Question

webwork 3 - topics 6 - 7: problem 4 (1 point)
find the maclaurin polynomial of degree 3 for the function $f(x)=e^{4x}$.
$t_3(x)=square+square x+square x^{2}+square x^{3}$
now use this polynomial to approximate $e^{8}$
$e^{8}approxsquare$

Explanation:

Step1: Recall Mac - Laurin formula

The Mac - Laurin series for a function $f(x)$ is given by $T_n(x)=\sum_{k = 0}^{n}\frac{f^{(k)}(0)}{k!}x^{k}$, where $f^{(k)}(x)$ is the $k$-th derivative of $f(x)$ and $f^{(0)}(x)=f(x)$. For $n = 3$, $T_3(x)=f(0)+f^{\prime}(0)x+\frac{f^{\prime\prime}(0)}{2!}x^{2}+\frac{f^{(3)}(0)}{3!}x^{3}$.

Step2: Find the derivatives of $f(x)=e^{4x}$

First derivative: $f^{\prime}(x)=\frac{d}{dx}(e^{4x}) = 4e^{4x}$. Second derivative: $f^{\prime\prime}(x)=\frac{d}{dx}(4e^{4x})=16e^{4x}$. Third derivative: $f^{(3)}(x)=\frac{d}{dx}(16e^{4x}) = 64e^{4x}$.

Step3: Evaluate the function and its derivatives at $x = 0$

$f(0)=e^{4\times0}=1$, $f^{\prime}(0)=4e^{4\times0}=4$, $f^{\prime\prime}(0)=16e^{4\times0}=16$, $f^{(3)}(0)=64e^{4\times0}=64$.

Step4: Construct the Mac - Laurin polynomial

$T_3(x)=1 + 4x+\frac{16}{2}x^{2}+\frac{64}{6}x^{3}=1 + 4x+8x^{2}+\frac{32}{3}x^{3}$.

Step5: Approximate $e^{8}$

We want to approximate $e^{8}$, and since $f(x)=e^{4x}$, when $e^{8}=e^{4\times2}$, we substitute $x = 2$ into $T_3(x)$.
$T_3(2)=1+4\times2 + 8\times2^{2}+\frac{32}{3}\times2^{3}=1 + 8+32+\frac{256}{3}=\frac{3 + 24+96 + 256}{3}=\frac{379}{3}\approx126.33$.

Answer:

$T_3(x)=1+4x + 8x^{2}+\frac{32}{3}x^{3}$; $e^{8}\approx126.33$