Brief insight. 3.1 Understand mathematical equations appropriate to the solving of general...
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![Page 1: Brief insight. 3.1 Understand mathematical equations appropriate to the solving of general engineering problems 3.2 Understand trigonometric functions.](https://reader038.fdocuments.net/reader038/viewer/2022110321/56649cf45503460f949c1b25/html5/thumbnails/1.jpg)
Advanced Maths for Engineers
Brief insight
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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
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indices
a2 x a3
= a2 + 3
a6
a4 ÷ a2
= a4 - 2
a2
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(a2)3 = a6
indices
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3a2b3 x 2a4b
Separate the terms 3 x 2 = 6
a2 x a4 = a6 b3 x b = b4
Answer = 6a6b4
indices
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Show that 43/2 = 8 43/2 means the square root of 4 cubed
The square root of 4 = 2, 23 = 8
indices
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N = ax
logaN = x
4 = 22
log24 = 2
8 = 23
Log28 = 3
Logs and indices
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Foil
(2x + 5)(3x + 2) = 6x2 + 4x + 15x +10 =
6x2 +19x+10
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6x + 3y = 9 2x + 3y = 1
4x = 8X = 2
Y = -1
Simultaneous equations
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Pythagoras and trigonometry
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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
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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
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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
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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
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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
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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
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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