Proof - Proof of : If for then . From the definition of the...

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Proof of : If for then . From the definition of the definite integral we have, Now, by assumption and we also have and so we know that So, from the basic properties of limits we then have, But the left side is exactly the definition of the integral and so we have, Proof of : If for then . Since we have then we know that on and so by Property 8 proved above we know that,
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We also know from Property 4 that, So, we then have, Proof of : If for then . Give we can use Property 9 on each inequality to write, Then by Property 7 on the left and right integral to get,
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Proof of : First let’s note that we can say the following about the function and the absolute value, If we now use Property 9 on each inequality we get, We know that we can factor the minus sign out of the left integral to get, Finally, recall that if then and of course this works in reverse as well so we then must have, Fundamental Theorem of Calculus, Part I If is continuous on [ a,b ] then, is continuous on [ a,b ] and it is differentiable on and that,
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Proof - Proof of : If for then . From the definition of the...

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