Calculus Second Semester: Math 3112

Problem 1. Evaluate

\begin{displaymath}\int_{0}^{\pi/4} (2x^2 -x)\tan^{-1}(x) dx \end{displaymath}


\begin{displaymath}\int \cos^2(x)\sin^4(x) dx \end{displaymath}

Problem 2. Find the limit of

$\displaystyle \lim_{x \rightarrow 0}
\frac{x^2- \sin^2(x)}{1 - x^2/2 - \cos(x)}$
$\displaystyle \lim_{x \rightarrow 0}
\frac{x^5(e^{x^3} - 1)}{\cos(x^2) -1 - x^4/2}$

Problem 3. Determine the convergence or divergence of

$\displaystyle \int_{0}^{1} \frac{1}{\sqrt{2x-x^2}}dx $. Find its value if it converges.
$\displaystyle \int_{0}^{\pi/2} \frac{\tan(x)}{x^{3/2}+x^2}dx
$\displaystyle \int_{0}^{\infty}
\frac{\sin(1/x)}{\sqrt{x} +x}dx$

Problem 4. Determine whether the following series are convergent..

$\displaystyle \sum_{n=1}^{\infty} \frac{(n+3)! - n!}{2^{n-1}}$
$\displaystyle \sum_{n=1}^{\infty} \frac{n^4 3^{2n+1}}{10^{n-1}}$.
Problem 5. Find the set of convergence for

\begin{displaymath}\sum_{n=0}^{\infty} \frac{\sqrt{n}}{n+3} x^n.\end{displaymath}

Problem 6. Let

\begin{displaymath}f(x) = \frac{e^x}{1-x} = a_0 + a_1 x + a_2 x^2 + ......\end{displaymath}

Find a0 , a1, a2, a3, a4.

Problem 7. Find f(9)(0) and f(10)(0), where

\begin{displaymath}f(x) = x \ln(1 + x^3).\end{displaymath}

Hint: In order to find the Taylor series of $\ln(1+x)$ at 0, use the derivative technique.

Problem 8. Find the first three terms of the series for the function

\begin{displaymath}f(x) = (1+x^2) \sqrt{1 - x},\end{displaymath}

at x0 = 0.

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