Let f [a, b] : → R be bounded and f (x) > a > 0, for all x € [a, b]. Show that if f is 1 Riemann integrable on [a, b] then 7 : [a, b] → R, (7)(x) = f() is also Riemann integrable on [a, b]. Hint: We show that given e > 0, there is a partition P of [a, b] such that U (7, P) – L (},P) < - < €. • Explain why we can find a partition P of [a, b] such that U (f, P) – L (ƒ, P) < a²€. • Show that M₂ (7) = m(f) and mi (}) = M₁(s). • Show that U (}, P) – L (7,P) ≤ & (U (ƒ, P) – L (ƒ,P)) < e.
Let f [a, b] : → R be bounded and f (x) > a > 0, for all x € [a, b]. Show that if f is 1 Riemann integrable on [a, b] then 7 : [a, b] → R, (7)(x) = f() is also Riemann integrable on [a, b]. Hint: We show that given e > 0, there is a partition P of [a, b] such that U (7, P) – L (},P) < - < €. • Explain why we can find a partition P of [a, b] such that U (f, P) – L (ƒ, P) < a²€. • Show that M₂ (7) = m(f) and mi (}) = M₁(s). • Show that U (}, P) – L (7,P) ≤ & (U (ƒ, P) – L (ƒ,P)) < e.
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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![Let \( f : [a, b] \rightarrow \mathbb{R} \) be bounded and \( f(x) > \alpha > 0 \), for all \( x \in [a, b] \). Show that if \( f \) is Riemann integrable on \( [a, b] \) then \( \frac{1}{f} : [a, b] \rightarrow \mathbb{R}, \left( \frac{1}{f} \right)(x) = \frac{1}{f(x)} \) is also Riemann integrable on \( [a, b] \).
_Hint:_ We show that given \( \epsilon > 0 \), there is a partition \( P \) of \( [a, b] \) such that \( U\left( \frac{1}{f}, P \right) - L\left( \frac{1}{f}, P \right) < \epsilon \).
- Explain why we can find a partition \( P \) of \( [a, b] \) such that \( U(f, P) - L(f, P) < \alpha^2 \epsilon \).
- Show that \( M_i \left( \frac{1}{f} \right) = \frac{1}{m_i(f)} \) and \( m_i \left( \frac{1}{f} \right) = \frac{1}{M_i(f)} \).
- Show that \( U\left( \frac{1}{f}, P \right) - L\left( \frac{1}{f}, P \right) \leq \frac{1}{\alpha^2} (U(f, P) - L(f, P)) < \epsilon \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2F6feb41ee-d61a-4fae-96fe-8168f5405b7c%2Fbhx4wu_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( f : [a, b] \rightarrow \mathbb{R} \) be bounded and \( f(x) > \alpha > 0 \), for all \( x \in [a, b] \). Show that if \( f \) is Riemann integrable on \( [a, b] \) then \( \frac{1}{f} : [a, b] \rightarrow \mathbb{R}, \left( \frac{1}{f} \right)(x) = \frac{1}{f(x)} \) is also Riemann integrable on \( [a, b] \).
_Hint:_ We show that given \( \epsilon > 0 \), there is a partition \( P \) of \( [a, b] \) such that \( U\left( \frac{1}{f}, P \right) - L\left( \frac{1}{f}, P \right) < \epsilon \).
- Explain why we can find a partition \( P \) of \( [a, b] \) such that \( U(f, P) - L(f, P) < \alpha^2 \epsilon \).
- Show that \( M_i \left( \frac{1}{f} \right) = \frac{1}{m_i(f)} \) and \( m_i \left( \frac{1}{f} \right) = \frac{1}{M_i(f)} \).
- Show that \( U\left( \frac{1}{f}, P \right) - L\left( \frac{1}{f}, P \right) \leq \frac{1}{\alpha^2} (U(f, P) - L(f, P)) < \epsilon \).
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