Question

Fun calculus Q

Answer 1

http://www.math.ntu.edu.tw/~cheng/teaching/IG_chapter12.pdf#page=64

Answer 2

I'm only in 8th grade! :(

Answer 3

Don't know anything about calculus, sorry mate....

Answer 4

Sorry! Didn't know you didn't want me to comment!

Answer 5

Okay, so we have the taylor series at \(0\).

Answer 6

Ya, so a Maclaurin
http://mathworld.wolfram.com/MaclaurinSeries.html

Answer 7

@maddiemsp
Click the delete button on your post if you want to delete it.

Answer 8

@satellite73

Answer 9

\[
f(x) = \sum_{n=0}^{\infty}f^{(n)}(0)\frac{x^n}{n!}
\]

Answer 10

Ok thanks...@confluxepic !

Answer 11

Also known as the Maclaurin series.

Answer 12

We have \(f^{n}(0) = 10^{-n}\)

Answer 13

\[
f(x) = \sum_{n=0}^{\infty}f^{(n)}(0)\frac{x^n}{n!} = \sum_{n=0}^{\infty}\frac{1}{10^n}\frac{x^n}{n!} = \sum_{n=0}^{\infty}\frac{(x/10)^n}{n!}
\]

Answer 14

Let \(y=x/10\): \[
\sum_{n=0}^{\infty}\frac{(x/10)^n}{n!} =\sum_{n=0}^{\infty}\frac{y^n}{n!} = e^y=e^{x/10}
\]

Answer 15

Sure enough: \[
f(0) = e^{0/10}=e^0=1
\]And \[
f'(0) = \frac{1}{10}e^{x/10}\Bigg|^{x=0} = \frac 1{10} = 0.1
\]