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The equation: x^3 - 12x + 6 has two turning points. Use calculus to find the positions and natures of these turning points.

To find the turning points we need to find when the differential of the equations with respect to x is equal to 0. (dy/dx = 3x 2 - 12 = 0) From this we find that the turning points happen when x = 2, x = -2 ...
JR
Answered by Jon R. • Maths tutor
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Differentiating equations of the type ln[f(x)]

To solve such equations we take advantage of log lawes to simplify the problem . E.g ln[sqrt(1-x 2 )] = ln[(1-x 2 ) 1/2 ] = 1/2ln[1-x 2 ] After simplifing the problem we can differentiate with respect to x y...
MS
Answered by Mousa S. • Maths tutor
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How do I integrate x/(x^2 + 3) ?

To solve this you need to integrate by substitution. You can spot this because the differential of the bottom of the fraction is a multiple of the top part, showing this quickly; if u = x 2 + 3 (the bottom p...
KM
Answered by Knox M. • Maths tutor
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integrate by parts the equation dy/dx = (3x-4)(2x^2+5).

The equation we use to integrate by parts is y = uv - ∫ v(du/dx) dx + c so we separate dy/dx into u=(3x-4) and dv/dx=(2x 2 +5) however we still need to find du/dx and v, by differentiating u (bring the power...
AH
Answered by Abby H. • Maths tutor
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A curve is defined by the parametric equations x=t^2/2 +1 and y=4/t -1. Find the gradient of the curve at t=2 and an equation for the curve in terms of just x and y.

To find the gradient of the curve at t=2 we need to find an expression for dy/dx and then substitute in for t=2. We can make use of the chain rule to find this expression because dy/dx = (dy/dt)/(dx/dt) and ...
SC
Answered by Steven C. • Maths tutor
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