$$ \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\abs}[1]{\left\lvert#1\right\rvert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\ang}[1]{\left\langle#1\right\rangle} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} $$
Due: February 9th, 2018

Math 142B Assignment 3

  1. Let's verify that $e^x$ can be defined with the usual properties that we expect. In this problem, be sure not to use any properties of the logarithm before establishing them. Define the function $L$ as follows: $$\begin{align*}L : (0, \infty) &\longrightarrow \R\\x&\longmapsto\int_1^x \frac1t\,dt.\end{align*}$$
    1. Explain why $L$ is differentiable.
    2. Show that $\displaystyle\lim_{t\to\infty}L(t) = \infty$. (Hint: compare $L(n)$ to the harmonic series $\sum \frac1k$, which we know diverges from 142A.)
    3. Show that $L(ab) = L(a) + L(b)$. (A question from the midterm may be useful here.)
    4. Conclude that $\displaystyle\lim_{t\to0^+}L(t) = -\infty$.
    5. The fact that $L$ is continuous means we know know that its range is $(-\infty, \infty)$. Show that $L$ is invertible. (Hint: there is a very useful criterion from 142A about monotonic functions and invertiblilty; it may be useful to show that $L$ is monotonic.)
    6. Define the function $E : \R \to (0, \infty)$ to be the inverse of $L$, so $E(L(x)) = x$ and $L(E(y)) = y$. Show that $E'(x) = E(x)$.
    7. Show that for any rational number $q \in \Q$, $E(q) = E(1)^q$. (Hint: use Part C; recall that in 142A we were able to prove that for any $\alpha \gt 0$ and any $x, y \in \R$, $\alpha^{x+y} = \alpha^x\alpha^y$.)
    8. Use the fact that for any $\alpha \gt 0$ the function $x \mapsto \alpha^x$ is continuous to explain why $E(x) = E(1)^x$ for all $x \in \R$.
    9. Show that the Taylor series for $E$ centred at $0$ converges to $E$ at every point.
    10. Conclude that $E(1) = \sum_{j=0}^\infty \frac1{j!}$. This number, $E(1)$, is the familiar constant $e$.
  2. Show that if $p$ is a polynomial such that $p(x_0) = 0$, then there is a polynomial $q$ so that $p(x) = (x-x_0)q(x)$. Conclude that a polynomial of degree $n$ has at most $n$ roots.
  3. Suppose that $f, g : I \to \R$ have $n+1$ continuous derivatives. Show that they have contact of order $n$ at $x_0 \in I$ if and only if $$\lim_{x\to x_0}\frac{f(x)-g(x)}{(x\color{red}{-x_0})^n} = 0.$$
  4. Bonus:

    Suppose that $R \gt 0$ and $f : (-R, R) \to \R$ is infinitely differentiable. Suppose further that for any $0 \lt r \lt R$ there is a sequence $(M_{r,k})_k$ such that for all $x \in (-r, r)$, $$\abs{f^{(k)}(x)} \leq M_{r,k} \qquad\qquad\text{and}\qquad\qquad \lim_{k\to\infty} M_{r,k}\frac{r^k}{k!} = 0.$$ Show that for every $x \in (-\color{red}{R}, \color{red}{R})$, $$\int_0^x f(t)\,dt = \sum_{k=1}^\infty f^{(k-1)}(0)\frac{x^k}{k!}.$$

    (Hint: first show that our assumptions bounding the derivatives of $f$ mean that the Taylor polynomials converge uniformly to $f$ in the sense of this 142A assignment; next show that if a sequence of functions $(f_n)$ converges uniformly to $f$ then the sequence of integrals $\int_a^b f_n$ converges to $\int_a^b f$; then put these two statements together.)