Use a Taylor polynomial for of degree 3 for the function to approximate about c = 1.

Question

Use a Taylor polynomial for of degree 3 for the function  to approximate  about c = 1.

Quizplus · Accepted Answer

Using the Taylor polynomial of degree 3 centered at c = 1, we have:
$P(x) = f(1) + f'(1)(x-1) + \frac{f''(1)}{2!}(x-1)^2 + \frac{f'''(1)}{3!}(x-1)^3$
where $f(x) = e^x$, $f'(x) = e^x$, $f''(x) = e^x$, and $f'''(x) = e^x$. 
Therefore, plugging in these values and simplifying, we get:
$P(x) = e + e(x-1) + \frac{e}{2}(x-1)^2 + \frac{e}{6}(x-1)^3$
Evaluating $P(1.05)$, we get:
$P(1.05) = e + e(0.05) + \frac{e}{2}(0.05)^2 + \frac{e}{6}(0.05)^3 \approx 1.0440$
Therefore, the best choice is B.

Use a Taylor Polynomial for of Degree 3 for the Function