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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?

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter6: Applications Of The Derivative
Section6.CR: Chapter 6 Review
Problem 48CR
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**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?**
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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