LFSE012 SciEng Maths A Final Exam Formula Page 1 of 6

LFSE012 Science Engineering Mathematics A FORMULA SHEET

LOGARITHMS AND EXPONENTIAL FUNCTIONS

General logarithms: aa n log nx x  

Common logarithms: 10 log x N N x  

Natural logarithms: log x

ee n n x ln n x  

Index laws: m n m + n

mm n

n

m n mn

0

n

n

1

nn

a . a a

a = a

a

(a ) = a

a 1

1a

a

a a

 

Logarithm laws:

a a a

a a a

p

a a

a a

a

a

log  m n = log  m + log  n

mlog = log  m log  n

n

log m = p log m

1log log m

m

log a = 1

log 1 = 0

Exponential growth and decay: 0 k tA A b 0

k tA A b

Power functions: y = Axn

Exponential functions: y = Anx

ANALYTIC GEOMETRY

Cartesian and Polar Co-ordinates

Cartesian co-ordinates: x y,

Polar co-ordinates: r ,

2 2 2

cos θ

sin θ

x r

y r

r x y

tan = y

x

LFSE012 SciEng Maths A Final Exam Formula Page 2 of 6

Straight line

Gradient of a straight line: 2 1

2 1

rise

run

y ym

x x

General form for a straight line: a x + b y + c = 0

Slope-intercept form: y = m x + c

Points-slope form: 1 1( )y y m x x

Parallel Lines: 1 2m m

Perpendicular Lines: 1

2

1m

m

Circle

Centre (h, k), radius r : 2 2 2 x h y k r ( ) ( )

General form for a circle: x2 + y

2 + a x + b y + c = 0

centre (b

2 2

a , ) radius =

2 2 4

4

a b c

Parabola

General form for a parabola: y = a x2 + b x + c

Axis of symmetry: 2

bx

a

Parabola, vertex at (0, 0): 2 4y ax Focus at (a, 0); directrix x a

2 4x ay Focus at (0, a); directrix y a

Parabola, vertex at (h, k): 2

4y k a x h Focus at ( , )h a k ; directrix x h a

2

4x h a y k Focus at ( , )h k a ; directrix y k a

Quadratic Formula: ax2 + bx + c = 0

2 4

2

b b acx

a

Ellipse

Ellipse, centre at (0, 0): 2 2

2 21

x y

a b Vertices at ( ,0), (0, )a b

Foci at 2 2( ,0)a b

Ellipse, centre at (h, k):

2 2

2 21

x h y k

a b

Vertices at ( , ), ( , )h a k h k b

2 2Foci at h ka b ( )( , )

LFSE012 SciEng Maths A Final Exam Formula Page 3 of 6

Hyperbola

Hyperbola, centre at (0, 0):

2 2

2 21

x y

a b- Vertices at ( ,0)a

Foci at (± 2 2a b( ) ,0) ; Asymptotes : b

y xa

Hyperbola, centre at (h, k):

2 2

2 21

x h y k

a b

Vertices at ( , )h a k

2 2Foci at ; ah kb ( )( , ) Asymptotes: b

y k x ha

TRIGONOMETRY

Angle measure: π = 180c

Reciprocal ratios: 1 1 1

cosecθ = ; secθ = ; cotθ = sinθ cosθ tanθ

sin θ = sin (1800 – θ) cos θ = cos(360

0 – θ) tan θ = tan (180

0 + θ)

Trigonometric identities :

2 2

2 2

2 2

sinAsin A + cos A = 1 tanA

cosA

1 + tan A = sec A

1 + cot A = cosec A

sin A + B = sinA cosB + cosA sinB

sin A B = sinA cosB cosA sinB

cos A + B = cosA cosB sinA sinB

cos A B = cosA cosB sinA sinB

tanA + tanBtan A + B =

1 tanAtanB

tanA tanBtan A B =

1 tanAtanB

2 2 2 2

2

sin 2A = 2 sinA cosA

cos 2A = 2 cos A 1 = 1 2 sin A = cos A sin A

2 tanAtan 2A =

1 tan A

2 sinA cos B = sin A+B sin A B

2 cosA sin B = sin A+B sin A B

2 cosA cos B = cos A+B cos A B

2 sinA sin B = cos A B cos A B

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1 1sinA + sinB = 2 sin A+B cos A B

2 2

1 1sinA sinB = 2 cos A+B sin A B

2 2

1 1cosA + cosB = 2 cos A+B cos A B

2 2

1 1cosA cos B = 2 sin A+B sin A B

2 2

Exact ratios:

θ sin θ cos θ tan θ

0 0 1 0

π

6

1

2 3

2

1

3

π

4

1

2

1

2

1

π

3 3

2

1

2

3

π

2

1

0

(undefined)

DIFFERENTIATION

First principles:

0 0

= lim limx x

f x x f xdy yf x

dx x x

Product rule:

If where and

then

       y uv u u x v v x

dy du dvv u

dx dx dx

Quotient rule:

2

If where and

then

       u

y u u x v v xv

du dvv u

dy dx dx

dx v

Chain rule:

If where theny y u u u x

dy dy du

dx du dx

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Standard Derivatives:

y f x or ( )

dyf x

dx or

c 0

nx 1nn x

ex ex

eax axa e

ln x 1

x

sin x cos x

sin ax a cos ax

cos x – sin x

cos ax –a sin ax

tan x 2sec x

tan ax 2 seca ax

1sin x 2

1

1 x

1cos x 2

1

1 x

1tan x 2

1

1 x

Higher order derivatives:

2

2

If then

first derivative is

second derivative is

y f x

dyf x

dx

d yf x

dx

Implicit differentiation: When differentiating of y with respect to x :

d d dy

f y f ydx dy dx

Parametric Equations:

If where ,  :y f x x x t y y t

.dy dy dt dy dt

dx dt dx dx dt

LFSE012 SciEng Maths A Final Exam Formula Page 6 of 6

APPLICATIONS OF DIFFERENTIATION

Velocity & Acceleration:

If the displacement of a body is x f t then

Velocity = dx

dt Acceleration =

2

2

d x

dt

Graph sketching:

For a graph of the function y f x :

Turning points : occur where 0dy

dx

Points of inflexion : occur where 2

20

d y

dx

Local minimums : occur where 0dy

dx and

2

20

d y

dx

Local maximums : occur where 0dy

dx and

2

20

d y

dx

Small Increments: dyy x

dx

Mensuration

Area of a trapezium: 1

( )2

a b h

Curved surface area of a cylinder: 2πrh

Volume of a cylinder: 2r h

Volume of a cone: 21

3r h

Volume of a pyramid: 1

3Ah

Volume of a sphere: 34

3r

Area of a triangle: 1

sin2

bc A

Sine rule: sin sin sin

a b c

A B C

Cosine rule: 2 2 2 2 cosc a b ab C