Theorem 6.1.2 (Differentiation of the Integral) Let f be a continuous function on (a, b], and let F(x) = S* f(t) dt for a <* < b. Then F is C' and F' = f. Theorem 6.1.3 (Integration of the Derivative) Let f be C' on (a, b] (since the interval is closed, the derivative f' at the endpoints is a one- sided derivative). Then f'(x) dx = f(b) – f(a). Derive the integration of the derivative theorem from the differ- entiation of the integral theorem. Can you prove the converse implication?
Theorem 6.1.2 (Differentiation of the Integral) Let f be a continuous function on (a, b], and let F(x) = S* f(t) dt for a <* < b. Then F is C' and F' = f. Theorem 6.1.3 (Integration of the Derivative) Let f be C' on (a, b] (since the interval is closed, the derivative f' at the endpoints is a one- sided derivative). Then f'(x) dx = f(b) – f(a). Derive the integration of the derivative theorem from the differ- entiation of the integral theorem. Can you prove the converse implication?
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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![**Theorem 6.1.2 (Differentiation of the Integral)**
Let \( f \) be a continuous function on \([a, b]\), and let \( F(x) = \int_a^x f(t) \, dt \) for \( a \leq x \leq b \). Then \( F \) is \( C^1 \) and \( F' = f \).
**Theorem 6.1.3 (Integration of the Derivative)**
Let \( f \) be \( C^1 \) on \([a, b]\) (since the interval is closed, the derivative \( f' \) at the endpoints is a one-sided derivative). Then \(\int_a^b f'(x) \, dx = f(b) - f(a)\).
Derive the integration of the derivative theorem from the differentiation of the integral theorem. Can you prove the converse implication?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85f1beef-b59a-44ac-a29d-5ba6518650e4%2F9fac8f77-f2bc-4cbf-8781-b158ee1577ea%2Fx7i8dw_processed.png&w=3840&q=75)
Transcribed Image Text:**Theorem 6.1.2 (Differentiation of the Integral)**
Let \( f \) be a continuous function on \([a, b]\), and let \( F(x) = \int_a^x f(t) \, dt \) for \( a \leq x \leq b \). Then \( F \) is \( C^1 \) and \( F' = f \).
**Theorem 6.1.3 (Integration of the Derivative)**
Let \( f \) be \( C^1 \) on \([a, b]\) (since the interval is closed, the derivative \( f' \) at the endpoints is a one-sided derivative). Then \(\int_a^b f'(x) \, dx = f(b) - f(a)\).
Derive the integration of the derivative theorem from the differentiation of the integral theorem. Can you prove the converse implication?
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