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\centerline{MATH 134 -- Calculus with Fundamentals 2}
\centerline{Practice Day on Consequences of FTC, ``Part I''}
\centerline{February 1, 2018}
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\noindent
{\it Background}
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Recall that the ``FTC, part I'' says:  If $f$ is continuous on $[a,b]$ and
$F$ is any function satisfying $\frac{d}{dx} F(x) = F'(x) = f(x)$ for all $x$ in $[a,b]$ (called
an \emph{antiderivative} of $f$), then
$$\int_a^b f(x)\ dx = F(b) - F(a).$$
This means that we have a ``shortcut method'' for computing $\int_a^b f(x)\ dx$
as long as we have or can find a suitable \emph{antiderivative} $F$ -- a function
with $F'(x) = f(x)$ for the function $f(x)$ we are integrating.
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\noindent
{\it Questions}

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\begin{enumerate}
\item[(1)] Verify that 
$$\frac{d}{dx}(e^{x^2}) = 2x e^{x^2}.$$
\item[(2)] Use part (1) to evaluate:
$$\int_0^1 2x e^{x^2}\ dx$$
\item[(3)] Verify that 
$$\frac{d}{dx}(\ln(x^2+5x+6)) = \frac{2x+5}{x^2 + 5x + 6}.$$
\item[(4)]  Use part (4) to evaluate:
$$\int_1^2 \frac{2x + 5}{x^2 + 5x + 6}\ dx.$$
\item[(5)]  Verify that 
$$\frac{d}{dx}\left(\frac{1}{2}\left(1 + \sin(x)\cos(x)\right)\right) = \cos^2(x)$$
(You'll need a trig identity here -- ask if you don't recall identities using
$\sin^2(x)$ and $\cos^2(x)$!)
\item[(6)]  Use part (5) to evaluate:
$$\int_0^{\pi/2} \cos^2(x)\ dx.$$

\end{enumerate}

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