Problem

The curve $C$ has parametric equations

$$x = \sinh t \quad\text{and}\quad y = t + \cosh t.$$

(a) Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ in terms of $t$.

(b) Show that

$$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = \frac{1 - \sinh t}{\cosh^3 t}.$$

(c) Find the Maclaurin’s series for $y$ in terms of $x$ up to and including the term in $x^2$.