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3. Let f be integrable on [a, b] and let F(x) = f f(t)dt (by Thcorem*, F is well-defined on [a, b]). (a) (b) is differentiable at a c and F'(c) = f(c). %3D Prove that F is continuous on [a, b). Assume that c E (a, b). If f is continuous at x = c, show F

3. Let f be integrable on [a, b] and let F(x) = f f(t)dt (by Thcorem*, F is well-defined on [a, b]). (a) (b) is differentiable at a c and F'(c) = f(c). %3D Prove that F is continuous on [a, b). Assume that c E (a, b). If f is continuous at x = c, show F

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section: Chapter Questions
Problem 52RE
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Definition: Let f: [a, b] -R be a function. We say that a number f(x)dx is the integral of f over [a, b] if the following condition is satisfied: Given any number > 0 there is a corresponding number N such that for every n 2 N, the partition P = {ro,..., Tn} of [a, b] with x;-i-1 = for all 1<i < n and any choice of a E [a;-1, xi], we have b-a IESs)Ax - / 5(#)dxr| < &• i=1 Theorem* (The Sub-interval Theorem) (a) Let f be integrable on [a, b], and let a <c< b. Then f is integrable on both [a, c) and [c, b], and f f(x)dx = f(x)dx + S° f(x)dx. (b) Let f be integrable on [a, c] and [c, b), where a <c < b. Then f is integrable on [a, b]. %3D Problem: 1. Let f be integrable on [a, b]. Show that there is a number M such that f(x)| < M for all a E [a, b]. 2. Suppose that f is continuous on [a, b] and set M maxrEla,b) (x)|. Prove that lim %3D p00 3. Let f be integrable on [a, b] and let F(x) = " f(t)dt (by Theorem*, F is well-defined on [a, b). (a.) (b) Prove that F is continuous on [a, b). Assume that cE (a, b). If f is continuous at a c, show F
Transcribed Image Text:Definition: Let f: [a, b] -R be a function. We say that a number f(x)dx is the integral of f over [a, b] if the following condition is satisfied: Given any number > 0 there is a corresponding number N such that for every n 2 N, the partition P = {ro,..., Tn} of [a, b] with x;-i-1 = for all 1<i < n and any choice of a E [a;-1, xi], we have b-a IESs)Ax - / 5(#)dxr| < &• i=1 Theorem* (The Sub-interval Theorem) (a) Let f be integrable on [a, b], and let a <c< b. Then f is integrable on both [a, c) and [c, b], and f f(x)dx = f(x)dx + S° f(x)dx. (b) Let f be integrable on [a, c] and [c, b), where a <c < b. Then f is integrable on [a, b]. %3D Problem: 1. Let f be integrable on [a, b]. Show that there is a number M such that f(x)| < M for all a E [a, b]. 2. Suppose that f is continuous on [a, b] and set M maxrEla,b) (x)|. Prove that lim %3D p00 3. Let f be integrable on [a, b] and let F(x) = " f(t)dt (by Theorem*, F is well-defined on [a, b). (a.) (b) Prove that F is continuous on [a, b). Assume that cE (a, b). If f is continuous at a c, show F
Prove that lim p00 3. Let f be integrable on [a, b] and let F(x) = J" f(t)dt (by Thcorem*, F is well-defined on [a, b]). (a) (b) is differentiable at a c and F'(c) = f(c). Prove that F is continuous on [a, b]. Assume that cE (a, b). If f is continuous at a c, show F 1
Transcribed Image Text:Prove that lim p00 3. Let f be integrable on [a, b] and let F(x) = J" f(t)dt (by Thcorem*, F is well-defined on [a, b]). (a) (b) is differentiable at a c and F'(c) = f(c). Prove that F is continuous on [a, b]. Assume that cE (a, b). If f is continuous at a c, show F 1
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