Trapezoidal Rule: Maximum error in approximation?

In summary, to find the maximum error in the approximation of ∫ from 0 to 3 of f(x)dx using T4, an upper bound on |f''(x)| over x ∈ [0,3] is needed. This can be found by finding an upper bound on K, which in this case would be 2.
  • #1
Jess Karakov
11
3

Homework Statement


Suppose that T4 is used to approximate the ∫ from 0 to 3 of f(x)dx, where -2 ≤ f ''(x) ≤ 1 for all x. What is the maximum error in the approximation?

Homework Equations


|ET|≤ (K(b-a)^3)/(12n^2)

The Attempt at a Solution


So I know how to find the error of the trapezoidal rule using the above equation, but I do not understand how to find the maximum error in an approximation.
To find the max error I would find the max/mins of f ''(x), right? But I don't know how to do that when f(x) is not given
 
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  • #2
Jess Karakov said:

Homework Statement


Suppose that T4 is used to approximate the ∫ from 0 to 3 of f(x)dx, where -2 ≤ f ''(x) ≤ 1 for all x. What is the maximum error in the approximation?

Homework Equations


|ET|≤ (K(b-a)^3)/(12n^2)

The Attempt at a Solution


So I know how to find the error of the trapezoidal rule using the above equation, but I do not understand how to find the maximum error in an approximation.
To find the max error I would find the max/mins of f ''(x), right? But I don't know how to do that when f(x) is not given

You can't. All you can do is find an upper bound on the absolute error, so your actual error may be a lot less than your bound. Just find an upper bound on ##|f''(x)|## over ##x \in [0,3]##.

Note: people hardly ever find best bounds by finding actual maxima of things like ##|f''(x)|##; typically, they are satisfied with decent bounds.
 
  • #3
So K would just be 2?
 
  • #4
Jess Karakov said:
So K would just be 2?

You tell me.
 
  • #5
yes...:wink:
 

Related to Trapezoidal Rule: Maximum error in approximation?

What is the Trapezoidal Rule?

The Trapezoidal Rule is a method for approximating the value of a definite integral by dividing the area under the curve into trapezoids and summing up their areas.

How is the maximum error in approximation calculated using the Trapezoidal Rule?

The maximum error in approximation using the Trapezoidal Rule can be calculated using the formula: E = -((b-a)^3 / 12n^2) * f''(c), where E is the maximum error, a and b are the limits of integration, n is the number of subintervals, and f''(c) is the second derivative of the function at some point c between a and b.

What is the significance of the maximum error in approximation?

The maximum error in approximation gives us an idea of how close our approximation is to the actual value of the integral. A smaller maximum error indicates a more accurate approximation.

How can the maximum error in approximation be reduced?

The maximum error in approximation can be reduced by decreasing the width of the subintervals, which means using a larger number of subintervals. This will result in a more accurate approximation, but it will also require more computational effort.

What are the limitations of the Trapezoidal Rule in terms of accuracy?

The Trapezoidal Rule is not very accurate for functions with rapidly changing slopes or high-order derivatives. It is also less accurate for functions with more complex shapes. In these cases, other methods such as Simpson's Rule or the Midpoint Rule may provide better approximations.

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