Question

Evaluate the limit as h goes toward zero for the following equation:

Answer 1

\[\frac{ 7\cos(\frac{ \pi }{ 6 }+h) - 7\cos(\frac{ \pi }{ 6 }) }{ h }\]

Answer 2

I tried solving it out and got 0, but this isn't correct.

Answer 3

So this is our limit definition for the derivative, ya? :)\[\large\rm \frac{d}{dx}7\cos(x)|_{x=\pi/6}\quad=\lim_{h\to0} \frac{ 7\cos(\frac{ \pi }{ 6 }+h) - 7\cos(\frac{ \pi }{ 6 }) }{ h }\]So if you know your derivative shortcut rules,
you can avoid all the mess.

Do you need to solve this through limit though?
I'm trying to remember the trick for this one :)

Answer 4

Yep! I need to solve it out to see if it equals one of these answers:
a) -7
b) -7/2
c) 7sqrt(3)/2
d) doesn't exist

Answer 5

Oh oh now I remember :)
We need to apply our Angle Sum Formula:\[\large\rm \cos\left(a+b\right)=\cos(a)\cos(b)-\sin(a)\sin(b)\]Yah, that'll get us on the right track.

Answer 6

@zepdrix Sorry I ditched you in the middle of the question zep. I got unfairly banned, no response so far though.

Answer 7

@zepdrix Can you quickly help before this account gets banned too?

Answer 8

http://openstudy.com/study#/updates/567ef3dbe4b032ed60ddcf6a

It might be a bit slow though

Answer 9

I can't take the derivative of cosine to get 7sin(x) and then solve from there?

Answer 10

\[\large\rm \lim_{h\to0}\frac{7\color{orangered}{\cos\left(\frac{\pi}{6}+h\right)}-\cos\left(\frac{\pi}{6}\right)}{h}\]Becomes,\[\large\rm \lim_{h\to0}\frac{7\left[\color{orangered}{\cos\left(\frac{\pi}{6}\right)\cos\left(h\right)-\sin\left(\frac{\pi}{6}\right)\sin\left(h\right)}\right]-7\cos\left(\frac{\pi}{6}\right)}{h}\]Sorry it took me a sec to write all that out XD
That step make sense hopefully?

Answer 11

Yes, that is your other, more straight forward option :)

Answer 12

Derivative of cos(x) is -sin(x) though,
don't forget the minus :)

Answer 13

Why cos(pi/6)cos(h)-sin(pi/6)sin(h)?

Answer 14

If you're allowed to use your derivative shortcuts, then we don't need to worry about it.

But that would be an important step in solving it through the limit.

Answer 15

After that we would group things up in a clever way, and apply some limit identities.

Answer 16

I applied the Cosine Angle-Sum Identity,
too confusing? :)

Answer 17

Or just confusing `why` we would need that?