Calculus II Notes 5.3

Calculus II Notes 5.3 - Calculus II Stewart Dr Berg Spring...

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Unformatted text preview: Calculus II- Stewart Dr. Berg Spring 2010 Page 1 5.3 5.3 The Fundamental Theorem of Calculus A revolution took place in calculus when it was realized by Isaac Barrow (1630- 1677), who was Newton’s teacher. that antiderivatives could be used to calculate definite integrals. The basic intuition behind this is remarkably simple. Consider the function F ( x ) = f ( t ) dt a x ∫ . If f ( t ) ≥ for x ≥ a and we restrict our attention to x ≥ a , then F is a function giving the area under the curve and between a and x . Since the rate of change of the area at x is precisely the height of the curve at x , then d dx F ( x ) = f ( x ) . Hence, F is an antiderivative for f . In fact, this turns out to be true without putting any restrictions on f or x , as long as f is integrable. This intuition is completely obvious if f is a constant, since, for example, if we assume f ( x ) = c and calculate the area F under the curve f and between a and x , then F ( x ) = c ( x − a ) = cx − ca so ′ F ( x ) = c = f ( x ) . For a non-constant function, applying this to a rectangle being used to approximate area locally gives an approximation to the rate of change of the area. Calculus II- Stewart Dr. Berg Spring 2010 Page 2 5.3 Since F ( x + h ) − F ( x ) ≈ h f ( x ) , then F ( x + h ) − F ( x ) h ≈...
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Calculus II Notes 5.3 - Calculus II Stewart Dr Berg Spring...

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