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\begin{center}
{\Large Warm-Up Problems

January 27, 2003}
\end{center}


\begin{enumerate}

\item  Let $f(x)=\sin(x)$.
  \begin{enumerate}
  \item  Find a formula for $F(x)=\int_0^x f(t)\; dt$.  (Notice the use of $x$ and $t$.)
  \item  Find $F'(x)$.  (\sol: $\sin x$)
  \end{enumerate}

\item  Let $f(x)=\sin(x)$.
  \begin{enumerate}
  \item  Find a formula for $F(x)=\int_0^{x^2} f(u)\; du$.
  \item  Find $F'(x)$.  (\sol: $2x \sin(x^2)$)
  \end{enumerate}

\item  Let $f(x)=\sin(x)$.
  \begin{enumerate}
  \item  Find a formula for $F(x)=\int_x^0 f(z)\; dz$.
  \item  Find $F'(x)$.  (\sol: $-\sin(x)$)
  \end{enumerate}

\item  Let $f(x)=\sin(x)$.
  \begin{enumerate}
  \item  Find a formula for $F(x)=\int_{2x+1}^{-\pi} f(y)\; dy$.
  \item  Find $F'(x)$.  (\sol: $-2\sin(2x+1)$)
  \end{enumerate}

\item  Let $f(x)=\sin(x)$.
  \begin{enumerate}
  \item  Find a formula for $F(x)=\int_{2x}^{\tan x} f(r)\; dr$.
  \item  Find $F'(x)$.  (\sol: $-2\sin(2x) + (\sec^2x) \sin(\tan x)$)
  \end{enumerate}

\item  Compute:
  \[
  \frac{d}{dx} \int_4^{x^3+2x} 4t+1 \; dt
  \]
  (\sol: $(3x^2+2)( 4(x^3+2x)+1)$)


\item  Can you find the pattern?  How about this one:
  \[
  \frac{d}{dx} \int_{0}^x \sin(t^2)e^{t^2} \; dt
  \]
  (\sol: $\sin(x^2)e^{x^2}$)

\item  Can you find the pattern?  How about this one:
  \[
  \frac{d}{dx} \int_{0}^{3x-4x^2} \sin(t^2)e^{t^2} \; dt
  \]
  (\sol: $(3-8x)\sin(x^2)e^{x^2}$)




\end{enumerate}



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