Analysis Problem?

Question

Assume G(X)=definite ∫ g(t)dt on the interval [a,x] is Riemann integrable on [a,b], then prove G(X) is well defined and prove or disprove G(X) is the antiderivative of g(t); i.e. prove the following statement: If g is continuous at x, then G ' (X) = g(x).

Answer 1

In other words, prove the second fundamental theorem of calculus. The proof really should be in your textbook, but if not, you can use mine:

http://www.math.utah.edu/~taylor/5_Integral.pdf (the proof you're looking for is on page 17)

Believe it or not, the documents at http://www.math.utah.edu/~taylor/foundations.html really were the only textbook we had in my analysis class. Which was nice, since I didn't have to spend any money on them.

Note that the hypothesis that g is continuous at x is important - if g is not continuous there it may not even have an antiderivative. For instance, consider the function g(x) = {1 if x=0, 0 if x≠0}. This function is integrable, but the derivative of [-1, x]∫g(x) dx at 0 is not g(0)=1, but rather 0. A more interesting example is h(x) = {1 if x≥0, 0 if x<0}. h(x) is integrable, but H(x) = [-1, x]∫h(x) dx = {0 if x<0, x if x≥0} is not differentiable at 0.

Answer 2

Dimensional prognosis includes assessment of gadgets. Like in case you have a velocity it ought to acquire in ft/sec. so which you realize that in case you're to discover an answer in inches, then you incredibly will would desire to transform with the aid of inches/ft. and in some style you will ought to multiply with the aid of seconds, so which you're able to would desire to transform seconds/hr or some thing like that. So the respond delivers a clue to what aspects will yield the impressive gadgets. It receives extra complicated once you get into capability and capability with the aid of fact the gadgets conversion are extra complicated yet this would desire to supply you with a clue.