Higher Engineering Mathematics

Ess-For-H8152.tex 19/7/2006 18: 2 Page 705

Essential formulae

Number and Algebra

Laws of indices:

am×an=am+n

am

an

=am−n (am)n=amn

a

m

n = n

√

am a−n=

1

an

a^0 = 1

Quadratic formula:

If ax^2 +bx+c=0 thenx=

−b±

√

b^2 − 4 ac

2 a

Factor theorem

If x=ais a root of the equation f(x)=0, then

(x−a) is a factor off(x).

Remainder theorem

If (ax^2 +bx+c) is divided by (x−p), the

remainder will be:ap^2 +bp+c.

or if (ax^3 +bx^2 +cx+d) is divided by (x−p), the

remainder will be:ap^3 +bp^2 +cp+d.

Partial fractions

Provided that the numeratorf(x) is of less degree

than the relevant denominator, the following iden-

tities are typical examples of the form of partial

fractions used:

f(x)

(x+a)(x+b)(x+c)

≡

A

(x+a)

+

B

(x+b)

+

C

(x+c)

f(x)

(x+a)^3 (x+b)

≡

A

(x+a)

+

B

(x+a)^2

+

C

(x+a)^3

+

D

(x+b)

f(x)

(ax^2 +bx+c)(x+d)

≡

Ax+B

(ax^2 +bx+c)

+

C

(x+d)

Definition of a logarithm:

Ify=axthenx=logay

Laws of logarithms:

log (A×B)=logA+logB

log

(

A

B

)

=logA−logB

logAn=n×logA

Exponential series:

ex= 1 +x+

x^2

2!

+

x^3

3!

+···

(valid for all values ofx)

Hyperbolic functions

sinhx=

ex−e−x

2

cosechx=

1

sinhx

=

2

ex−e−x

coshx=

ex+e−x

2

sechx=

1

coshx

=

2

ex+e−x

tanhx=

ex−e−x

ex+e−x

cothx=

1

tanhx

=

ex+e−x

ex−e−x

cosh^2 x−sinh^2 = 11 −tanh^2 x=sech^2 x

coth^2 x− 1 = cosech^2 x