Mathematics

Problem

Define

F(x)=\int_{0}^{x} e^{t^{2}}\,dt.

The Maclaurin series for

e^{u}

\sum_{n=0}^{\infty}\dfrac{u^{n}}{n!}

for all

u

. (a) Write the first four non-zero terms of the series for

e^{t^{2}}

. (b) Using the series from part (a), obtain the first four non-zero terms of the Maclaurin series for

F(x)

. (c) Use the series from part (b) to approximate

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^{-6}

of the exact value.

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