Question

Show that function F(x) = x sin x is an antiderivative of f(x) = sin x +x cos x. Determine the antiderivative of f(x) which passes point (0, 3).

Answer 1

show? as in derive by the rules or by first principles?

Answer 2

Also, how do you correctly say "pass a point" mathematically in english?

Answer 3

By first principles, please.

Answer 4

passes thru the point im assuming is what it means

Answer 5

Right, thanks for that!

Answer 6

first principles are a pain with trig becasue there is alot of hand waiving involved; like a magician or a hypnotist .... its true because i say so type thing

Answer 7

there is no good way to prove sin(x)/x goes to 1 as x goes to 0 :)

Answer 8

Right, well then I can just derive F(x) and prove it that way.

Answer 9

\[\lim_{h ->0}\frac{sin(x+h)-sin(x)}{h}\]
to derive by first principles is still deriving :) the "rules" provide established shortcuts

Answer 10

Heh, I understand. :)

Answer 11

Would it be ok to integrate f(x) instead?

Answer 12

This is actually really easy, thanks for the help though. :)

Answer 13

but for practice:

\[\frac{sin(x)cos(h)+sin(h)cos(x)-sin(x)}{h}\]
\[\frac{sin(x)(cos(h)-1)+sin(h)cos(x)}{h}\]
\[\frac{sin(x)(cos(h)-1)}{h}+\frac{sin(h)cos(x)}{h}\]
\[\lim_{h->0}\left( sin(x)\frac{(cos(h)-1)}{h}+cos(x)\frac{sin(h)}{h}\right)\]
and for the hand waiving part:

[cos(h)-1]/h -> 0 ; and sin(h)/h -> 1 leaveing us with ...

[sin(x)]' = cos(x) ... tada lol

Answer 14

haha, thanks! how do you break the LaTeX lines like you did there?

Answer 15

integrate [sin(x)+xcos(x)] and see if it produces [x sin(x)] should be prove enough

Answer 16

using the delimiters [ ] make seperate lines.

Answer 17

\[stuff in here\*]
^ remove * to format

Answer 18

\[\pi = 3\]
\[0 = 0\]

Answer 19

can u integrate x*cosx

Answer 20

for inline; change the [ ] to ( )

Answer 21

spose im typing along and I want it in the sentence: \(\pi \ r^2\). then I use the ( ) delimiters

Answer 22

Like this \(\pi = 3\)? Awesome!

Answer 23

lol; and if you want to get real fancy: \(\color{purple}{\text{just add some color}}\) :)

Answer 24

Thanks so much! I'm very new to LaTeX. :)

Answer 25

Sriram, you can give me an example of how, if you want a medal. :)

Answer 26

\begin{array}l\color{red}{\normalsize\text{T}}\color{orange}{\normalsize\text{a}}\color{#9c9a2e}{\normalsize\text{s}}\color{green}{\normalsize\text{t}}\color{blue}{\normalsize\text{e}}\color{purple}{\normalsize\text{ }}\color{purple}{\normalsize\text{t}}\color{red}{\normalsize\text{h}}\color{orange}{\normalsize\text{e}}\color{#9c9a2e}{\normalsize\text{ }}\color{green}{\normalsize\text{r}}\color{blue}{\normalsize\text{a}}\color{purple}{\normalsize\text{i}}\color{purple}{\normalsize\text{n}}\color{red}{\normalsize\text{b}}\color{orange}{\normalsize\text{o}}\color{#9c9a2e}{\normalsize\text{w}}\color{green}{\normalsize\text{!}}\color{blue}{\normalsize\text{!}}\color{purple}{\normalsize\text{ }}\color{purple}{\normalsize\text{l}}\color{red}{\normalsize\text{o}}\color{orange}{\normalsize\text{l}}\color{#9c9a2e}{\normalsize\text{}}\end{array}

Answer 27

\[\color{red}{test}\]

Answer 28

\color{yourcolor}{yourtext}
So for instance: \color{red}{yourtext}

Answer 29

http://assets.openstudy.com/updates/attachments/4e0771810b8bd74af49beb9d-amistre64-1309110788233-fader.html

Answer 30

Actually you need to write the text outside of equation mode
\color{yourcolor}{\text{yourtext}}
\[\color{blue}{\text{This is is the result}}\]

Answer 31

Yea, I noticed I can study the LaTeX source by right clicking on the text and choosing 'show source'. :)