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Really need help understanding this limit/derivative question...
The limit
Lim x approachs 5pi CosX +1 / x5pi
represents the derivative of some function f(x) at some number a. Find f and a .
I don't even understand the wording of this question...
 one year ago
 one year ago
Really need help understanding this limit/derivative question... The limit Lim x approachs 5pi CosX +1 / x5pi represents the derivative of some function f(x) at some number a. Find f and a . I don't even understand the wording of this question...
 one year ago
 one year ago

This Question is Closed

zepdrixBest ResponseYou've already chosen the best response.1
Remember the limit definition of a derivative?\[\large f'(x)=\lim_{h \rightarrow 0}\frac{f(xh)f(x)}{h}\]Well there is also another definition that we see less often, of this form,\[\large f'(x)=\lim_{x \rightarrow a}\frac{f(x)f(a)}{xa}\]This is the one that we want to analyze.
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
If we compare this to the limit we were given, we can see that is looks like our A value will be 5pi yes? \[\large \lim_{x \rightarrow 5\pi}\frac{\cos x +1}{x5\pi}\]
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
The top is still a little tricky though, we have to sort it out.
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
wait the second one just looks like the mean value theorem? it oddly looks like f(a)b/ab = f'(c)
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
oh nevermind me theres a limit sorry im a bit tired
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
heh, yah it looks similar :)
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
\[\large \cos(5\pi)=?\]
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
So yeah, but we're looking for the original function is that it or looking for the derivative? and cos5pi would be... .96 ? But I think i did it in degree so it might be wrong.
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
That's one of your special angles that you're going to want to remember. It will produce the same value as Pi. 5pi is Pi with an extra spin around the circle. 1 yes?
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
With 2 extra spins* my bad.
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
I'll ask you one thing before we continue, when taking pi in a derivative, do we ALWAYS use it in radians? I mean sometimes it works when I don't put it in my calculator as a radian
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
If you're dealing with Pi, then yes you need to be in radians :o You could convert to degrees if radians are confusing you though.
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
i mean pi i mean cos and sin... wow I' m sounding stupid
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
5pi is the same as 180 degrees.
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
so in the question am I using L'hospital rule or finding the limit? that's where I'm lost. the question is really confusing . And alright thanks for explaining the pi. :)
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
No L'Hop. We're relating a weird looking limit back to the Limit Definition of a Derivative. We need to match up the pieces so we can see what the original function was. So far we've established that our A value is 5pi. If you're unsure about that, compare the form of our limit with the Definition,\[\large \large f'(x)=\lim_{x \rightarrow a}\frac{f(x)f(a)}{xa}\qquad\qquad \rightarrow \qquad \qquad \lim_{x \rightarrow 5\pi}\frac{\cos x +1}{x5\pi}\]
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
Yeah it's represented by 5pi?
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
so i have to equate both limit? and finding my h?
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
no h, we're using the second definition that i posted, not the one involving h.
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
It's another form of the limit definition. It comes up less often.
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
My bad I understood we had to manipulate it back to the other form
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
Then when its ask to find F, am I suppose to find cosx?
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
sorry, the wording is really throwing me off
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
If we can show that the limit matches the DEFINITION, then we can show what our F is. So we've established that 5pi matches the A we're looking for. We've also shown that cos(5pi)=1. If we can somehow find a 1 in the top of that fraction, we can make it look like the Definition.
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
Oh wow. I understand. This was a very basic question but I have never seen that definition in my entire class. Thanks a lot! :)
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
\[\large \frac{\cos x+1}{x5\pi} \qquad =\qquad \frac{\cos x(1)}{x5\pi} \qquad = \qquad \frac{\cos x(\cos 5\pi)}{x5\pi}\]
 one year ago

zepdrixBest ResponseYou've already chosen the best response.1
Make sense? :) k cool!
 one year ago

MarcLeclairBest ResponseYou've already chosen the best response.0
Thanks a lot you're patient!
 one year ago
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