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Problem 2: For the following statements, answer true or false. If the statement is true, provide a short reason. If the statement is false, provide a counterexample (or explain why). (a) If f is integrable on [a, b] and F,G are two different anti-derivatives of f, then F(x) = G(x) +c for all x E [a, b and some constant c E R. %3D (b) If 9. g(x) dx, then every upper sum for f is an upper sum for g. f(x) dr = | (c) If f, g are polynomials such that g(x) 0 for all x E (a, b], then f/g is integrable on [a, b]. (d) If f is continuous, y is differentiable, y' is integrable, and y(a) = y(b), then: %3D True | f(p(x))g'(x)dx = 0. xp(x),6((x)d)f (e) If f is continuous, f(-1) = -1Wand f(1) = 1, then there exsts a c> 0 such that: %3D rc f (x) dx = 0. %3D

Problem 2: For the following statements, answer true or false. If the statement is true, provide a short reason. If the statement is false, provide a counterexample (or explain why). (a) If f is integrable on [a, b] and F,G are two different anti-derivatives of f, then F(x) = G(x) +c for all x E [a, b and some constant c E R. %3D (b) If 9. g(x) dx, then every upper sum for f is an upper sum for g. f(x) dr = | (c) If f, g are polynomials such that g(x) 0 for all x E (a, b], then f/g is integrable on [a, b]. (d) If f is continuous, y is differentiable, y' is integrable, and y(a) = y(b), then: %3D True | f(p(x))g'(x)dx = 0. xp(x),6((x)d)f (e) If f is continuous, f(-1) = -1Wand f(1) = 1, then there exsts a c> 0 such that: %3D rc f (x) dx = 0. %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 2: For the following statements, answer true or false. If the statement is true, provide a short reason. If the statement is false, provide a counterexample (or explain why). (a) If f is integrable on [a, b] and F,G are two different anti-derivatives of f, then F(x) = G(x) +c for all x E [a, b and some constant c E R. %3D (b) If 9. g(x) dx, then every upper sum for f is an upper sum for g. f(x) dr = | (c) If f, g are polynomials such that g(x) 0 for all x E (a, b], then f/g is integrable on [a, b]. (d) If f is continuous, y is differentiable, y' is integrable, and y(a) = y(b), then: %3D True | f(p(x))g'(x)dx = 0. xp(x),6((x)d)f (e) If f is continuous, f(-1) = -1Wand f(1) = 1, then there exsts a c> 0 such that: %3D rc f (x) dx = 0. %3D
Transcribed Image Text:Problem 2: For the following statements, answer true or false. If the statement is true, provide a short reason. If the statement is false, provide a counterexample (or explain why). (a) If f is integrable on [a, b] and F,G are two different anti-derivatives of f, then F(x) = G(x) +c for all x E [a, b and some constant c E R. %3D (b) If 9. g(x) dx, then every upper sum for f is an upper sum for g. f(x) dr = | (c) If f, g are polynomials such that g(x) 0 for all x E (a, b], then f/g is integrable on [a, b]. (d) If f is continuous, y is differentiable, y' is integrable, and y(a) = y(b), then: %3D True | f(p(x))g'(x)dx = 0. xp(x),6((x)d)f (e) If f is continuous, f(-1) = -1Wand f(1) = 1, then there exsts a c> 0 such that: %3D rc f (x) dx = 0. %3D
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