brief insight. 3.1 understand mathematical equations appropriate to the solving of general...
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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
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