

\DefineExercise{E4-8_WarmUp1}{
  \begin{enumerate}[label=(\alph*)]
  \item Write down a particular antiderivative of $3x^4$.  (It should \emph{not} have an arbitrary constant in it.)
  \item Write down a different particular antiderivative of $3x^4$.
  \end{enumerate}
}

\DefineExercise{E4-8_WarmUp2}{
  \begin{enumerate}[label=(\alph*)]
  \item Write down an antiderivative of $e^{2x}$.
\item Write down $\int e^{2x} dx$ (the \emph{indefinite integral}) of $e^{2x}$.
\item Write down a particular antiderivative of $e^{2x}$, $F(x)$, with the property that $F(1)=2$.
  \end{enumerate}
}

\DefineExercise{E4-8_Main1}{
  \begin{enumerate}[label=(\alph*)]
  \item Write down a particular antiderivative of $3x^4+4x^3-\cos x$.

\texttt{Make sure to take the derivative of your answer to check that you're correct.}

  \item Write down a particular antiderivative of $4e^{2x}+\frac{2}{1+x^2}-3\sec x+2^x$.
  \end{enumerate}
}

\DefineExercise{E4-8_Main2}{
Your friend proposes a product rule for antiderivatives:
\begin{quotation}
  If $F(x)$ is the antiderivative of $f(x)$ and $G(x)$ is the antiderivative of $g(x)$ then $F(x)G(x)$ should be the antiderivative of $f(x)g(x)$.
\end{quotation}
Demonstrate that this isn't always true.
}

\DefineExercise{E4-8_Main3}{
  \begin{enumerate}[label=(\alph*)]
  \item Find the indefinite integral $\int \frac{4}{x^2} dx$.
  \item Use Part 2 of the Fundamental Theorem of Calculus to find the definite integral $\int_1^2 \frac{4}{x^2}dx$.
  \item Explain why Part 2 of the Fundamental Theorem of Calculus doesn't tell you how to find $\int_{-1}^2\frac{4}{x^2} dx$.  (You'll learn how to solve this integral next semester.)
  \end{enumerate}
}

\DefineExercise{E4-8_Main4}{
    \begin{tikzpicture}
  \begin{axis}[width=240, height =200, xmax=2, ymax=2, xmin=-2, ymin=-2, xlabel=$x$, ylabel=$\frac{x^3}{4}$, axis x line = middle, axis y line=middle]
\addplot[smooth, samples=1000, domain=-2:2] {x*x*x/4};
\end{axis}
    \end{tikzpicture}

  \begin{enumerate}[label=(\alph*)]
\item Find $\int_{-2}^{2}|\frac{x^3}{4}|dx$.  (Careful: you don't know any antiderivative for $|\frac{x^3}{4}|$!)
\item Find $\int_{-\pi}^{\pi/3}|\sin x|dx$.
  \end{enumerate}
}

\DefineExercise{E4-8_Main5}{
If $F'(t)$ is the \emph{rate of change} at time $t$ then $\int_a^b F'(t) dt$ is the \emph{net change}---the overall amount of change---between times $a$ and time $b$.
  \begin{enumerate}[label=(\alph*)]
  \item We throw a rock straight up into the air at time $0$.  At time $t$, the velocity of the rock is $v(t)$.  (We don't know an exact formula for $v(t)$.)  Write down an equation involving $v$, the time $t$, and an integral which will be true when the rock returns to its starting height (and at no other time except perhaps time $0$).
  \item From the information given, is it possible to write down an equation that will be true when the rock hits the ground?
  \item What is the difference between these two examples?  What does it have to do with the constant of integration?
  \end{enumerate}
}

\DefineExercise{E4-8_Main6}{
  \begin{enumerate}[label=(\alph*)]
  \item If $F(x)=\int_0^x t^2 dt$, find a polynomial which is equal to $F(x)$.
  \item If $G(z)=F(z^2)$, find $G(z)$ as a polynomial.
  \item Find $G'(x)$.
  \end{enumerate}
}

