121616949-math.157 - 7.2 The Fundamental Theorem of...

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7.2 The Fundamental Theorem of Calculus 143 be the same as these problems, in the sense that when we try to approximate solutions we end up with mathematics that looks like the two examples, though of course the function involved will not always be so simple. Even better, we now see that while the second problem did not appear to be amenable to approach one, it can in fact be solved in the same way. The reasoning is this: we know that problem one can be solved easily by finding a function whose derivative is 3 t . We also know that mathematically the two problems are the same, because both can be solved by taking a limit of a sum, and the sums are identical. Therefore, we don’t really need to compute the limit of either sum because we know that we will get the same answer by computing a function with the derivative 3 t or, which is the same thing, 3 x . It’s true that the first problem had the added complication of the “10”, and we certainly need to be able to deal with such minor variations, but that turns out to be quite simple.
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