Calculating the Derivative of g(t) using Integration - Question #9 MSFE Prereq

[xchangz]

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]

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 :).

What have you tried? Before we can give you any help, you must show some effort at solving your problem.

 

[xchangz]

What have you tried? Before we can give you any help, you must show some effort at solving your problem.

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]

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!

Look under "variable limits" here. You basically want to combine the usual FTC with the chain rule (since there is t-dependence in both the limits and the integrand).

 

[xchangz]

Look under "variable limits" here. You basically want to combine the usual FTC with the chain rule (since there is t-dependence in both the limits and the integrand).

Thanks a lot! But I do not totally understand the proof. Under the general form of variable limits, which is similar to my problem, I understand everything up until the part with the mean value theorem. I don't understand what went on there. Though this provides me with a better insight on the problem.