 # Practice Problems for Module 12

Sections 11.10 and 11.11

Directions. The following are review problems. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. You can also follow the link for the full video page to find related videos.
1. Find the Maclaurin series of $$f(x)=x^2\cos^2(3x)$$.​

$$\displaystyle f(x)=\frac{x^2}{2}+\sum_{n=0}^\infty \frac{(-1)^n6^{2n}}{2\cdot(2n)!} x^{2n+2}$$

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2. Express $$\displaystyle \int_0^{2/3} \! x^3\sin\left(x^2\right) \, dx$$ as an infinite series.​

$$\displaystyle \int_0^{2/3} \! x^3\sin\left(x^2\right) \, dx=\sum_{n=0}^\infty \frac{(-1)^n}{(4n+6)(2n+1)!} \left(\frac{2}{3}\right)^{4n+6}$$
Video Errata: The upper limit of integration is $$2/3$$, but the presenter writes $$3/2$$ throughout the problem. The final answer is correct if you replace the $$3/2$$ with $$2/3$$.

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3. Find the Taylor series of $$\displaystyle f(x)=\frac{3}{(4x-5)^6}$$ centered at $$x=2$$.

$$\displaystyle f(x)=\sum_{n=0}^\infty \frac{(-1)^n\cdot(n+1)(n+2)(n+3)(n+4)(n+5)\cdot4^n}{120\cdot3^{n+5}}(x-2)^n$$

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4. Find the Taylor series of $$\displaystyle f(x)=\left(x^2-x\right)e^x$$ centered at $$x=1$$.

$$\displaystyle f(x)=\sum_{n=0}^\infty \frac{n^2e}{n!}(x-1)^n$$

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5. Find the $$4^{\textrm{th}}$$ degree Taylor polynomial, $$T_4(x)$$, for $$\displaystyle f(x)=e^{-3x}-2-3x$$ centered at $$x=4$$.

\begin{align}\displaystyle T_4(x)=&\left(e^{-12}-14\right)-\left(3e^{-12}+3\right)(x-4)+\frac{9}{2}e^{-12}(x-4)^2\\&-\frac{9}{2}e^{-12}(x-4)^3+\frac{27}{8}e^{-12}(x-4)^4\end{align}

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