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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2019

Paper 4

\b[No.] Define
$$
f_n(x)=\sqrt{x^2+\frac1n}.
$$
Each $f_n$ is a <differentiable function> on $[-1,1]$, while
$$
0\leq f_n(x)-|x|
=\frac{1/n}{\sqrt{x^2+1/n}+|x|}
\leq\frac1{\sqrt n}.
$$
\b[Thus $f_n$ converges uniformly to the <absolute value function> $|x|$, which is not differentiable at $0$. Uniform convergence alone therefore does not preserve differentiability.]

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 3E / iii / Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 3E / iii / Solution

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Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 3E / iii / Solution

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