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{\Large \textbf{STA 256f19 Assignment One: Calculus Review}}%\footnote{Copyright information is at the end of the last page.}
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\noindent
These homework problems are not to be handed in.  They are preparation for Term Test 1 (and the rest of the course).

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\item $ \displaystyle  \int_{1}^{3} \frac{1}{t^3} dt $ \hspace{10mm} [answ: 4/9]

\item $ \displaystyle  \int_{0}^{\infty} e^{-\theta x} dx $,
      where $\theta > 0$. \hspace{10mm} [answ: 1/$\theta$]
\item $ \displaystyle  \int_{0}^{\infty} x e^{-x} dx $ \hspace{10mm}  [answ: 1] 
\item $ \displaystyle  \frac{d}{dx} (xe^x)  $ \hspace{10mm} [answ: $(1+x)e^x$]

\item  $ \displaystyle  \frac{d}{dt} \ln(1+e^x)  $\hspace{10mm} [answ: $ \frac{e^x}{1+e^x}$ ]

\item Find the maximum or minimum of 
   $ \displaystyle  f(x) = e^{-\frac{1}{2}(x-\mu)^2} $  \hspace{10mm} [answ: max at $x=\mu$]
\item $ \displaystyle  \sum_{k=0}^{\infty} \frac{1}{2^k} $ \hspace{10mm} [answ: 2]
\item For $-1<a<1$ , find
       $ \displaystyle  \sum_{k=j}^{\infty} a^k $ \hspace{10mm} [answ: $ \displaystyle \frac{a^j}{1-a}$; prove it.] 
\item For $\lambda>0$, find
       $ \displaystyle  \sum_{k=0}^{\infty} 
       \frac{\lambda^k e^{-\lambda}}{k!} $ \hspace{10mm} [answ: 1] 
\item Show $\displaystyle \lim_{n \rightarrow \infty}\left(1 + \frac{x}{n}\right)^n = e^x$. Hint: Use natural logs and L'H\^{o}pital's rule. 

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This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Mathematical and Computational Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website: 
     
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\href{http://www.utstat.toronto.edu/~brunner/oldclass/256f19} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/256f19}}
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\item Let $f(x,y) = $
   \(  \left\{ \begin{array}{ll} % ll means left left
   x y^2 & \mbox{for } x < y  \\
   0     & \mbox{elsewhere} 
   \end{array}  \right. \).
   Find $ \displaystyle  \int_{0}^{1} \int_{0}^{1} f(x,y)\,dy\,dx $. \hspace{10mm} [answ: 1/10, \textbf{not} 1/6]