(Homework Help)
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==Homework Help==
 
==Homework Help==
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Hello, this is gary from ma181.  let's solve the extra credit problem.
 +
Here is the problem in italics:
 +
 +
"Extra Credit Problem
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--------------------
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 +
Suppose that  f(x)  is continuously differentiable
 +
on the interval  [a,b].  Let  N  be a positive integer
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and let  M = Max { |f'(x)| : a <= x <= b }.  Let
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h = (b-a)/N  and let  R_N  denote the "right endpoint"
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Riemann Sum for the integral
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I = int( f(x), x=a..b).
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 +
In other words,
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R_N = sum( f(a + n*h)*h , n=1..N ).
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Explain why the error, E = | R_N - I |, satisfies
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E < M(b-a)^2/N."
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So what does this equation "E < M(b-a)^2/N" mean.  This reads that the error is less than the Maximum value of the derivative of the function of x multiplied by the interval squared from x=a to x=b all divided by the total number of subintervals N.
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 +
I don't understand why this must be true.  Maybe I'm wrong, but if f(x) were a horizontal line, wouldn't E=0 and M(b-a)^2/N also be =0.  That would mean it is a false statement that E < M(b-a)^2/N.  Are we to assume that E <= M(b-a)^2/N?
  
 
==Interesting Articles about Calculus==
 
==Interesting Articles about Calculus==

Revision as of 12:05, 16 September 2008

Math 181 Honors Calculus

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Homework Help

Hello, this is gary from ma181. let's solve the extra credit problem. Here is the problem in italics:

"Extra Credit Problem


Suppose that f(x) is continuously differentiable on the interval [a,b]. Let N be a positive integer and let M = Max { |f'(x)| : a <= x <= b }. Let h = (b-a)/N and let R_N denote the "right endpoint" Riemann Sum for the integral

I = int( f(x), x=a..b).

In other words,

R_N = sum( f(a + n*h)*h , n=1..N ).

Explain why the error, E = | R_N - I |, satisfies

E < M(b-a)^2/N."

So what does this equation "E < M(b-a)^2/N" mean. This reads that the error is less than the Maximum value of the derivative of the function of x multiplied by the interval squared from x=a to x=b all divided by the total number of subintervals N.

I don't understand why this must be true. Maybe I'm wrong, but if f(x) were a horizontal line, wouldn't E=0 and M(b-a)^2/N also be =0. That would mean it is a false statement that E < M(b-a)^2/N. Are we to assume that E <= M(b-a)^2/N?

Interesting Articles about Calculus

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