Taylor Series Quiz

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1. Find the 3rd-order Taylor polynomial $p_3(x)$ for $f(x) = \sqrt{x}$ centered at $a = 1$.

   (a) $1 + \frac{1}{2}(x-1) + \frac{1}{8}(x-1)^2 + \frac{1}{16}(x - 1)^3$
   (b) $1 + \frac{1}{2}(x-1) + \frac{1}{8}(x-1)^2 + \frac{3}{16}(x - 1)^3$
   (c) $\frac{1}{2\sqrt{x}} + \frac{1}{2}$
   (d) $1 + \frac{1}{2}(x-1) + \frac{3}{8}(x-1)^2 + \frac{5}{16}(x - 1)^3$

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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. Estimate $\cos{(0.1)}$ using the 4th-degree Taylor polynomial $p_4(x)$ for $\cos{x}$.

   (a) $0.9$
   (b) $0.099$
   (c) $0.995$
   (d) $1.005$

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

   (a) $(x-1) - \frac{1}{2}(x-1)^2$
   (b) $1 + 2(x-1) - \frac{3}{2}(x-1)^2$
   (c) $1 + (x-1) - \frac{3}{2}(x-1)^2$
   (d) $(x-1) - \frac{3}{2}(x-1)^2$

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5. Find the Maclaurin series for $f(x) = \frac{3}{3+x}$.

   (a) $\sum_{n=0}^\infty 3(3+x)^n$
   (b) $\sum_{n=0}^\infty (-1)^n3^nx^n$
   (c) $\sum_{n=0}^\infty \dfrac{x^n}{3^n}$
   (d) $\sum_{n=0}^\infty \dfrac{(-1)^nx^n}{3^n}$

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