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MathSofiya
Group Title
Please help....Please help....Taylor series
f(x)=sinx a=1 n=3
\[0.8 \le x \le 1.2\]
\[sinx\approx T_3(x)=sin(1)+\frac{cos(1)}{1}(x1)\frac{sin(1)}{2}(x1)^2\frac{cos(1)}{6}(x1)^3\]
(b) Use Taylor's Inequality to estimate the accuracy of the approximation \[f(x)\approx T_n(x)\] lies in the given interval.
\[\leftR_3(x)\right\le \frac{M}{4!} {\leftx1\right}^4\]
I'm almost there.....
 2 years ago
 2 years ago
MathSofiya Group Title
Please help....Please help....Taylor series f(x)=sinx a=1 n=3 \[0.8 \le x \le 1.2\] \[sinx\approx T_3(x)=sin(1)+\frac{cos(1)}{1}(x1)\frac{sin(1)}{2}(x1)^2\frac{cos(1)}{6}(x1)^3\] (b) Use Taylor's Inequality to estimate the accuracy of the approximation \[f(x)\approx T_n(x)\] lies in the given interval. \[\leftR_3(x)\right\le \frac{M}{4!} {\leftx1\right}^4\] I'm almost there.....
 2 years ago
 2 years ago

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MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
here is an example that me and @smoothmath did http://openstudy.com/study#/updates/500af8b7e4b0549a892f4a6d
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Which part is giving you trouble?
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
It looks like you made some mistakes in constructing the Taylor Series.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
i see it
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Kind of a lot of mistakes. Let me see you try again.
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
\[\large T_3 = f(a) +\frac{f'(a)}{1!}*(xa) + \frac{f''(a)}{2!}*(xa)^2 + \frac{f'''(a)}{3!}*(xa)^3\]
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
I fixed it
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
That's just by the definition of the Taylor series.
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Much better =)
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Okay, that's the first part. Now for the second part, we need to get M.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
\[f^4(x)=sinx\le M\]
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
\[\leftR_3(x)\right\le \frac{sinx}{4!} {\leftx1\right}^4\] shoot me
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Hold on. We need to get an actual value of M. Remember how we get that?
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
it's the absolute value of f^4 (x)
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
don't give up on me quite yet please..... sin(1) or I have to do something with \[x \ge 0.8\]
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Let's step back, slow down, and understand what it is that we're doing. Just some theory.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
\[f^{(n+1)}\le f(lower limit)<\]
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
\[f^{(n+1)}\le f(lower limit)<M\]
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
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.
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
The basic statement is: \( \large f^{n+1} \le M\) Therefore, \(R_n < \text{some thing dependent on M}\)
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
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.
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
ceiling*
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
yep. So we pick the lowest limit that we're allowed 0.8 in this case
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
I don't think you're right.
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Let me talk about the maximum of a function for a bit.
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Okay, I'm thinking just a few examples will help you to see what it is we're doing.
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
dw:1342915184406:dw
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Text on right says max=10
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
dw:1342915323520:dw
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
dw:1342915451038:dw
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Tell me that you understand. Because I can't make the words to explain this.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
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.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
That M allows us to put a cap on the error
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
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.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
That value M is greater than the next derivative up.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
hold on smooth
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Well, let me plot the 4th derivative for us.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
no hold on
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
http://www.wolframalpha.com/input/?i=plot+sin%28x%29+on+x%3D.5+to+x%3D1.5
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
x=1.2 will give me the ultimate cap. \[f^4(x)=sinx\le sin(1.2)<1.932\]
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
ultimate cap for our interval
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
sin(1.2) is not 1.932.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
0.932 :P
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
That's impossible, since sin oscillates between 1 and 1.
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Good good =)
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
typo dude...chill out
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
and I didn't look at your wolfram ;P
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
Victory?
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Almost there.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
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
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
remember the interval \[7 \le x \le 9\]
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
I do remember. Please realize that the maximum is not always going to be at one of the endpoints.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
I understand that now
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
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.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
Yes sir.
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Okay good =)
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
\[\leftR_3(x)\right\le \frac{.932}{4!} {\left1.21\right}^4\]
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
\[\leftR_3(x)\right\le 0.000054\]
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
Looks good, pal.
 2 years ago

MathSofiya Group TitleBest ResponseYou've already chosen the best response.1
Thank ya! You're amazingly patient. Thank you soo much John. :)
 2 years ago

SmoothMath Group TitleBest ResponseYou've already chosen the best response.1
My pleasure! =D
 2 years ago
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