Probability
The Analysis of Data, volume 1
Differentiation: Exercises
$
\def\P{\mathsf{\sf P}}
\def\E{\mathsf{\sf E}}
\def\Var{\mathsf{\sf Var}}
\def\Cov{\mathsf{\sf Cov}}
\def\std{\mathsf{\sf std}}
\def\Cor{\mathsf{\sf Cor}}
\def\R{\mathbb{R}}
\def\c{\,|\,}
\def\bb{\boldsymbol}
\def\diag{\mathsf{\sf diag}}
\def\defeq{\stackrel{\tiny\text{def}}{=}}
$
D.2. Exercises
- 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 $(\cos (x))'=-\sin(x)$.
- Find a concrete example where the Taylor polynomial $P_k$ of $f$ does not converge to $f$ as $k\to\infty$.
- 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 $Q$?
- 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.
- 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?