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Problem

Show that $ \frac{1}{2} (T_n + M_n) = T_{2n} $.

08:38

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Problem 48 Hard Difficulty

Show that if $ f $ is a polynomial of degree 3 or lower, then Simpson's Rule gives the exact value of $ \int_a^b f(x)\ dx $.


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Related Topics

Integration Techniques

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01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

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27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

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Watch More Solved Questions in Chapter 7

Problem 1
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Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
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Problem 12
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Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
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Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
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Problem 48
Problem 49
Problem 50

Video Transcript

Okay. This question wants us to prove that Simpson's rule gives the exact area under the curve for pollen. No meals with degree of that, most three. So initially at this problem statement, you might not know it too. D'oh! Because it seems quite complicated. But there's a little trick we know using the error expression for the Simpsons rule. So based on the book, we're given this formula that the air from the Simpson approximation is at most this constant K times B minus eight of the fifth over 1 80 end of the fourth Okay, s. So how does that help us? Well, we're also told that Kay is any constant bigger than the fourth derivative of X for all ex. So we can also say that Kay is proportional to the maximum value of F quadruple prime of X and f of X. Is any cubic excused plus B x squared plus C x plus D. So what's the fourth derivative of a cubic polynomial? Well, if you differentiate x cubed four times, he'd zero drift. Durand shed X squared four times you zero x four tons of zero. And a constant, of course, is here. So the fourth derivative of our function zero. And since r K is anything bigger than our equal to the max value of our fourth derivative, well, we could just say Okay, equal zero for Q Bix. So air of the Simpson approximation is listening equal to zero. So we can conclude that since the error zero, the rule must give the exact value of the integral. So again, a very tough looking problem. Fun. We just used the fact that this constant K has to be proportional to the max value of the fourth derivative, which we know is zero. So the error must be zero. And if there's zero air, it means that Simpson's role exactly gives us theatre girls value, which is a pretty neat fact.

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Calculus: Early Transcendentals

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Related Topics

Integration Techniques

Top Calculus 2 / BC Educators
Grace He

Numerade Educator

Anna Marie Vagnozzi

Campbell University

Heather Zimmers

Oregon State University

Kristen Karbon

University of Michigan - Ann Arbor

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

Video Thumbnail

27:53

Basic Techniques

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

Join Course
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