theoi prove the mea theorem. 44. A baseball is hit upward and its height at time t in seconds is given by H(t) = 100t – 161² feet a. Find the velocity of the baseball after t seconds. b. Find the time at which the velocity of the ball is 0. c. Find the height of the ball at which the velocity is 0.
theoi prove the mea theorem. 44. A baseball is hit upward and its height at time t in seconds is given by H(t) = 100t – 161² feet a. Find the velocity of the baseball after t seconds. b. Find the time at which the velocity of the ball is 0. c. Find the height of the ball at which the velocity is 0.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question 44
![### Calculus Problems and Graph Analysis
#### Functions and Derivatives
For each of the functions given in Problems 29 to 34, find intervals where the function \( f \) is increasing and intervals where \( f \) is decreasing.
**29.** \( f(x) = x^2 - x + 1 \)
**30.** \( f(x) = 5 - x^2 \)
**31.** \( f(x) = x^3 + x \)
**32.** \( f(x) = 8 - x^3 \)
**33.** Let \( f \) be the function for which the graph of the derivative \( y = f'(x) = g(x) \) is given by:
- **Graph Description:** The graph shows a curve that crosses the x-axis at multiple points, indicating changes in intervals of increase and decrease.
**34.** Let \( f \) be the function for which the graph of the derivative \( y = f'(x) = g(x) \) is given by:
- **Graph Description:** This graph also features a curve crossing the x-axis at several points, revealing intervals where the function is increasing or decreasing.
#### Level 2: Applied and Theory Problems
**41.** Let \( f \) be differentiable on the interval \((a, b)\). Use the mean value theorem to prove if \( f' < 0 \) on \([a, b]\), then \( f \) is decreasing on \((a, b)\).
**42.** **Rolle’s Theorem:** Let \( f \) be differentiable on \((a, b)\) and continuous on \([a, b]\). Assume \( f(a) = f(b) = 0 \). Without using the mean value theorem, argue that there exists a \( c \) in \((a, b)\) such that \( f'(c) = 0 \).
**43.** Use Rolle’s theorem to prove the mean value theorem.
**44.** A baseball is hit upward and its height at time \( t \) in seconds is given by:
\[ H(t) = 100t - 16t^2 \] feet
- **a.** Find the velocity of the baseball after \( t \) seconds.
- **b.** Find the time at which the velocity of the ball is 0.
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe4db1e84-75df-498d-9686-5cabfd0e0c2f%2F0b4dd567-b205-4e6d-92a6-5e79a14e4b34%2Fcsvd077d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Problems and Graph Analysis
#### Functions and Derivatives
For each of the functions given in Problems 29 to 34, find intervals where the function \( f \) is increasing and intervals where \( f \) is decreasing.
**29.** \( f(x) = x^2 - x + 1 \)
**30.** \( f(x) = 5 - x^2 \)
**31.** \( f(x) = x^3 + x \)
**32.** \( f(x) = 8 - x^3 \)
**33.** Let \( f \) be the function for which the graph of the derivative \( y = f'(x) = g(x) \) is given by:
- **Graph Description:** The graph shows a curve that crosses the x-axis at multiple points, indicating changes in intervals of increase and decrease.
**34.** Let \( f \) be the function for which the graph of the derivative \( y = f'(x) = g(x) \) is given by:
- **Graph Description:** This graph also features a curve crossing the x-axis at several points, revealing intervals where the function is increasing or decreasing.
#### Level 2: Applied and Theory Problems
**41.** Let \( f \) be differentiable on the interval \((a, b)\). Use the mean value theorem to prove if \( f' < 0 \) on \([a, b]\), then \( f \) is decreasing on \((a, b)\).
**42.** **Rolle’s Theorem:** Let \( f \) be differentiable on \((a, b)\) and continuous on \([a, b]\). Assume \( f(a) = f(b) = 0 \). Without using the mean value theorem, argue that there exists a \( c \) in \((a, b)\) such that \( f'(c) = 0 \).
**43.** Use Rolle’s theorem to prove the mean value theorem.
**44.** A baseball is hit upward and its height at time \( t \) in seconds is given by:
\[ H(t) = 100t - 16t^2 \] feet
- **a.** Find the velocity of the baseball after \( t \) seconds.
- **b.** Find the time at which the velocity of the ball is 0.
-
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