1. ## MacLaurin polynomial

Let .
Find the MacLaurin polynomial of degree 5 for .

Use this polynomial to estimate the value of .

I don't understand this. For the MacLaurin polynomial do I take derivative until the 5th derivative?

2. Originally Posted by superman69
Let .
Find the MacLaurin polynomial of degree 5 for .

Use this polynomial to estimate the value of .

I don't understand this. For the MacLaurin polynomial do I take derivative until the 5th derivative?
$\displaystyle F'(x) = e^{-4x^4} \approx 1 - 4x^4$

$\displaystyle F(x) \approx x - \frac{4x^5}{5}$

approximate $\displaystyle F(0.16)$

3. Originally Posted by superman69
Let .
Find the MacLaurin polynomial of degree 5 for .

Use this polynomial to estimate the value of .

I don't understand this. For the MacLaurin polynomial do I take derivative until the 5th derivative?
Yup.

$\displaystyle F(0)=0$

$\displaystyle F'(x)=e^{-4x^4}, F'(0)=1$

$\displaystyle F''(x)=-16x^3e^{-4x^4}, F''(0)=0$

So the polynomial of degree 2 is $\displaystyle f_2(x)=x$.

Just keep going til you hit 5; then plug in 0.16.

4. yeah okay i got that part how do i do the first part Find the MacLaurin polynomial of degree 5 for F(x) I spended about 45 mins on that and still couldn't find the correct answer

5. But this is what I got so far.

6. Originally Posted by superman69

But this is what I got so far.
Should be $\displaystyle 1+(-4x^4)+\frac{(-4x^4)^2}{2!}+\frac{(-4x^4)^3}{3!}...$

Then integrate to find the series for $\displaystyle F(x)$, just like Skeeter said. Note that you only need the first two terms of this sequence, since all the rest have $\displaystyle \text{deg}\geq5$.

$\displaystyle \int 1-4x^4\,dx=x-\frac{4x^5}{4}$