A function f(x) can be approximated using a polynomial Pn ...
BC: Q402.CH9B – Taylor Series: Error Analysis (LESSON 3)
THM E1: If a series for [pic]is strictly alternating and decreasing in absolute value to zero, the error in using [pic]to approximate [pic]is less than or equal to the first omitted non-zero term:
[pic]with [pic]
THM E2: Taylor’s Formula with Remainder
The error in using [pic]to approximate [pic]is equal to [pic] for some value c for [pic].
[pic] with [pic] for [pic]
Random Theorem: If x is any real number, then [pic]
1. Use the first two nonzero terms of the Maclaurin series to approximate sin(0.1). Estimate the error.
2. Approximate [pic]to four decimal places. Estimate the error.
3. Let f be the function defined by [pic].
a) Write the first four terms and the general term of the Taylor series expansion of f(x) about [pic].
b) Use the result from part (a) to find the first four terms and the general term of the series expansion about [pic]for [pic].
c) Use the series in part (b) to compute a number that differs from [pic]by less than 0.05. Justify your answer.
4. The Taylor series about [pic] for a certain function f converges to f(x) for all x in the interval of convergence. The nth derivative of f at x = 5 is given by [pic], and [pic].
a) Write the third-degree Taylor polynomial for f about x = 5.
b) Find the radius of convergence of the Taylor series for f about x = 5.
c) Show that the sixth-degree Taylor polynomial for f about x = 5 approximates f(6) with error less than [pic]
5. The function f is defined by the power series [pic] for all real numbers.
(a). Find [pic] and [pic]. Determine whether f has a local maximum, a local minimum, or neither at [pic]. Give a reason for your answer.
(b). Show that [pic] approximates [pic]with error less than [pic].
(c) Show that [pic]is a solution to the differential equation [pic].
CALCULATOR ACTIVE
6. Use the first two nonzero terms of the Maclaurin series to approximate sin(x). Estimate the maximum error if |x| ................
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