## anonymous one year ago Critical numbers help? I will reward with a medal!

1. anonymous

$\sin^2x+cosx$ $0<x<2\pi$

2. zepdrix

Get that derivative? :U

3. anonymous

sinx (2cosx -1)

4. zepdrix

Mmm ok looks good! :)

5. zepdrix

$\large\rm \sin x (2\cos x -1)=0$Apply your Zero-Factor Property$\large\rm \sin x=0\qquad\qquad\qquad 2\cos x-1=0$

6. zepdrix

sine is zero at what angles?

7. anonymous

I'm not sure how to get that answer.

8. zepdrix

9. anonymous

I was horrible at memorizing trig stuff. I enjoyed solving certain types of problems with them, but quite frankly I don't remember much of anything except for a few identities.

10. anonymous

Is it a particular formula, or is it just something you'd have to memorize?

11. zepdrix

Ugh OpenStudy freezing on me again.....

12. anonymous

It does that to me a lot.

13. anonymous

(this may seem like a stupid question, but which part is sine?

14. zepdrix

|dw:1444434263248:dw|

15. zepdrix

|dw:1444434276482:dw|sine is the y-coordinates

16. zepdrix

So we want to know which angles give us a y-coordinate of 0

17. anonymous

So the 180 angle

18. zepdrix

Hmm there's another one :d

19. anonymous

Oops! Didn't see that. The 0 angle

20. zepdrix

Oh wait... was your interval really given like this? $$\large\rm 0<x<2\pi$$ with strict inequality on BOTH sides?

21. anonymous

That's what it looked like.

22. zepdrix

Oh ok :) Then I guess we can throw out the 0, since it's not in our interval. So then, $$\large\rm \sin x=0$$ implies that our angle $$\large\rm x=\pi$$. Ok great, we've found one critical point.

23. zepdrix

$\large\rm 2\cos x-1=0$Solving for cos x,$\large\rm \cos x=\frac{1}{2}$

24. zepdrix

25. anonymous

So 60 degrees for cosine?

26. zepdrix

|dw:1444434707066:dw|

27. anonymous

I'm not sure why I keep looking only in the first section. Thank you again for pointing that out.

28. anonymous

60 and 300

29. zepdrix

Degrees are icky :) lol You gotta try to get comfortable with radians XD

30. zepdrix

Ok great so we've found all of our critical points!! $\large\rm x=\frac{\pi}{3},~\pi,~\frac{5\pi}{3}$

31. zepdrix

Here is the graph in case you wanted to see it, it's pretty cool https://www.desmos.com/calculator/6sbfzaocym You can clearly see a hill top at pi/3 and 5pi/3 and the bottom of a valley at the pi.

32. anonymous

Wow! I never knew you could graph something like this on desmos. That's very interesting.

33. anonymous

My connection on this site is horrible. sorry

34. anonymous

Thank you for your help! (I'll probably be back on here soon with more questions)

35. zepdrix

Yah it's a little complicated to graph just a small interval on desmos, you have to put it into set brackets { } and the interval comes first, then colon, then your function.

36. anonymous

Could you help me with one last thing really quickly?

37. zepdrix

sure

38. anonymous

I know the derivative is sinx (2cosx -1) but I can't exactly figure out how to get that.

39. zepdrix

oh you cheated on the first part? come on broski -_-

40. anonymous

No, I didn't cheat.

41. anonymous

The assignment gives me the derivative

42. zepdrix

Oh :O

43. zepdrix

$\large\rm y=(\sin x)^2+\cos x$$\large\rm y'=2(\sin x)(\sin x)'+(\cos x)'$Start by applying power rule to the sine function. Then we have chain rule as well.

44. zepdrix

You really have to remember your sine and cosine derivatives to get through this one :)

45. zepdrix

$\large\rm y'=2(\sin x)(\cos x)+(-\sin x)$

46. zepdrix

And then factor a sin x out of each term.

47. anonymous

Oh okay! I see now. Thank you. :)

48. zepdrix

:D