3. Let f(x) = :{₁ I; 2x; (0 ≤ x < 1) (1 ≤ x ≤2) i) Is the function f(x) continuous on [0, 2]? Explain (a graph is not a proof!) ii) Compute AND graph the function F(x) = f* f(t)dt Hint: Imitate the proof of the problem in the notes/video about the relationship between integration and differentiation. ii) Is the function F(x) continuous at x = 1? iii) Does F'(1) exist? Explain. (Hint: Compute the derivative of F(r) from the left and right at x = 1 iv) Does iii) violate the FTC II that states F'(x) = f(x) for all x at which f is continuous?
3. Let f(x) = :{₁ I; 2x; (0 ≤ x < 1) (1 ≤ x ≤2) i) Is the function f(x) continuous on [0, 2]? Explain (a graph is not a proof!) ii) Compute AND graph the function F(x) = f* f(t)dt Hint: Imitate the proof of the problem in the notes/video about the relationship between integration and differentiation. ii) Is the function F(x) continuous at x = 1? iii) Does F'(1) exist? Explain. (Hint: Compute the derivative of F(r) from the left and right at x = 1 iv) Does iii) violate the FTC II that states F'(x) = f(x) for all x at which f is continuous?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem 3**
Let: \[ f(x) =
\begin{cases}
x; & (0 \le x < 1) \\
2x; & (1 \le x \le 2)
\end{cases}
\]
**i) Is the function \( f(x) \) continuous on \([0, 2]\)? Explain (a graph is not a proof!)**
**ii) Compute AND graph the function** \[ F(x) = \int_{0}^{x} f(t) \, dt \]
Hint: Imitate the proof of the problem in the notes/video about the relationship between integration and differentiation.
**iii) Is the function \( F(x) \) continuous at \( x = 1 \)?**
**iv) Does \( F'(1) \) exist? Explain. (Hint: Compute the derivative of \( F(x) \) from the left and right at \[ x = 1 \]
**v) Does iii) violate the FTC II that states \( F'(x) = f(x) \) for all \( x \) at which \( f \) is continuous?**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f647d9f-6880-40dd-a686-e7d1674263c8%2Fb0f1164f-178d-4b21-8479-e75b99f793e9%2Fhd55aps_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3**
Let: \[ f(x) =
\begin{cases}
x; & (0 \le x < 1) \\
2x; & (1 \le x \le 2)
\end{cases}
\]
**i) Is the function \( f(x) \) continuous on \([0, 2]\)? Explain (a graph is not a proof!)**
**ii) Compute AND graph the function** \[ F(x) = \int_{0}^{x} f(t) \, dt \]
Hint: Imitate the proof of the problem in the notes/video about the relationship between integration and differentiation.
**iii) Is the function \( F(x) \) continuous at \( x = 1 \)?**
**iv) Does \( F'(1) \) exist? Explain. (Hint: Compute the derivative of \( F(x) \) from the left and right at \[ x = 1 \]
**v) Does iii) violate the FTC II that states \( F'(x) = f(x) \) for all \( x \) at which \( f \) is continuous?**
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