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AP Calculus BC – Chapter 9 - Infinite Series 9.3: Taylor’s Theorem

AP Calculus BC – Chapter 9 - Infinite Series 9.3: Taylor’s Theorem. Goals : Approximate a function with a Taylor polynomial. Analyze the truncation error of a series using graphical methods or the Remainder Estimation Theorem. Use Euler’s formula to relate the functions sinx, cosx, and e x.

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AP Calculus BC – Chapter 9 - Infinite Series 9.3: Taylor’s Theorem

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  1. AP Calculus BC – Chapter 9 - Infinite Series 9.3: Taylor’s Theorem Goals: Approximate a function with a Taylor polynomial. Analyze the truncation error of a series using graphical methods or the Remainder Estimation Theorem. Use Euler’s formula to relate the functions sinx, cosx, and ex.

  2. Taylor’s Theorem with Remainder: Theorem: If f has derivatives of all orders in an open interval I containing a, then for each positive integer n and for each x in I, where for some c between a and x.

  3. Notes on Taylor’s Theorem: The first equation in Taylor’s Theorem is Taylor’s formula. The function Rn(x) is the remainder of order n or the error term for the approximation of f by Pn(x) over I. It is also called the Lagrange form of the remainder, and bounds on Rn(x) found using this form are Lagrange error bounds.

  4. Convergence: If Rn(x) —> 0 as n —> ∞ for all x in I, we say that the Taylor series generated by f at x = a converges to f on I, and we write

  5. Remainder Estimation Theorem: Remainder Estimation Theorem: If there are positive constants M and r such that |f(n+1)(t)|≤ Mrn+1 for all t between a and x, then the remainder Rn(x) in Taylor’s Theorem satisfies the inequality If these conditions hold for every n and all the other conditions of Taylor’s Theorem are satisfied by f, then the series converges to f(x).

  6. Euler’s Formula: Euler’s Formula: eix = cosx + i sinx. Euler’s Formula: eiπ + 1 = 0.

  7. Assignments: • CW1: Harmonic Series worksheet. • CW2: Explorations 1 & 2. • HW 9.3: #3-24 (every 3rd), 25.

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