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First simplify the equation of the curve y= x^4 - 3x^3 .The gradient is the differential.To differentiate, bring down the power and take one from it.x^4 becomes 4x^3-3x^3 becomes (-3x3)= -9x^2dy/dx= 4x^3 ...

Firstly, to be able to find the distance between the two points we must find the y-coordinates of each point by substituting in the x values. For x=4y=3/4x4-2=1For x=8y=3/4x8-2=4Now that we know the coord...

To first find the centre of the circle, the key fact to remember is that the radius of a circle is normal (perpendicular) to any tangent. This means if you take the normals to both of these tangents at th...

If you are comfortable with differentiation. You can take the second derviatve of the equation of the cruve and plug in the x value of the curve. Based on this answer you can determine if it's a maximum, ...

You can rearrange the equation to get 2cos^2(x)+2sin^2(x) = 2. This can be factorised to get cos^2(x) + sin^2(x) = 1 which is a known identity. We can use the fact that 2cos^2(x) = 2sin^2(x) - 2 and subst...