## MathSofiya Group Title Please help....Please help....Taylor series f(x)=sinx a=1 n=3 $0.8 \le x \le 1.2$ $sinx\approx T_3(x)=sin(1)+\frac{cos(1)}{1}(x-1)-\frac{sin(1)}{2}(x-1)^2-\frac{cos(1)}{6}(x-1)^3$ (b) Use Taylor's Inequality to estimate the accuracy of the approximation $f(x)\approx T_n(x)$ lies in the given interval. $\left|R_3(x)\right|\le \frac{M}{4!} {\left|x-1\right|}^4$ I'm almost there..... 2 years ago 2 years ago

1. MathSofiya Group Title

here is an example that me and @smoothmath did http://openstudy.com/study#/updates/500af8b7e4b0549a892f4a6d

2. SmoothMath Group Title

Which part is giving you trouble?

3. SmoothMath Group Title

It looks like you made some mistakes in constructing the Taylor Series.

4. MathSofiya Group Title

where?

5. MathSofiya Group Title

i see it

6. SmoothMath Group Title

Kind of a lot of mistakes. Let me see you try again.

7. SmoothMath Group Title

$\large T_3 = f(a) +\frac{f'(a)}{1!}*(x-a) + \frac{f''(a)}{2!}*(x-a)^2 + \frac{f'''(a)}{3!}*(x-a)^3$

8. MathSofiya Group Title

I fixed it

9. SmoothMath Group Title

That's just by the definition of the Taylor series.

10. SmoothMath Group Title

Much better =)

11. SmoothMath Group Title

Okay, that's the first part. Now for the second part, we need to get M.

12. MathSofiya Group Title

$f^4(x)=sinx\le M$

13. MathSofiya Group Title

$\left|R_3(x)\right|\le \frac{sinx}{4!} {\left|x-1\right|}^4$ shoot me

14. SmoothMath Group Title

Hold on. We need to get an actual value of M. Remember how we get that?

15. MathSofiya Group Title

it's the absolute value of f^4 (x)

16. MathSofiya Group Title

don't give up on me quite yet please..... sin(1) or I have to do something with $x \ge 0.8$

17. SmoothMath Group Title

Let's step back, slow down, and understand what it is that we're doing. Just some theory.

18. MathSofiya Group Title

$f^{(n+1)}\le f(lower limit)<$

19. MathSofiya Group Title

$f^{(n+1)}\le f(lower limit)<M$

20. SmoothMath Group Title

Here'es the idea. We've made a taylor polynomial to approximate the function. Now we're trying to make a statement about how accurate the approximation is. Taylor's inequality allows us to do that. Here's how. We look at the derivative one higher than our approximation, and we look at it on the interval that we're interested in. If we can pick out a number, call it M, that the derivative is less than on the whole interval, then that allows us to use that M to limit the error.

21. SmoothMath Group Title

The basic statement is: $$\large f^{n+1} \le M$$ Therefore, $$R_n < \text{some thing dependent on M}$$

22. SmoothMath Group Title

The lower the M I'm able to pick, the better. Why? Because it allows me to put a lower cieling on the possible error.

23. SmoothMath Group Title

ceiling*

24. MathSofiya Group Title

yep. So we pick the lowest limit that we're allowed 0.8 in this case

25. SmoothMath Group Title

I don't think you're right.

26. SmoothMath Group Title

Let me talk about the maximum of a function for a bit.

27. MathSofiya Group Title

ok

28. SmoothMath Group Title

Okay, I'm thinking just a few examples will help you to see what it is we're doing.

29. SmoothMath Group Title

|dw:1342915184406:dw|

30. SmoothMath Group Title

Text on right says max=10

31. SmoothMath Group Title

|dw:1342915323520:dw|

32. SmoothMath Group Title

|dw:1342915451038:dw|

33. SmoothMath Group Title

Tell me that you understand. Because I can't make the words to explain this.

34. MathSofiya Group Title

I'm really sorry, John. I'm really trying to understand this. Ok here is what I understand. I see that you have drawn three unique functions, and each function was given an interval, and a maximum was chosen. The maximum that you have chosen is the M. The derivative should be less than M on the whole interval.

35. MathSofiya Group Title

That M allows us to put a cap on the error

36. MathSofiya Group Title

What we're trying to do with M is to put a cap on the error. That's why we need to find an actual value for M.

37. MathSofiya Group Title

That value M is greater than the next derivative up.

38. MathSofiya Group Title

hold on smooth

39. SmoothMath Group Title

Well, let me plot the 4th derivative for us.

40. MathSofiya Group Title

no hold on

41. SmoothMath Group Title
42. MathSofiya Group Title

x=1.2 will give me the ultimate cap. $f^4(x)=sinx\le sin(1.2)<1.932$

43. MathSofiya Group Title

ultimate cap for our interval

44. SmoothMath Group Title

sin(1.2) is not 1.932.

45. MathSofiya Group Title

0.932 :P

46. SmoothMath Group Title

That's impossible, since sin oscillates between 1 and -1.

47. SmoothMath Group Title

Good good =)

48. MathSofiya Group Title

typo dude...chill out

49. MathSofiya Group Title

and I didn't look at your wolfram ;P

50. MathSofiya Group Title

Victory?

51. SmoothMath Group Title

Almost there.

52. MathSofiya Group Title

The example is what confused me. They picked 7 instead of 9....but 1/7 is greater than 1/9 that's why they picked 7

53. MathSofiya Group Title

remember the interval $7 \le x \le 9$

54. SmoothMath Group Title

I do remember. Please realize that the maximum is not always going to be at one of the endpoints.

55. MathSofiya Group Title

I understand that now

56. SmoothMath Group Title

You really have to look at what the function is. It's going to be different every time and you have to analyze the function to figure out where the upper bound is.

57. MathSofiya Group Title

Yes sir.

58. SmoothMath Group Title

Okay good =)

59. MathSofiya Group Title

$\left|R_3(x)\right|\le \frac{.932}{4!} {\left|1.2-1\right|}^4$

60. MathSofiya Group Title

yes?

61. MathSofiya Group Title

$\left|R_3(x)\right|\le 0.000054$

62. SmoothMath Group Title

Looks good, pal.

63. MathSofiya Group Title

Thank ya! You're amazingly patient. Thank you soo much John. :)

64. SmoothMath Group Title

My pleasure! =D