Taylor Series Quiz

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) $1 + \frac{1}{2}(x-1) + \frac{3}{8}(x-1)^2 + \frac{5}{16}(x - 1)^3$

Find the Maclaurin series for $f(x) = \pi x^2 - e^2 x + e^\pi$.

(a) $2\pi x - e^2$

(b) $\pi x^2 - e^2 x + e^\pi$

(c) $2\pi x - 2ex$

(d) $2\pi x - 2ex + \pi e^{\pi - 1}$

Find the Maclaurin series for $\ln{\sqrt{4 - x^2}}$.

(a) $\sum_{n=1}^\infty (-1)^n4^nx^{2n}$

(b) $\frac{\ln{4}}{2} - \frac{1}{2}\sum_{n=1}^\infty \dfrac{x^{2n}}{n 4^n}$

(c) $-\frac{1}{2}\sum_{n=1}^\infty \dfrac{x^{2n}}{n 4^n}$

(d) $1 - \frac{1}{2}\sum_{n=1}^\infty \dfrac{x^{2n}}{n 4^n}$

Find the sum $\sum_{n=2}^\infty \dfrac{(-2)^n}{n!}$.

(a) $e^{-2}$

(b) $e^{-2} - 1$

(c) $e^{-2} - 3$

(d) $e^{-2} + 1$

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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