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Further Mathematics

A Level

The function f is defined for x > 0 by f (x) = x^1n x. Obtain an expression for f ′ (x).

By taking the natural log on both sides we can see that: ln(f(x)) = ln(x)^2 This is a more familiar expression that we know how to differentiate LHS: f '(x)/f(x), RHS: 2*ln(x)/x By re...

Answered by Steven E. • Further Mathematics tutor

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Given z=cosx+isinx, show cosx=1/2(z+1/z)

We first consider 1/z=(cosx+isinx)^(-1). Application of De Moivre's theorem for integer n: (cosx+isinx)^(n)=cosnx+isinnx yields the result 1/z=cosx-isinx. Addition of the two forms z and z^(-1) steers us ...

Answered by Chris S. • Further Mathematics tutor

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Find the square root of complex number 3 + 4i

Strategy: write down an equation satisfied by the square root, and solve it algebraically. Method: square root x+iy satisfies (x+iy)² = 3 + 4i. Expand: x

Answered by Jakob S. • Further Mathematics tutor

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Show that cosh^2(x)-sinh^2(x)=1

In exponential form, sinh^2(x)=1/4(e^2x+e^-2x-2), cosh^2(x)=1/4(e^2x+e^-2x+2). Therefore cosh^2(x)-sinh^2(x)=1/4(2-(-2))=4/4=1, as required.

Answered by Tutor80806 D. • Further Mathematics tutor

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Why is the argument of a+bi equal to arctan(b/a)?

Think about the point a+bi on the complex plane. Specifically, a is how far along the x (real) axis, and b is how far up the y (imaginary) axis the point is. If you draw a line connecting the origin and t...

Answered by Martin S. • Further Mathematics tutor

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