121616949-math.159 - t Now the theorem THEOREM 7.3...

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7.2 The Fundamental Theorem of Calculus 145 want to compute lim n →∞ n - 1 X i =0 f ( t i t, it no longer matters what f ( t ) stands for—it could be a speed, or the height of a curve, or something else entirely. We know that the limit can be computed by finding any function with derivative f ( t ), substituting a and b , and subtracting. We summarize this in a theorem. First, we introduce some new notation and terms. We write Z b a f ( t ) dt = lim n →∞ n - 1 X i =0 f ( t i t if the limit exists. That is, the left hand side means, or is an abbreviation for, the right hand side. The symbol R is called an integral sign , and the whole expression is read as “the integral of f ( t ) from a to b .” What we have learned is that this integral can be computed by finding a function, say F ( t ), with the property that F ( t ) = f ( t ), and then computing F ( b ) - F ( a ). The function F ( t ) is called an antiderivative
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Unformatted text preview: t ). Now the theorem: THEOREM 7.3 Fundamental Theorem of Calculus Suppose that f ( x ) is con-tinuous on the interval [ a, b ]. If F ( x ) is any antiderivative of f ( x ), then Z b a f ( x ) dx = F ( b )-F ( a ) . Let’s rewrite this slightly: Z x a f ( t ) dt = F ( x )-F ( a ) . We’ve replaced the variable x by t and b by x . These are just different names for quantities, so the substitution doesn’t change the meaning. It does make it easier to think of the two sides of the equation as functions. The expression Z x a f ( t ) dt is a function: plug in a value for x , get out some other value. The expression F ( x )-F ( a ) is of course also a function, and it has a nice property: d dx ( F ( x )-F ( a )) = F ± ( x ) = f ( x ) ,...
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