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\title{\vspace*{-1.5in}\textbf{Math 2300--007: Exam III Study Guide}}
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\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\chead{April 13, 2018}
\rhead{Exam III Study Guide}
\vspace{-.5in}
\noindent
\textbf{Topics List:}
\begin{itemize}
\item \S8.5: Power Series 
\begin{itemize}
\item Interval of convergence (Don't forget the endpoints!)
\item Radius of convergence (always symmetric around center)
\item Ratio test
\end{itemize}

\item \S8.6: Representations of Functions as Power Series
\begin{itemize}
\item Making new power series from geomentric series by...
\begin{itemize}
\item Substitution
\item Differentiation
\item Integration
\item Multiplication by a power of $x$
\end{itemize}

\item Can ``steal'' the interval of convergence from the geometric series, but \textit{need to check the endpoints!}
\end{itemize}

\item \S8.7: Taylor and Maclaurin Series
\begin{itemize}
\item Taylor polynomials/series
\begin{itemize}
\item Find from scratch using $f$ and it's derivatives evaluated at ``center'' $a$
\item Find by modifying the Taylor polynomials/series of common functions
\end{itemize}
\item Remember Maclauren series for $\frac{1}{1-x}$, $e^x$, $\sin{x}$, $\cos{x}$, $\arctan{x}$, $\ln(1+x)$.

\item Can use Taylor polynomials/series in place of ``complicated'' functions when e.g.
\begin{itemize}
\item integrating
\item finding limits
\item estimating values such as $\sqrt{2}$ or $\ln(3)$ or $e$.
\end{itemize}

\item Taylor's Inequality for estimating error. Use in several ways:
\begin{itemize}
\item to estimate error in estimate of e.g. $e$ or $\sqrt{5}$.
\item to find $n$ so that the error in your approximation is within a certain amount
\item to prove that a Taylor series for $f(x)$ converges to $f(x)$ in the interval of convergence
\end{itemize}
 
\end{itemize}

\item \S7.1, 7.2, 7.3: Differential Equations (equations involving derivatives)
\begin{itemize}
\item Checking that an equation is a solution to a differential equation
\item Slope fields (graphical perspective)
\item Euler's method (numerical perspective)
\item Separation of variables (analytic perspective)
\item Vocabulary: equilibrium solution (stable, unstable), autonomous
\end{itemize}
\end{itemize}
\newpage
\noindent
\textbf{Selected Example Problems Provided by Class}\vskip12pt
\begin{enumerate}
\item Evaluate $\int e^{-x^2}\,dx$ as in infinite series. 
\item Use an infinite series to find $\displaystyle \lim_{x\to 0} \frac{e^x-1-x}{x^2}$.
\item \begin{enumerate}[(a)]
\item Find the 4th degree polynomial for $f(x) = \sqrt{x}$ centered at $a=1$ by differentiating and using the general form of Taylor Polynomials. 
\item Use the previous answer to find the 4th degree Taylor Polynomial for $g(x) = \sqrt{1-x}$ centered at $a=0$.
\item Use the previous answer to find a 3rd degree Taylor Polynomial for $h(x) = \frac{1}{\sqrt{1-x^2}}$
\end{enumerate}

\item Determine the value of the sum $\displaystyle \sum_{n=0}^\infty\frac{5\cdot 3^n}{(n+1)!}$.

\item Find the interval of convergence of $\displaystyle \sum_{n=1}^\infty \frac{(2x+1)^n}{n\cdot 3^n}$. What is the radius of convergence?

\item Use Euler's method with a step size of $0.3$ to estimate $y(0.9)$, where $y(x)$ is the solution to the initial-value problem
\[
y' = x^2, \ y(0) = 1.
\] 

\item Solve the differential equation 
\[
f'(x) = f(x)(1-f(x)),\ f(0) = \frac{1}{3}.
\]

\item Find the Taylor series for $f(x) = \frac{7}{x^4}$ about $x=-3$.

\item Show that the Maclauren series for $\cos(x)$ converges to $\cos{x}$. 

\item  Estimate $\ln(1.4)$ to be within $0.0005$.

\item Solve the differential equation $\frac{dv}{dt} = e^{v+2t}$.

\item Determine the Taylor Series for $f(x) = x^6e^{2x^3}$ about $x =0$.

\item A function $y(t)$ satisfies the differential equation \[\frac{dy}{dt} = y^4-6y^3+5y^2.\] What are the constant solutions of the equation? For what values of $y$ is $y(t)$ increasing? Decreasing?

\item Suppose $\displaystyle \sum_{n=0}^\infty c_nx^n$ converges when $x=-4$ and diverges when $x=6$. What can be said about the convergence or divergence of the following series? 
\begin{enumerate}[(a)]
\item $\displaystyle \sum_{n=0}^\infty c_n$
\item $\displaystyle \sum_{n=0}^\infty c_n8^n$
\item $\displaystyle \sum_{n=0}^\infty c_n(-3)^n$
\item $\displaystyle \sum_{n=0}^\infty (-1)^nc_n9^n$
\end{enumerate}

\item How large should $n$ be to estimate $e$ to within six decimal places (0.0000005) using a Taylor polynomial centered at $a=0$?

\item Determine the value of the sum $\displaystyle \sum_{n=0}^\infty \frac{2^n}{n!}$

\item What are the radius and interval of convergence of the series $\displaystyle \sum_{n=1}^\infty \frac{x^n}{\sqrt{n}}$?

\item Solve the differential equation \[\frac{dy}{dx} = \frac{te^t}{y\sqrt{1+y^2}}\]

\item Find the 10th degree Taylor Polynomial centered at $a=1$ for the function $f(x) = 2x^2-x+1$.

\item Find the sum of the series $\displaystyle \sum_{n=0}^\infty \frac{(-1)^n\pi^{2n}}{6^{2n}(2n)!}$.

\item Sketch the direction field for the differential equation $y' = xy-x^2$. Then, sketch the solution curve that passes through the point $(0,1)$.

\item Use a 5th degree Taylor polynomial centered at 0 to estimate $\sin(3^\circ)$. What does Taylor's inequality say about the error?

\end{enumerate} 

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