Taylor Series Quiz

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1. Find the 2nd-order Taylor polynomial $p_2(x)$ for $f(x) = \tan{x}$ centered at $a = \frac{\pi}{4}$.

   (a) $2 + (x - \pi/4) + (x - \pi/4)^2$
   (b) $1 + 2(x - \pi/4) + 2(x - \pi/4)^2$
   (c) $1 + (x - \pi/4) + 2(x - \pi/4)^2$
   (d) $(x - \pi/4) + 2(x - \pi/4)^2$

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2. Find the Taylor series for $f(x) = x^2 - 2x$ centered at $a = 1$.

   (a) $(x-1)^2 - 2(x-1) + 1$
   (b) $(x-1)^2$
   (c) $(x-1)^2 - 1$
   (d) $(x-1)^2 - 2(x-1)$

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3. Suppose $f(0) = 2, f'(0) = 6, f''(0) = -8$. What is the 2nd degree Taylor polynomial for $f(x)$ centered at $0$? (Assume that $f$ is infinitely differentiable).

   (a) $p_2(x) = 1 + x - \frac{1}{3}x^2$
   (b) $p_2(x) = 2 + 3x - \frac{4}{3}x^2$
   (c) $p_2(x) = 2 + 6x - 4x^2$
   (d) $p_2(x) = 2 + 6x - 8x^2$

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4. Find the 2nd-order Taylor polynomial $p_2(x)$ for $f(x) = x\sin{x}$ centered at $x = \pi$.

   (a) $x^2 - \frac{1}{6}x^4$
   (b) $x^2$
   (c) $-\pi(x-\pi) - (x-\pi)^2$
   (d) $\pi(x-\pi) - 2(x-\pi)^2$

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5. Find a Maclaurin series for $e^{2x} - x^2$.

   (a) $\sum_{n=1}^\infty \dfrac{x^{2n}}{n!}$
   (b) $1 + \sum_{n=2}^\infty \dfrac{x^{2n}}{n!}$
   (c) $\sum_{n=0}^\infty \dfrac{x^{2n}}{n!}$
   (d) $1 + 2\sum_{n=1}^\infty \dfrac{x^{2n}}{n!}$

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