AP Calculus BC 10.15 Representing Functions As Power Series Questions

Review power-series operations by deriving expansions through algebra, substitution, differentiation and integration while tracking intervals.

Covered syllabus points

LIM-8.G—Represent a given function as a power series

[Maximum number: 2]

The Taylor series for a function f about x=4 is given by $\sum_{n=1}^{\infty} \frac{(x-4)^{n+1}}{(n+1) 3^{n}}=\frac{(x-4)^{2}}{2 \cdot 3}+\frac{(x-4)^{3}}{3 \cdot 3^{2}}+\frac{(x-4)^{4}}{4 \cdot 3^{3}}+\cdots+\frac{(x-4)^{n+1}}{(n+1) 3^{n}}+\cdots$ and converges to f(x) on its interval of convergence.

Find the first three nonzero terms and the general term of the Taylor series for $f^{\prime}$ , the derivative of f, about x=4.

B Find the first three nonzero terms and the general term of the Taylor series for $f^{\prime}$ , the derivative of f,

about x=4.

$f^{\prime}(x)=\frac{(x-4)}{3}+\frac{(x-4)^{2}}{9}+\frac{(x-4)^{3}}{27}+\cdots+\frac{(x-4)^{n}}{3^{n}}+\cdots$	First three terms	Point 6 (P6)
		General term	Point 7 (P7)

Scoring Notes for Part B

- P6 is earned by presenting the first three nonzero terms in a list or as part of a polynomial or series.

- P7 is earned by identifying the correct general term (either individually or as part of a polynomial or

series).

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Covered syllabus points

Question 1