AP Calculus BC 10.14 Taylor Series Forms Overview
Review Taylor-series forms by recognising standard expansions and transforming them with substitution, derivatives or integrals.
- Syllabus
- Effective Fall 2025
- Course
- AP Calculus BC
Covered syllabus points
10.14 Finding Taylor or Maclaurin Series for a Function question 1
[Maximum number: 3]
A function f has derivatives of all orders for -1<x<1. The derivatives of f satisfy the conditions above. The Maclaurin series for f converges to f(x) for |x|<1.
Show that the first four nonzero terms of the Maclaurin series for f are x−2x2+3x3−4x4, and write the general term of the Maclaurin series for f.
f(0)=0
f′(0)=1f′′(0)=−1(1)=−1f′′′(0)=−2(−1)=2f(4)(0)=−3(2)=−6
The first four nonzero terms are
0+1x+2!−1x2+3!2x3+4!−6x4=x−2x2+3x3−4x4.
The general term is n(−1)n+1xn.
10.14 Finding Taylor or Maclaurin Series for a Function question 2
[Maximum number: 1]
The Taylor series for a function f about x=4 is given by
∑n=1∞(n+1)3n(x−4)n+1=2⋅3(x−4)2+3⋅32(x−4)3+4⋅33(x−4)4+⋯+(n+1)3n(x−4)n+1+⋯ and converges to f(x) on
its interval of convergence.
The Taylor series for f′ described in part B is a geometric series. For all x in the interval of
convergence of the Taylor series for f′, show that f′(x)=7−xx−4.
C The Taylor series for f′ described in part B is a geometric series. For all x in the interval of convergence
of the Taylor series for f′, show that f′(x)=7−xx−4.
| The Taylor series for f′ is a geometric series with first term 3x−4 and common ratio 3x−4. f′(x)=1−3x−43x−4=3−(x−4)x−4=7−xx−4 | Verification | Point 8 (P8) |
|---|---|---|
| Scoring Notes for Part C | ||
- A response of f′(x)=1−3x−43x−4 is sufficient to earn P8.
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