121616949-math.160 - Z b a f t dt = lim n →∞ n-1 X i =0...

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146 Chapter 7 Integration since F ( a ) is a constant and has derivative zero. In other words, by shifting our point of view slightly, we see that the odd looking function G ( x ) = Z x a f ( t ) dt has a derivative, and that in fact G ( x ) = f ( x ). This is really just a restatement of the Fundamental Theorem of Calculus, and indeed is often called the Fundamental Theorem of Calculus. To avoid confusion, some people call the two versions of the theorem “The Fundamental Theorem of Calculus, part I” and “The Fundamental Theorem of Calculus, part II”, although unfortunately there is no universal agreement as to which is part I and which part II. Since it really is the same theorem, differently stated, some people simply call them both “The Fundamental Theorem of Calculus.” THEOREM 7.4 Fundamental Theorem of Calculus Suppose that f ( x ) is con- tinuous on the interval [ a, b ] and let G ( x ) = Z x a f ( t ) dt. Then G ( x ) = f ( x ). We have not really proved the Fundamental Theorem. In a nutshell, we gave the following argument to justify it: Suppose we want to know the value of
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Unformatted text preview: Z b a f ( t ) dt = lim n →∞ n-1 X i =0 f ( t i )Δ t. We can interpret the right hand side as the distance traveled by an object whose speed is given by f ( t ). We know another way to compute the answer to such a problem: find the position of the object by finding an antiderivative of f ( t ), then substitute t = a and t = b and subtract to find the distance traveled. This must be the answer to the original problem as well, even if f ( t ) does not represent a speed. What’s wrong with this? In some sense, nothing. As a practical matter it is a very convincing arguement, because our understanding of the relationship between speed and distance seems to be quite solid. From the point of view of mathematics, however, it is unsatisfactory to justify a purely mathematical relationship by appealing to our un-derstanding of the physical universe, which could, however unlikely it is in this case, be wrong....
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