Section6_3 - Section 6.3 The Fundamental Theorem Part I...

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Unformatted text preview: (3/20/08) Section 6.3 The Fundamental Theorem, Part I Overview: The Fundamental Theorem of Calculus shows that differentiation and integration are, in a sense, inverse operations. It is presented in two parts. We previewed Part I in Section 6.1 and prove it in this section. It deals with integrals of derivatives. Part II, which will be covered in Section 6.4, involves derivatives of integrals Topics: • Another look at Section 6.1 • The Fundamental Theorem of Calculus, Part I Another look at Section 6.1 In the first section of this chapter we used the following result in applications involving continuous functions whose derivatives are step functions. Theorem 1 Suppose that a function y = F ( x ) is continuous on a finite closed interval [ a,b ] and that its derivative r = F prime ( x ) is a step function on [ a,b ] . Then the region between the graph r = F prime ( x ) and the x-axis for a ≤ x ≤ b consists of a finite number of rectangles, and the change in the function’s value from x = a to x = b is given by F ( b )- F ( a ) = bracketleftBigg The area of all rectangles above the x-axis bracketrightBigg- bracketleftBigg The area of all rectangles below the x-axis bracketrightBigg . (1) For the continuous function y = F ( x ) whose derivative is the step function in Figure 1, for instance, this theorem states that F ( b )- F ( a ) is equal to the area of the two rectangles above the x-axis minus the area of the three rectangles below the x-axis. x r r = F prime ( x ) a b FIGURE 1 The Fundamental Theorem of Calculus, Part I We know from Theorem 2 of Section 6.2 that the difference of areas described in Theorem 1 is equal to the integral of F prime ( x ) from a to b . Consequently, equation (1) can be rewritten in the form, F ( b )- F ( a ) = integraldisplay b a F prime ( x ) dx. (2) 198 Section 6.3, The Fundamental Theorem, Part I p. 199 (3/20/08) Equation (2) for functions with step-function derivatives is a special case of the following general result. Theorem 2 (The Fundamental Theorem of Calculus, Part I) Suppose that y = F ( x ) is continuous and its derivative r = F prime ( x ) is piecewise continuous on an interval containing a and b . Then integraldisplay b a F prime ( x ) dx = F ( b )- F ( a ) (3a) or, in Leibniz notation integraldisplay b a dF dx dx = F ( b )- F ( a ) . (3b) Proof: We can establish this theorem for general functions that satisfy its hypotheses by using the Mean Value Theorem from Section 4.1. Recall that the Mean Value Theorem states that if y = F ( x ) is continuous on an interval [ a,b ] and its derivative exists for all x with a < x < b , then there is at least one point c with a < c < b such that the average rate of change F ( b )- F ( a ) b- a of F for a ≤ x ≤ b equals its (instantaneous) rate of change F prime ( c ) at that point: F ( b )- F ( a ) b- a = F prime ( c ) . (4) The geometric interpretation of the Mean Value Theorem is illustrated in Figure 2. Since the average rate of change of F is the slope of the secant line through the points at...
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Section6_3 - Section 6.3 The Fundamental Theorem Part I...

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