The discussion revolves around calculating the derivative of a function defined as an integral with variable limits, specifically g(t) = integral from 0 to t f(t,x)dx. This involves concepts from calculus, particularly the fundamental theorem of calculus and its application to functions of multiple variables.
The discussion is ongoing, with participants sharing their attempts and insights. Some guidance has been provided regarding the combination of the fundamental theorem and the chain rule, but there remains a lack of complete understanding regarding the proof and its implications.
Participants express confusion over the application of the mean value theorem in the context of variable limits, indicating a need for further clarification on this aspect of the problem.
What have you tried? Before we can give you any help, you must show some effort at solving your problem.xchangz said:Compute the derivative with respect to t of g(t) = integral from 0 to t f(t,x)dx
http://www.ieor.columbia.edu/pdf-files/MSFE_Prereq.pdf
or question #9 from the website above. Thanks in advance :).
Mark44 said:What have you tried? Before we can give you any help, you must show some effort at solving your problem.
xchangz said:Well what I've tried so far is that I know that the fundamental theorem of calculus derivative with respect to t of integral from 0 to t f(x) dx is just f(t). However, that won't work in this case because we have the introduction of a 2nd variable t. It won't be just f(t,t). If I assume f(t,x) = t^2 +3tx + x^2 just as a dummy definition as an experiment, I get that g'(t) = 17t^2 / 2 but if we assume it is f(t,t), it will be 5t^2 so the two answers don't match up. I've been staring at this problem for hours now!
gabbagabbahey said:Look under "variable limits" here. You basically want to combine the usual FTC with the chain rule (since there is [itex]t[/itex]-dependence in both the limits and the integrand).