函数 $\mathrm f$ 满足 $\mathrm f''(x)=\mathrm f(x)$。

用数学归纳法证明：对于每一个正整数 $n$，

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

一次归纳要跨越两个导数阶数，因此从假设式连续求导两次，每次都用f的二阶导数等于自身。

For $n=1$, the product rule gives

which is the required expression, so the base case holds.

Assume that, for some positive integer $k$,

Differentiating once and using $f''=f$,

Differentiating again,

Since $2k+1=2(k+1)-1$, this is the required formula for $n=k+1$. The formula holds for $n=1$ and truth for $n=k$ implies truth for $n=k+1$, so it follows by induction for every positive integer $n$.