Problem

A curve has parametric equations

$$x = t^3 - t^2 + t - 1 \qquad\text{and}\qquad y = te^t.$$

(a) Show that 1 is the only real value of $t$ for which $x = 0$.

(b) Show that

$$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{(t + 1)e^t}{3t^2 - 2t + 1}.$$

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