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#6 FRQ AP CALC BC 2022

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#6 FRQ AP CALC BC 2022
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  • Taylor Swift The Teacher
  • The function f is defined by the power series f(x) = x - x^3/3 + x^5/5 - x^7/7 + ... + (-1)^n*x^(2n+1)/(2n+1) + ... for all real numbers x for which the series converges.
  • Today we will be learning about power series!
  • 1
  • 2
  • Using the ratio test, find the interval of convergence of the power series for f. Justify your answer.
  • This is part A of the free response question.
  • How do we use the ratio test???
  • 3
  • lim | (-1)^(n+1)*x^(2n+3) * 2n+1 |
  • n-inf.
  • This is how we use the ratio test. Next we need to find the IOC using | x^2 | 1.
  • 2n+3
  • How do we find the interval of convergence???
  • (-1)^n*x^(2n+1)
  • = | x^2 | 1
  • 4
  • @ x = -1@ x = 1
  • Since | x^2 | 1 is -1 x 1, we have to test if f converges at x = -1 and x =1.
  • both converge by the alternating series test
  • 5
  • Show that | f(1/2) - 1/2 | 1/10. Justify your answer.
  • This is part B of the free response question.
  • How do we show that???
  • 6
  • Approximating f(1/2) with 1/2 uses the first term of the series, therefore error is... | f(1/2) - 1/2 | (1/2)^3 / 3 = 1/24 1/10using the magnitude of the fist term left off as error bound.
  • At x = 1/2, the series for f(x) is an alternating series with terms that decrease in magnitude to 0, therefore a converging alternate series.
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