## anonymous 5 years ago find mean value? 2cos2x+sinx=0 a= -pi\2, b=pi\2

1. anonymous

2. myininaya

{f(b)-f(a)}/{b-a}={f(pi/2)-f(-pi/2)}/{pi}={[2cos(pi)+sin(pi/2)]-[2cos(-pi)+sin(-pi/2)]}/{pi} ={2(-1)+1-2(-1)-(-1)}/pi=(-2+1+2+1)/pi=2/pi but f'(x)=4sin(2x)+cosx f'(c)=4sin(2c)+cos(c) set f'(c)=2/pi and solve for c

3. anonymous

i got a value of c

4. myininaya

ok cool!

5. anonymous

i got ! 4sin2c+(1\90)=cosc now how can be find value of c

6. myininaya

4sin(2c)+cos(c)=2/pi

7. anonymous

8. myininaya

-0.04535615498 is what I got using a calculator so the problem says to use the mean value thm?

9. anonymous

how can you find the value of c . give me the method?

10. anonymous

$Average = \frac{1}{b-a}\int\limits_{a}^{b}2\cos 2x +\sin x dx$

11. anonymous

dumcow i don't understand this method . give me another method?

12. anonymous

are you looking for the mean value of the function from a to b or does it say to use mean value theorem to find point that equals avg rate of change

13. anonymous

but my teacher says that u can use mean value theorm

14. myininaya
15. myininaya

m(b-a)=int(f(x),a..b) m is the mean value

16. myininaya

cow has already said this though

17. myininaya

do you know how to integrate?

18. anonymous

no i don't know?

19. anonymous

Okay, By the mean value theorem. The mean value is [f(b)-f(a)]/[b-a]. $f(b)=f(\pi/2)=2\cos \pi +\sin(\pi/2)=-2+1=-1$ $f(a)=f(-{\pi \over 2})=2\cos (-\pi)+\sin (-\pi/2)=-2-1=-3$

20. anonymous

Therefore, ${f(b)-f(a) \over b-a}={-1-(-3) \over {\pi \over 2}-{-\pi \over 2}}={2 \over \pi}$

21. anonymous

Does that help?

22. anonymous

its right but how we can find the value of c?

23. anonymous

This value, we just found (2/pi) is equal to f'(c). You can use this relation to find c.

24. anonymous

fc= -4sin2c+cosc

25. anonymous

-4sin2c+cosc=pi\2 so how we can find C?

26. anonymous

Solve the equation, you may get more than one value for c. Take only the value that is in the given interval (-pi/2,pi/2).

27. anonymous

how can it solve the equa? i don't understand this equation to find the value for C

28. myininaya

i would graph it and approximate the solution

29. myininaya

you could also use newton's method

30. anonymous

You never solved quadratic equation?! Hmm I think myininaya got a point. It's difficult to solve it using identities. Probably graphing is a good method.

31. anonymous

Just gimme a minute.

32. anonymous

BRB

33. anonymous

BRB what?

34. myininaya

means be right back

35. anonymous

ok

36. anonymous

no body can solve this question?

37. anonymous

oops myininaya has already solved the problem. Sorry I didn't see that.

38. myininaya

lol

39. anonymous

you are give my proper metjhod to find the valuc of C?

40. anonymous

as myininaya, we can estimate the value by graphing.

41. anonymous

c will be around 1.66

42. anonymous

you said that C1.66 . how you can find tell me?

43. anonymous

Wait. this value of c is out side our interval. The value of c, that's in the interval is around 0.045

44. myininaya

look for x intercepts of f'(c)=2/pi

45. anonymous

$f'(c)={f(b)-f(a) \over b-a}$ this is the formula.

46. anonymous

ok i know thic formula i completed.. -4sin2c+cosx=2\pi after what can i do i don't understand?

47. myininaya

48. myininaya

this is what I got when I used newton's method to find c

49. anonymous

f(x)=2cos2x+sinx andf(x)=-4sin2x+cosx

50. myininaya

oops i forgot about the 2/pi you try i have to go

51. anonymous

ok thanks!