Probability

The Analysis of Data, volume 1

Differentiation: Exercises

D.2. Exercises

1. Finish the proof of Proposition D.1.6. Find the Taylor expansion of $\sin (x)$ and $\cos (x)$ around $\alpha=0$. Use your Taylor expansions to prove that $(\sin(x))'=\cos(x)$ and $(\cos (x))'=-\sin(x)$.
2. Find a concrete example where the Taylor polynomial $P_k$ of $f$ does not converge to $f$ as $k\to\infty$.
3. Find a function whose derivative is the polynomial $Q(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$. Are there other functions whose derivatives equal $Q$?
4. The chapter describes differentiable functions as smooth, and non-differentiable functions as non-smooth. The function $f(x)=|x|$ is smooth everywhere except at $x=0$ where it has a sharp corner. Prove that $|x|$ is differentiable everywhere except at 0. Argue informally why the characterization of differentiable functions as smooth is appropriate.
5. Express $(f\circ g\circ h)'(x)$ in terms of $f(x),g(x),h(x)$ and their derivatives. Can you generalize your result to a composition of an arbitrary number of differentiable functions?