Differentiating inverse functions

Given that y = sin^-1(x²), find

In this question you must show detailed reasoning.

Use de Moivre’s theorem to determine constants A, B and C such that

sin⁴θ ≡ A cos 4θ + B cos 2θ + C.

The function f is defined by

The diagram shows the curve with equation y = for 0 ⩽ x < 1 and the asymptote x = 1. The region R is the unbounded region between the curve, the x-axis, the line x = 0 and the line x = 1.

You are given that the area of R is finite.

Determine the exact area of R.

form.

A surge in the current, I units, through an electrical component at a time, t seconds, is to be modelled. The surge starts when t = 0 and there is initially no current through the component. When the current has surged for 1 second it is measured as being 5 units. While the surge is occurring, I is modelled by the following differential equation.

By using an integrating factor show that, according to the model, while the surge is occurring,

I is given by I = (2t - t²)(sin^-1(t - 1) + 5).

The surge lasts until there is again no current through the component.

Determine the length of time that the surge lasts according to the model.

Determine, according to the model, the rate of increase of the current at the start of the surge.

Give your answer in an exact form.

You are given that f(x) = tan^-1(1 + x).

Find the value of f (0).

Determine the value of f ′ (0).

Hence find the Maclaurin series for f(x) up to and including the term in x².