Which of the following is the degree 3 Taylor polynomial centered at $a = 0$ for $f ( x ) = \cos ( 2 x )$ ?
1) $1 - ( 2 x ) ^ { 2 } + 16 x ^ { 4 }$ 2) $1 - 2 x ^ { 2 } + \frac { 8 } { 3 } x ^ { 4 }$ 3) $1 - \frac { ( 2 x ) ^ { 2 } } { 2 ! }$
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Question

Which of the following is the degree 3 Taylor polynomial centered at $a = 0$ for $f ( x ) = \cos ( 2 x )$ ?
1) $1 - ( 2 x ) ^ { 2 } + 16 x ^ { 4 }$ 2) $1 - 2 x ^ { 2 } + \frac { 8 } { 3 } x ^ { 4 }$ 3) $1 - \frac { ( 2 x ) ^ { 2 } } { 2 ! }$ 
A)None
B)1
C)2
D)3
E)1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Quizplus · Accepted Answer

The degree 3 Taylor polynomial for $f(x) = \cos(2x)$ centered at $a = 0$ is $1 - \frac{(2x)^2}{2!}$ because the Taylor series for $\cos(x)$ is $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$, and substituting $2x$ for $x$ gives $1 - \frac{(2x)^2}{2!} + \frac{(2x)^4}{4!} - \cdots$. Since we are looking for the degree 3 polynomial, we only go up to the $x^2$ term, making option 3 correct.

Which of the Following Is the Degree 3 Taylor Polynomial a=0a = 0a=0

Which of the Following Is the Degree 3 Taylor Polynomial $a = 0$