The calculated value of $e^{2x}$ using Taylor's series with the first four terms for $x = 1/2$ is ______
Official Answer Key:
2.665 to 2.668
Step-by-Step Solution & Explanation
The Maclaurin expansion of the exponential function is:
e^u = 1 + u + u²/2! + u³/3! + u⁴/4! + …
Here u = 2x, and at x = 1/2 we have u = 1.
First four terms:
1 + 1 + 1²/2! + 1³/3!
= 1 + 1 + 0.5 + 0.16667
= 2.667
(The exact value e¹ = 2.71828; truncation after four terms leaves an error of about 0.051.)
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