Define F(x)= _ 0 ^ x e^ t^ 2 ,dt

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Define F(x)=∫0xet2 dt.F(x)=\int_{0}^{x} e^{t^{2}}\,dt. The Maclaurin series for eue^{u} is ∑n=0∞unn!\sum_{n=0}^{\infty}\dfrac{u^{n}}{n!} for all uu. (a) Write the first four non-zero terms of the series for et2e^{t^{2}}. (b) Using the series from part (a), obtain the first four non-zero terms of the Maclaurin series for F(x)F(x). (c) Use the series from part (b) to approximate F(0.3)F(0.3). Give your answer to six decimal places. (d) Explain why the magnitude of the next term in the series provides an upper bound for the error in the approximation found in part (c). Determine whether your approximation is within 10−610^{-6} of the exact value.

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