Advanced Maths for Engineers

Brief insight

3.1 Understand mathematical equations appropriate to the solving of general engineering problems

3.2 Understand trigonometric functions and equations

3.3 Understand differentiation and integration

3.4 Understand complex numbers

Advanced Maths for Engineers

indices

a2 x a3

= a2 + 3

a6

a4 ÷ a2

= a4 - 2

a2

(a2)3 = a6

indices

3a2b3 x 2a4b

Separate the terms 3 x 2 = 6

a2 x a4 = a6 b3 x b = b4

Answer = 6a6b4

indices

Show that 43/2 = 8 43/2 means the square root of 4 cubed

The square root of 4 = 2, 23 = 8

indices

N = ax

logaN = x

4 = 22

log24 = 2

8 = 23

Log28 = 3

Logs and indices

Foil

(2x + 5)(3x + 2) = 6x2 + 4x + 15x +10 =

6x2 +19x+10

 6x + 3y = 9    2x + 3y = 1

4x = 8X = 2

Y = -1

Simultaneous equations

Pythagoras and trigonometry

sec x  =    1               cos x

cosec x =    1                     sin x cot x =      1       =   cos x               tan x         sin x

sin x = tan x cos x

Trigonometric functions

y = x2 + 4x Calculate dy/dx when x = 3 dy/dx = 2x + 4 = 10 y = 6x3 + 2x2 +3 Calculate dy/dx when x = 2

dy/dx = 18x + 4x = 44

TASTE OF CALCULUS

Distance / Time graph

The gradient represents the change in distance with respect to time dy/dx

Speed is the differential of distance

Acceleration is the differential of speed

Maximum and minimum values

Let's use for our first example, the equation 2X2 -5X -7 = 0

The derivative dy/dx = 4x -5 = 0

4x = 5 x = 5÷4 = 1.25

Y = 2*(1.25)2 -5*1.25 -7Y = -10.125

At minimum value

Maximum and minimum values

Y = -4X2 + 4X + 13 = 0

dY/dX = -8X + 4

X = 4 ÷ -8 = -0.5

Y = -4*(.5*.5)2 +4*.5 + 13Y = 14

At Maximum value

Complex numbers

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, where i2 = −1

When a Real number is squared the result is always non-negative. Imaginary numbers ofthe form bi are numbers that when squared result in a negative number.

Complex numbers