Find the fourth Maclaurin polynomial $P _ { 4 } ( x )$ for the function $f ( x ) = x \cos x$
A) $1 - x - \frac { 1 } { 3 } x ^ { 3 }$
B) $x - \frac { 1 } { 3 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 4 }$
C) $x - \frac { 1 } { 2 } x ^ { 3 }$
D) $1 + x + \frac { 1 } { 3 } x ^ { 3 }$

Question

Quizplus · Accepted Answer

The Maclaurin series for $f(x) = x\cos(x)$ is found by evaluating the derivatives of $f(x)$ at $x = 0$ and applying the formula for the Maclaurin series. The derivatives are: $f(x) = x\cos(x)$, $f'(x) = \cos(x) - x\sin(x)$, $f''(x) = -2\sin(x) - x\cos(x)$, $f'''(x) = -3\cos(x) + x\sin(x)$, and so on. Evaluating these at $x = 0$ gives $f(0) = 0$, $f'(0) = 1$, $f''(0) = 0$, $f'''(0) = -3$, and $f''''(0) = 0$. Thus, the fourth Maclaurin polynomial is $P_4(x) = 0 + 1\cdot x + 0\cdot \frac{x^2}{2!} -3\cdot \frac{x^3}{3!} + 0\cdot \frac{x^4}{4!} = x - \frac{1}{2}x^3$.

Find the Fourth Maclaurin Polynomial P4(x)P _ { 4 } ( x )P4​(x)

Find the Fourth Maclaurin Polynomial $P _ { 4 } ( x )$