2unit Formulae

8/12/2019 2unit Formulae

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MATHEMATICS REVISION OF FORMULAE AND RESULTS

Surds

ab =ab

a

b=a

b

(a)2=aAbsolute Value

a= a if a 0a=a if a


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Trigonometric Results

sin=opposite

hypotenuse (SOH)

cos=adjacent

hypotenuse (CAH)

tan=opposite

adjacent (TOA)

Complementary ratios:

sin90 =coscos90 =sintan90 =cot

sec90 =coseccosec(90 )=sec

Pythagorean Identities

sin2+cos2=1

1+cot2=cosec2

tan2+1=sec2

tan=sin

cos and cot=

cos

sin

The Sine Rule

a

sinA=

b

sinB=

c

sinC

The Cosine Rule

a2=b2+c2 2bcCosA

CosA=b

2+c2a2

2bc

The Area of a Triangle

Area=1

2abSinC

The Quadratic Polynomial

The general quadratics is: y=ax2+bx+c

The quadratic formula is: x=bb24ac

2a

The discriminant is: =b2 4ac

If 0 the roots are real

If


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Differentiation

First Principles:

f' (x) = limh

f(x+ h) f(x)h

or

f' (c) = limx

c

f

(x) f(c)h

Ify=xn thendy

dx=nxn1

Chain Rule:d

dxf (u)=f '(u) du

dx

Product Rule: Ify=uvthendy

dx=

udv

dx+v

du

dx

Quotient Rule: Ify=u

vthen

dy

dx=

v

du

dx udv

dx

v2

Trigonometric Functions:

d

dxsinx= cosx

d

dxcosx= sinx

d

dxtanx= sec2x

Exponential Functions:

d

dx ef(x)

=f '(x)ef(x)

d

dxax= ax.lna

Logarithmic Functions:d

dxlog

ef(x)= f '(x)

f (x)

Geometrical Applications of Differentiation

Stationary points:dy

dx=0

Increasing function:dy

dx> 0

Decreasing function:dy

dx< 0

Concave up:d

2y

dx2< 0

Concave down:d

2y

dx2> 0

Minimum turning point:dy

dx=0and

d2y

dx2> 0

Maximum turning point:dy

dx=0and

d2y

dx2< 0

Points of inflexion:d

2y

dx2=0and concavity changes

about the point.

Horizontal points of inflexion:dy

dx=0and

d2y

dx2= 0and

concavity changes about the point.


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Approximation Methods

The Trapezoidal Rule:

fxdx= h2y

0+y

n+ 2y

1+y

2+y

3+ +y

n1b

a

Simpsons Rule:

fxdx= h3y

0+y

n+ 4y

1+y

3++ 2y

2+y

4+

b

a

In both rules, h=ba

nwhere nis the number of strips.

Integration

Iff(x) 0 for axb, the area bounded by the

curvey=f(x), thex-axis andx=aandx=bis given

by fxdxba

.

The volume obtained by rotating the curvey=f(x)about the x-axis betweenx=aandx=bis given by

fx2ba

Ifdx

dx

=x

n theny=xn+1

n+1

Ifdx

dx= ax+bn then y =ax+ bn

a(n + 1)

Trigonometric Functions:

sinxdx = cosx+C

cosxdx = sinx+C

sec2xdx

= tanx+ C

Exponential Functions:

eaxdx= eaxa

+ C and axdx= 1ln a

.ax

Logarithmic Functions:

f' (x)f(x) dx= logex +C

Sequences and Series

Arithmetic Progression

d= U2 U1Un=a+n 1dSn=

n

2 [2a+n 1d]Sn=

n

2[a

+l] wherelis the last term

Geometric Progression

r=U2

U1

Un=arn1

Sn=arn1 1 =

a1rn1r

S=a

1 r

The Trigonometric Functions

radians = 180

Length of an arc: l=r

Area of a sector: A=1

2r2

Area of a segment: A=1

2r2( sin)

[In these formulae, is measured in radians.]

Small angle results:

sinx0

cosx1

tanx

0lim

x 0sinx

x= lim

x 0tanx

x= 1

Fory=sin

nxandy=cos

nxthe period is2

n

For y=sinnxthe period is

n


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Logarithmic and Exponential Functions

The Index Laws:

ax ay= ax+y

axay= axy

ax

y

= axy

ax= 1ax

ax

y= axy a0=1

The logarithmic Laws:

If logab=c then ac

=b

logax + log

ay= log

axy

logax log

ay= log

axy

logan + nlog

ax

logaa=1 and log

a1 =0

The Change of Base Result:

logab=

logeb

logea=

log10b

log10a

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