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Ideally to find the exact area under the curve, we would integrate the function and substitute in the bounds given. However, using the trapezium rule gives an approximation whereby using more trapezia inc...

We know that 1. sin(a+b) = sin(a)cos(b)+sin(b)cos(a) and 2. sin(a-b) = sin(a)cos(b)-sin(b)cos(a) Add equations 1. and 2. sin(a+b)+sin(a-b) = 2sin(a)cos(b)+sin(b)cos(a)-sin(b)cos(a) = 2sin(a)cos(b) Let x=a...

For the line passing through A and B: m = (y2-y1)/(x2-x1) = (-6-4)/(7-3) = -5/2

For the perpendicular line: m = -1/(-5/2) = 2/5 

y - y1 = m*(x - x1)  >>  y - 4 = (2/5)*(x - 3)  >>...

y = (3x⁴ - 18)/x

The gradient of a tangent to a curve is equal to dy/dx 

However, we must simplify this equation before we can differentiate it;

y = 3x³ - 18/x =...

y= x³ +5x² -6x +4

dy/dx = 3x² +10x -6

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