Explain why or why not Determine whether the following stateme...

Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(b) If ƒ is a linear function on the interval [a,b] , then a midpoint Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.

Accepted Answer

Hello, in this video, we are told to suppose that the function G of X is equal to 5 x minus 2 is defined on the interval from 3 to 7. For any positive integer N, does the midpointrman sum using Nubintervals yield the exact same value as the definite integral from 3 to 7 of G of XDX? Now, in order to verify this problem, we are going to use the definition of the definite integral. That is given as the following. The definite integral from A to B of G of XDX should have the same value as the summation from I is equal to 1 ton of G of Xi multiplied by delta X. Here, delta X is defined as B minus A divided by N, and because we are using the midpoint rule, Xi is defined. As A plus I minus 1/2 delta X. So let's go ahead and first start this problem by solving for the definite integral of the given function. Now, just to list out a few things, the given function G of X is equal to 5 X minus 2, and we are working on the interval from 3 to 7. So the definite integral of G of X is going to be the definite integral from 3 to 7 of 5 X minus 2D X. By taking the antiderivative of each term in the definite integral, we will be left with 5 halves X squared, minus 2 X, and this will be evaluated from 3 to 7. And by using the fundamental theorem of calculus part 2, this will give us the result, 5 halves multiplied by 7 squared, minus 2 multiplied by 7. This quantity subtract. 5 halves multiplied by 3 squared, minus 2 multiplied by 3. This will further simplify to be 245 divided by 2 minus 14 minus 45 divided by 2 minus 6, and by further simplifying, this is going to give us a final solution of 92. So we know that the definite integral for G of X is going to have the value of 92. And so now we want to verify that the summation of G of X is going to have the same value. Now, in order to define the summation, the first thing we want to do is we want to divine delta X. Now, delta X, again, is our larger boundary minus the smaller boundary divided by N. That is going to be 7 divided by 7 minus 3 divided by N, which is going to give us a delta X value of 4 divided by N. So this is going to be Delta X for the summation. Now we need to define X I. Now, X sub I is going to be our lower boundary, which is 3, plus I minus 12 multiplied by delta X, which is 4 divided by N. This is going to be the X sub I that we plug into the summation. So now if we take these two pieces of information, the summation for G of X is going to be the following. We will have the summation from I is equal to 1 ton of the function G of Xi. That is going to be the quantity 5 multiplied by 3 plus I minus 1/2. Multiplied by delta X, which is 4 divided by N. So this will be just the substitution for X inside of the G of X function, then it is 5 X minus 2, so we subtract 2. And this quantity itself is going to be multiplied by delta X, which is 4 divided by N. Now, the next thing we want to do is we want to go ahead and simplify the summation. First, let's just go ahead and push the summation into the value 3 plus I minus 1/2 multiplied by 4 divided by N. So, by applying that summation, we know that at least that part of the summation is going to be the summation from Is equal to 1 ton of 3. Plus the quantity I minus 1/2 multiplied by 4 divided by N. Now, by distributing the summation and simplifying, we can simplify this to be 3 N plus 4 divided by N multiplied by the summation from I is equal to 1 ton of the quantity I minus 12. Now the summation. From 1 ton of just I is going to simplify. To be N multiplied by N + 1 divided by 2, and the summation of just a constant. is just going to simplify to be the constant multiplied by N. So this is going to further simplify to be 3N + 4 divided by N multiplied by N multiplied by N + 1 divided by 2 minus N divided by 2, which is going to further simplify to be 3N + 4 N divided by 2. Which gives us 5 N. And so if we were to resubstitute this back into the original summation, we will be left with 5. Multiplied by 5 N minus 2N. Multiplied by 4 divided by N. This will further simplify to be 23 multiplied by 4, which is going to give us the value of 92. And that means that the summation of G of X is going to have the same value as the definite integral of G of X. And with that being said, the solution to this problem is going to be a, it is true. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(b) If ƒ is a linear function on the interval [a,b] , then a midpoint Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.

Key Concepts

Linear Functions

Riemann Sums

Definite Integrals

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