Chapter 3: Applications of Differentiation
Section 3.3: Taylor Polynomials
Example 3.3.2
With f and P3 as in Example 3.3.1, obtain R3x and use it to estimate the largest difference between f and P3 on −1,1. Hence, obtain the maximum value of R4 = f4c4!x4, which in turn requires finding the maximal value of f4c for −1≤c≤1.
Find the actual value of the largest difference between f and P3 on −1,1.
Solution
The maximum of f−P3 is bounded by the maximum of R4 , which, on −1,1 is ⅇ4!≐0.1133.
The actual maximum of f−P3 occurs at one of ±1, as per Figure 3.3.1.
Write P3x=… Context Panel: Assign Function
P3x=1+x+12x2+16x3→assign as functionP3
Write the absolute value of the difference at x=−1 Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻5
f−1−P3−1 = ⅇ−1−13→at 5 digits0.03455
Write the absolute value of the difference at x=1 Context Panel: Evaluate and Display Inline
f1−P31 = ⅇ−83→at 5 digits0.0516
The largest difference between f and P3 on −1,1 is approximately 0.0516, and this largest difference occurs at x=1. The estimate of this largest difference using R3 is 0.1133. In general, the largest difference between f and one of its Taylor polynomials occurs at the endpoint of an interval, and the actual difference is smaller than the estimated difference using Rn.
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