Find the Taylor polynomial of order 3 generated by f at a.
-$$f ( x ) = x ^ { 3 } , a = 10$$
A) $P _ { 3 } ( x ) = 6 + 3 ( x - 10 ) + ( x - 10 ) ^ { 2 } + ( x - 10 ) ^ { 3 }$
B) $P _ { 3 } ( x ) = 1000 + 100 ( x - 10 ) + 100 ( x - 10 ) ^ { 2 } + ( x - 10 ) ^ { 3 }$
C) $P _ { 3 } ( x ) = 1000 + 300 ( x - 10 ) + 30 ( x - 10 ) ^ { 2 } + ( x - 10 ) ^ { 3 }$
D) $\mathrm { P } _ { 3 } ( \mathrm { x } ) = 4000 + 300 ( \mathrm { x } - 10 ) + 20 ( \mathrm { x } - 10 ) ^ { 2 } + ( \mathrm { x } - 10 ) ^ { 3 }$

Question

Quizplus · Accepted Answer

The Taylor series of order 3 generated by f at a=10 is given by:
$$P_3(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3$$
$$=10^3+3(10)^2(x-10)+6(10)(x-10)^2+(x-10)^3$$
$$=1000+300(x-10)+30(x-10)^2+(x-10)^3$$
Option C matches the above equation, hence it is the correct choice.

Find the Taylor Polynomial of Order 3 Generated by F f(x)=x3,a=10f ( x ) = x ^ { 3 } , a = 10f(x)=x3,a=10

Find the Taylor Polynomial of Order 3 Generated by F $f ( x ) = x ^ { 3 } , a = 10$