Unformatted text preview: Z b a f ( t ) dt = lim n →∞ n1 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 understanding of the physical universe, which could, however unlikely it is in this case, be wrong....
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 Spring '07
 JonathanRogawski
 Math, Calculus, Derivative, Fundamental Theorem Of Calculus

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