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\flushleft\underline{Homework 11 --- MATH 4590 Spring 2018} \\
\begin{enumerate}[1.]
\item In this problem we will sketch the graphs of $Q_n(x)$ (from 16 April slides).
\begin{enumerate}
\item First compute $\displaystyle\int_{-1}^1 (1-x^2)^n \mathrm{d}x$ for $n=0,1,2,3,4,5$ (Using WolframAlpha for this is OK). This defines the constants $c_n = \dfrac{1}{\displaystyle\int_{-1}^1 (1-x^2)^n \mathrm{d}x}$. 
\item Write down $Q_n(x)=c_n (1-x^2)^n$ for $n=0,1,2,3,4,5$. 
\item Draw $Q_n(x)$ (from 16 April slides) on $[-1,1]$ for $n=0,1,2,3,4,5$ (ok to use WolframAlpha or Desmos for this). 
\end{enumerate}
\item In this problem you will argue that $(1-x^2)^n \geq 1-nx^2$ as needed in the proof of the Weierstrass approximation theorem (16 April slides). To do it, consider the function  $g \colon [0,1] \rightarrow \mathbb{R}$ defined by
$$(*) \hspace{35pt} g(x) = (1-x^2)^n - 1+nx^2.$$
\begin{enumerate}
\item Differentiate this function. 
\item Is the derivative positive or negative? 
\item What does that mean (in a calc 1 sense)? 
\item What is the value of $g(0)$? 
\item Combining your answers to part (c) and part (d), what must you conclude about $g(x)$ for $x \in [0,1]$?
\item What do you do with your answer to (e) to arrive at the desired inequality $(*)$?
\end{enumerate}

\item It is said in the 16 April slides that for any $0<\delta<1$, the function $Q_n \colon [0,1] \rightarrow \mathbb{R}$ defined by $Q_n(x)=c_n (1-x^2)^n$ obeys $Q_n \rightrightarrows 0$ on $[\delta,1]$. In this problem, we investigate that further.
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\item It was shown in the slides that $c_n < \sqrt{n}$. Therefore
$$Q_n(x) \leq \sqrt{n} (1-x^2)^n.$$
Use L'H\^{o}pital's rule to argue that for any $x \in [\delta,1]$,
$$\displaystyle\lim_{n \rightarrow \infty} Q_n(x) = 0.$$
(\textit{hint}: rewrite the function as $\dfrac{\sqrt{n}}{(1-x^2)^{-n}}$ before trying L'H\^{o}pital's rule. Also be careful when differentiating... it is \textbf{not} a ``power rule" since you are differentiating w.r.t $n$.)
\item Does $Q_n \rightrightarrows 0$ on $[0,1]$? Why or why not? (A graphical argument about ``$\epsilon$-tubes" combined with the graphs in question 1 is ok)
\item Why do we know that $Q_n \rightrightarrows 0$ on $[\delta,1]$? (again, a graphical argument is ok)
\end{enumerate}

\item Recall the definition of Bernstein polynomials from the 16 April slides. Let $f(x)=\cos(2\pi x)$. Find and plot the first 5 Bernstein polynomials associated with $\cos(2\pi x)$ on the interval $[0,1]$.
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