Evaluate the function $f ( x ) = \cos x$ by a Taylor polynomial of degree 4 centered at $a = 0$, and $x = \frac { \pi } { 4 }$.
Select the correct answer.
A) $\quad 0.7074$
B) $4.2074$
C) $3.2074$
D) $2.2074$
E) $1.2074$

Question

Quizplus · Accepted Answer

The Taylor polynomial of degree 4 centered at $a=0$ is given by

$$T_4(x)= \cos a + \sin a (x-a) -\frac{\cos a}{2!}(x-a)^2-\frac{\sin a}{3!}(x-a)^3+\frac{\cos a}{4!}(x-a)^4$$

Substituting $a=0$ and $x=\frac{\pi}{4}$, we get

$$T_4\left(\frac{\pi}{4}\right)= \cos 0 + \sin 0 \left(\frac{\pi}{4}-0\right) -\frac{\cos 0}{2!}\left(\frac{\pi}{4}-0\right)^2-\frac{\sin 0}{3!}\left(\frac{\pi}{4}-0\right)^3+\frac{\cos 0}{4!}\left(\frac{\pi}{4}-0\right)^4$$

Simplifying, we get

$$T_4\left(\frac{\pi}{4}\right)=1-\frac{\pi^2}{32}+\frac{\pi^4}{4! \cdot 2^4}= 0.7074$$

Therefore, the best choice is (A) with a value of 0.7074.

Evaluate the Function f(x)=cos⁡xf ( x ) = \cos xf(x)=cosx By a Taylor Polynomial of Degree 4 Centered At

Evaluate the Function $f ( x ) = \cos x$ By a Taylor Polynomial of Degree 4 Centered At