(a) Let $x(t)$ and $\phi_{n}(t)$, for $n=0,1,2, \ldots$, be real-valued functions on $\mathbb{R}$.

(i) Define what it means for the sequence $\left\{\phi_{n}(t)\right\}_{n=0}^{\infty}$ to be an asymptotic sequence as $t \rightarrow \infty$.

(ii) Define what it means for $x(t)$ to have the asymptotic expansion

$$x(t) \sim \sum_{n=0}^{\infty} a_{n} \phi_{n}(t) \quad \text { as } \quad t \rightarrow \infty$$

(b) Use the method of stationary phase to calculate the leading-order asymptotic approximation as $x \rightarrow \infty$ of

$$I(x)=\int_{0}^{1} \sin \left(x\left(2 t^{4}-t^{2}\right)\right) d t$$

[You may assume that $\int_{-\infty}^{\infty} e^{i u^{2}} d u=\sqrt{\pi} e^{i \pi / 4}$.]

(c) Use Laplace's method to calculate the leading-order asymptotic approximation as $x \rightarrow \infty$ of

$$J(x)=\int_{0}^{1} \sinh \left(x\left(2 t^{4}-t^{2}\right)\right) d t$$

[In parts (b) and (c) you should include brief qualitative reasons for the origin of the leading-order contributions, but you do not need to give a formal justification.]