive side

x → c +¿ lim ¿

0.99

-0.0001

0.999

≈1

¿

x

-0.001

f(x)

1.1.1.1 Limit Theorem x → c−¿ x → c+¿ =lim ¿ ¿

0.5

1.5

0.01

1.01

0.001

1.001

0.0001

1.0001

lim f ( x ) =L , if and only if lim ¿ x→ c

¿

1.1.2 Analytical Method lim ( x+ 1) x→ 0

≈1 As “x” approaches “0” from the negative side

1.2 Properties of Limit Function

−¿

x→c lim ¿

1.2.1 The Limit of a Constant is itself

¿

x

x=0 , lim of x is ( 0+1 ) or 1

f(x)

lim k=k , where k is only constant x→ c

lim 20=20 -0.5

0.5

x→ 0

1.2.2 The Limit of a Function is “c” -0.01

0.9

lim x=c

lim f ( x ) f (x ) lim ( )= x→ c x→ c g ( x ) lim g ( x )

x→ c

lim x=8

x →c

x→ 8

1.2.3 Constant Multiple Theorem

lim [f ( x ) ] p=+[lim f ( x ) ] p

lim k∗f ( x)=k∗lim f (x) x→ c

x→ c

1.2.8 Radical/ Root Theorem lim √n f ( x )= n lim g ( x )

x 8(¿ ¿ 2+ x ) lim ¿

x→ c

x →1

x (¿¿ 2+ x) 8* lim ¿ x→ 1

x (¿¿ 2+ x) lim ¿ x→ 1

1 (¿¿ 2+1) ¿ =2

=8*2 | =16

1.2.4 Addition Theorem lim [f ( x )+ g ( x ) ]=lim f ( x ) + lim g ( x ) x→ c

x →c

1.2.5 Multiplication Theorem lim [f ( x )∗g ( x ) ]=lim f ( x )∗lim g ( x ) x→ c

x→c

x→ c

“k” means constant

x→ c

1.2.7 Power Theorem

x →c

1.2.6 Division Theorem

x →c

√

x →c