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# Calculus 2: test 3

This is a test for Calculus 2: course.

1. Find the Taylor polynomial of order $2$ centered at $c=\pi/2$ of the function $$f(x)=\sin^{2}x.$$ How accurate is this approximation on the interval $[0,\pi]?$
2. Find the radius and the interval of convergence of the series $$\sum\dfrac{(x-1)^{n}}{\sqrt{n}2^{n}}.$$
3. Find the Maclauren series of the function $$f(x)=\dfrac{x^{2}}{1-x^{2}}$$ and its interval of convergence.
4. Use the fact that $$\dfrac{d}{dx}xe^{x^{2}}=e^{x^{2}}+2x^{2}e^{x^{2}}$$ to find the power series representation of the latter function.
5. Identify the following quadratic curve $$y^{2}-4y=x+5,$$ find its focus/foci and sketch it.
6. Suppose the parametric curve is given by $$x=e^{-t},y=e^{3t}.$$ Eliminate the parameter, simplify and sketch the graph of the function.

1. Let $$I=\int_{0}^{8}f(x)dx.$$ (a) Use the graph of $y=f(x)$ on the right to estimate $L_{4},M_{4},R_{4}.$
(b) Compare them to $I$.
2. Evaluate $$\int_{-\infty}^{\infty}xe^{-x^{2}}dx.$$
3. Find the area of the surface of revolution around the $x$-axis obtained from $y=\sqrt{x},4\leq x\leq9.$
4. Find the centroid of the region bounded by the curves $y=4-x^{2},y=x+2$.
5. Describe the motion of a particle with position $(x,y),$ where $x=2+t\cos t,$ $y=1+t\sin t,$ as $t$ varies within $[0,\infty)$.
6. Suppose the parametric curve is given by $x=te^{t},y=\sin2t,-\pi\leq t\leq3.$ Set up, but do not evaluate, the integrals that represent (a) the arc-length of the curve$,$ (b) the area of the surface obtained by rotating the curve about the $x$-axis.
7. Plot the curve $r=2\cos(3\theta)$. Find the line(s) through the origin tangent to the curve.