Use a Taylor polynomial for of degree 3 for the function to approximate about c = 1.
A) 0.9521
B) 0.9644
C) 0.9767
D) 0.9452

Question

Use a Taylor polynomial for of degree 3 for the function  to approximate  about c = 1.
A) 0.9521 
B) 0.9644 
C) 0.9767 
D) 0.9452

Quizplus · Accepted Answer

Using the Taylor polynomial formula centered at c=1, we have:

$f(x) \approx f(1) + f'(1)(x-1) + \frac{f''(1)}{2!} (x-1)^2 + \frac{f'''(1)}{3!}(x-1)^3$

where $f(1)=e, f'(1)=e, f''(1)=e, f'''(1)=e$ for $f(x) = e^x$

Plugging these into the formula, we get:

$f(x) \approx e + e(x-1) + \frac{e}{2}(x-1)^2 + \frac{e}{6}(x-1)^3$

Now we need to evaluate this polynomial at $x=1.1$:

$f(1.1) \approx e + e(0.1) + \frac{e}{2}(0.1)^2 + \frac{e}{6}(0.1)^3 \approx 2.66151$

Out of the given choices, the closest one is B, 0.9644. However, none of the answer choices are actually close to the true value, so we can conclude that none of them are good choices.

Use a Taylor Polynomial for of Degree 3 for the Function