Section6_4 - Section 6.4 The Fundamental Theorem Part II...

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Unformatted text preview: (3/20/08) Section 6.4 The Fundamental Theorem, Part II Overview: We discussed Part I of the Fundamental Theorem of Calculus in the last section. We establish Part II of the theorem here. We also show how Part II can be used to prove Part I and how it can be combined with the Chain Rule to find derivatives of integrals with functions as limits of integration. Then we discuss a definition of the natural logarithm as an integral. Topics: • The Fundamental Theorem, Part II • Another proof of Part I of the Fundamental Theorem • Derivatives of integrals with functions as limits of integration • Defining the natural logarithm as an integral The Fundamental Theorem, Part II Part I of the Fundamental Theorem of Calculus that we discussed in Section 6.3 states that if F is continuous and its derivative F prime is piecewise continuous on an interval containing a and b , then the integral of F prime from a to b equals the change in F across the interval: integraldisplay b a F prime ( x ) dx = F ( b )- F ( a ) . (1) Part II of the Fundamental Theorem deals with derivatives of definite integrals with respect to variable upper limits of integration: Theorem 1 (The Fundamental Theorem, Part II) If f is continuous on an open interval I containing the point a , then the function integraldisplay x a f ( t ) dt is differentiable on I and for all x in I , d dx integraldisplay x a f ( t ) dt = f ( x ) . (2) Proof: We define F ( x ) = integraldisplay x a f ( t ) dt for x in the interval I . We will prove the theorem by showing that for all x in I , F prime ( x ) = f ( x ) . (3) We suppose first that f ( t ) is nonnegative on I and that x is greater than a , as in Figure 1. Then F ( x ) is the area of the region between the graph of y = f ( t ) and the t-axis for a ≤ t ≤ x and (2) states that the rate of change F prime ( x ) of the area with respect to x equals the height f ( x ) of the region at its right side. 209 p. 210 (3/20/08) Section 6.4, The Fundamental Theorem, Part II Region of area Region of area Region of area F ( x ) F ( x + Δ x ) F ( x + Δ x )- F ( x ) FIGURE 1 FIGURE 2 FIGURE 3 For small positive Δ x , F ( x + Δ x ) = integraldisplay x +Δ x a f ( t ) dt is the area of the region between the graph and the t-axis for a ≤ t ≤ x + Δ x in Figure 2, so that the difference F ( x + Δ x )- F ( x ) = integraldisplay x +Δ x x f ( t ) dt is the area of the region between the graph and the t-axis for x ≤ t ≤ x + Δ x in Figure 3. Because f is continuous on the finite closed interval x ≤ t ≤ x + Δ x , it has—by the Extreme Value Theorem of Section 3.2—a minimum value m (Figure 4) and a maximum value M (Figure 5) in the interval. Rectangle of area m Δ X Rectangle of area M Δ X FIGURE 4 FIGURE 5 The rectangle in Figure 4 has area m Δ x and is contained in the region of Figure 3, while the rectangle in Figure 5 has area M Δ x and contains the region of Figure 3. Therefore, m Δ x ≤ F ( x + Δ x )- F ( x ) ≤ M Δ x....
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This note was uploaded on 09/13/2010 for the course MATH Math 20A taught by Professor Eggers during the Summer '08 term at UCSD.

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Section6_4 - Section 6.4 The Fundamental Theorem Part II...

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