Let f(x) = $e ^ { x / 2 }$ . Determine the second Taylor polynomial of f(x) at x = 2.
A) 1 + $\frac { 1 } { 2 }$ (x - 2) + $\frac { 1 } { 8 }$ $( x - 2 ) ^ { 2 }$
B) 1 - $\frac { 1 } { 2 }$ (x - 2) + $\frac { 1 } { 8 }$ $( x - 2 ) ^ { 2 }$
C) 1 + $\frac { 1 } { 2 }$ (x - 2) + $\frac { 1 } { 6 }$ $( x - 2 ) ^ { 2 }$
D) e + $\frac { \mathrm { e } } { 2 }$ (x - 2) + $\frac { \mathrm { e } } { 8 }$ $( x - 2 ) ^ { 2 }$

Question

Quizplus · Accepted Answer

The second Taylor polynomial of $f(x) = e^{x/2}$ at $x = 2$ is found by evaluating the function and its derivatives at $x = 2$, then using the formula for the Taylor series expansion. At $x = 2$, $f(2) = e^{2/2} = e$, $f'(x) = \frac{1}{2}e^{x/2}$ so $f'(2) = \frac{1}{2}e$, and $f''(x) = \frac{1}{4}e^{x/2}$ so $f''(2) = \frac{1}{4}e$. The second Taylor polynomial is $f(2) + f'(2)(x - 2) + \frac{f''(2)}{2!}(x - 2)^2 = e + \frac{e}{2}(x - 2) + \frac{e}{8}(x - 2)^2$.

Let F(x) = ex/2e ^ { x / 2 }ex/2 Determine the Second Taylor Polynomial of F(x) at X

Let F(x) = $e ^ { x / 2 }$ Determine the Second Taylor Polynomial of F(x) at X