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Consider the functions f(x) = −x^3 + 2x^2 + 3x and g(x) = −x^3 + 3x^2 − x + 3. (a) Find df/dx (x) and hence show that f(x) has turning points at when x = 2 /3 ± √ 13/ 3 . [5] (b) Find the points where f(x) and g(x) intersect. [4]

a) First differentiate f(x) using standard polynomial derivative rules which gives, df/dx=-3x^2+4x+3. The derivative function gives the gradient of f for any value of x. Turning points occur when the grad...

Answered by George Alexander L. • Maths tutor

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Solve the following: sinx - cosx = 0 for 0≤x≤360

We know that sinx/cosx = tanx. Therefore we can write sinx - cosx = 0 as sinx = cosx . By diving both sides by cosx, we get tanx = 1. By taking tan inverse of both sides, we can see that for 0≤x≤360, we g...

Answered by Aaman K. • Maths tutor

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Why does integration by parts work?

Recall the product rule for differentiation: the derivative of uv is equal to u'v+uv'.If we use the fact that integration reverses differentiation (so the integral of f' is f), then we calculate that uv i...

Answered by Thomas B. • Maths tutor

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A curve has equation 2(x^2)+3x+10. What is the gradient of the curve at x=3

Answered by Tom E. • Maths tutor

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A curve has equations: x=2sin(t) and y=1-cos(2t). Find dy/dx at the point where t=pi/6

Since this question concerns parametric's, one may move to eliminate t from the equation to calculate dy/dx directly. However, in this case it is much easier to use the chain rule and realise that dy/dx=d...

Answered by Cameron H. • Maths tutor

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