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Find the Taylor Series expansion of tan(x) about π/4 up to the term in terms of (x-π/4)^3.

Firstly, set f(x) = tan(x). We can then differentiate this 3 times in order to find f'(x), f''(x) and f'''(x). This will require use of the chain rule and the product rule. We can find:f'(x) = sec²

Answered by Jack M. • Further Mathematics tutor

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find all the roots to the equation: z^3 = 1 + i in polar form

1st write 1+ i in polar form, sketch a diagram to find the angle ( =pi/4) and find the modulus ( sqrt(2))z^3 = Sqrt(2) E^ipi/4This is true for all equivalent solutions (add 2kpi) z^3 = sqrt(2)e^(pi/4 +2kp...

Answered by Rajan M. • Further Mathematics tutor

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Find the eigenvalues and eigenvectors of the matrix M , where M{2,2} = (1/2 2/3 ; 1/2 1/3) Hence express M in the form PDP^-1 where D is a diagonal matrix.

To start off, It's worth noting the definition of eigenvalues: for a Matrix A (n x n), it's ith eigenvalue (λ_i) is defined as the scalar constant the ith eigenvector (v_i) is multipli...

Answered by Liam T. • Further Mathematics tutor

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Find the general solution of the second order differential equation y''(t)+y(t) = 5exp(2t)

The general solution y_GS(t) to a differential equation in the form ay''(t)+by'(t)+cy(t)=f(t) is the sum of the complementary function y_CF(t) and a <...

Answered by Alex C. • Further Mathematics tutor

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