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MATH 2A / Fall Quarter of 2006 \hfill September 19, 2006

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Last Name: \hfill First Name: \hfill SID: \hfill {}

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1. (4 points) Find the derivatives of the following functions:
\begin{enumerate}
\item[i)] $f(x) = \sin^2 x + \log x^3 - \ln \frac{x}{2}$
\item[ii)] $g(x) = \alpha \cot x + \sqrt{x} + \sqrt[7]{x^{x+1}}$, where $\alpha \neq 0$
\end{enumerate}

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2. (3 points) Evaluate the following integral:
$$ \int_0^{\infty} e^{x^2} \, dx \,\,\,\,\,\, \mbox{or} \,\,\,\,\,\, \int_0^{\infty} \exp(x^2) \, dx $$

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3. (3 points) Prove that $\sin x \le x - \frac{x^3}{3!}$ for all real numbers $x$.

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