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Uniform Convergence & Integration

In Analysis we had this as a question. The solution is the second picture and the question is the first. I have looked at the solution and still cannot make sense of it. I understand the conditions we are stating in order to apply the FTC however I cannot see the integral with c instead of x is formed and that this must be equal to the 'c derivative.' Could anyone give a more detailed breakdown of the solution please?
Assignment 3 Q11.png30.6KB
Assignment 3 Q11 A.png27.5KB
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Avatar for davros
davros
16
Original post by VincentCheung
In Analysis we had this as a question. The solution is the second picture and the question is the first. I have looked at the solution and still cannot make sense of it. I understand the conditions we are stating in order to apply the FTC however I cannot see the integral with c instead of x is formed and that this must be equal to the 'c derivative.' Could anyone give a more detailed breakdown of the solution please?


It's a long time since I did any of this formally, and I usually head for the Wikipedia article if I need to calculate the derivative of an integral, but are you just confused by the labelling of the variables in the hint? a and b are clearly taking the place of the limits, but in the original integral y is the dummy variable whereas in the hint x is the dummy variable!
Yes the notation has confused me a quite a bit, with a and b I am fine. I am confused mainly about the sentence 'the c-derivative of the integral ____ is equal to the integral of the c-derivative of the integrand.' As in I understand the intuition but when it comes to formulating this with the function I have no clue how to start
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Avatar for davros
davros
16
Original post by VincentCheung
Yes the notation has confused me a quite a bit, with a and b I am fine. I am confused mainly about the sentence 'the c-derivative of the integral ____ is equal to the integral of the c-derivative of the integrand.' As in I understand the intuition but when it comes to formulating this with the function I have no clue how to start


Does this link help at all?

http://en.wikipedia.org/wiki/Differentiation_under_the_integral_sign

Scroll down and work through the examples.
That has helped a lot, thank you! That formula makes so much more sense than the hint..

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