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\begin{document}
	
	\mbox{}
	\title{Problem Set 20: Some Applications of Taylor Series}
	
	\maketitle	
	
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	%\section{Introduction} \label{sec:intro}
	\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.
		
	\begin{problem}
		Evaluate the indefinite integral as an infinite series.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			
			\item $\int \sqrt{1+x^3}\,dx$;
			
			\item $\int x^2 \sin(x^2)\,dx$;
			
			\item $\int \frac{\cos x - 1}x\,dx$;
			
			\item $\int \tan^{-1}(x^2)\,dx$.
		
			
		\end{enumerate}
	\end{problem}
	
	\begin{problem}
		Use series to approximate the definite integral to within the indicated accuracy. 
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			
			\item $\int_0^{1/2} x^3 \tan^{-1} x\,dx$ \quad (four decimal places);
			
			\item $\int_0^1 \sin(x^4)\,dx$ \quad (four decimal places);
			
			\item $\int_0^{0.5} x^2e^{-x^2}\,dx$ \quad ($|\text{error}| < 0.001$).
			
		\end{enumerate}
	\end{problem}
	
	\begin{problem}
		Let $f(x) = (1 + x^3)^{30}$. Compute $f^{(57)}$ and $f^{(58)}$.
	\end{problem}
	
	\begin{problem}
		Find the Taylor polynomial $T_3(x)$ for $f(x) = \cos x$ at $a = \pi/2$. Estimate $\cos 80^\circ$ correct to five decimal places.
	\end{problem}
	
	\begin{problem}
		Approximate $f(x) = x \sin x$ at $0$ by its Taylor polynomial $T_4(x)$ when $-1 \le x \le 1$. Estimate the potential error of this approximation.
	\end{problem}
	
	\begin{thebibliography}{20}
		
		\bibitem{jS15} J.~Stewart: \emph{Single Variable Calculus} 8th Edition, Cengage Learning, Boston 2015.
		
	\end{thebibliography}
	
\end{document}