Problem

Figure 1 shows a sketch of the curve $C$ with parametric equations

$$x=5+2\tan t \qquad y=8\sec^2 t \qquad -\frac{\pi}{3}\leq t\leq \frac{\pi}{4}$$

(a) Use parametric differentiation to find the gradient of $C$ at $x=3$

The curve $C$ has equation $y=\mathrm{f}(x)$, where $\mathrm{f}$ is a quadratic function.

(b) Find $\mathrm{f}(x)$ in the form $a(x+b)^2+c$, where $a$, $b$ and $c$ are constants to be found.

(c) Find the range of $\mathrm{f}$.