Suppose F(t) has the derivative f(t) shown below, and F(0) 3 2 1 -1 -2 -3 F(2) = F(9) = 1 2 3 4 5 6 7 8 9 = 3. Find values for F(2) and F(9)
Suppose F(t) has the derivative f(t) shown below, and F(0) 3 2 1 -1 -2 -3 F(2) = F(9) = 1 2 3 4 5 6 7 8 9 = 3. Find values for F(2) and F(9)
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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![### Instruction: Analyzing Derivative and Integrating to Find Function Values
Suppose \( F(t) \) has the derivative \( f(t) \) shown below, and \( F(0) = 3 \). Find values for \( F(2) \) and \( F(9) \).
#### Graph Explanation:
The given graph is a plot of the derivative function \( f(t) \). The x-axis represents the variable \( t \) ranging from 0 to 9, and the y-axis represents \( f(t) \) values ranging from -3 to 3.
Key points on the graph:
- From \( t = 0 \) to \( t = 1 \), \( f(t) \) increases linearly from \( 0 \) to \( 2 \).
- From \( t = 1 \) to \( t = 3 \), \( f(t) \) decreases linearly from \( 2 \) to \( -1 \).
- From \( t = 3 \) to \( t = 5 \), \( f(t) \) decreases linearly from \( -1 \) to \( -3 \).
- From \( t = 5 \) to \( t = 9 \), \( f(t) \) is constant at \( 0 \).
#### Integration of \( f(t) \) to Find \( F(t) \):
The value of \( F(t) \) can be found by integrating \( f(t) \) and using the initial condition \( F(0) = 3 \).
1. **From \( t = 0 \) to \( t = 2 \):**
- The function \( f(t) = 2t \).
- The area under the curve (integral of \( f(t) \)): \(\int_0^2 2t dt = 2 \).
2. **From \( t = 2 \) to \( t = 3 \):**
- The function \( f(t) = -3(t-1) + 2 \).
- The area under the curve (integral of \( f(t) \)): -1/2.
So, from the cumulative integration:
\[ F(2) = F(0) + \int_0^2 f(t) dt = 3 + 1 \cdot 2 = 5 \]
3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9659bf0-1b68-44e7-9939-e8c666150000%2F5d00910a-d524-4ab8-bbba-c0de596f6807%2Fd8z69lfh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Instruction: Analyzing Derivative and Integrating to Find Function Values
Suppose \( F(t) \) has the derivative \( f(t) \) shown below, and \( F(0) = 3 \). Find values for \( F(2) \) and \( F(9) \).
#### Graph Explanation:
The given graph is a plot of the derivative function \( f(t) \). The x-axis represents the variable \( t \) ranging from 0 to 9, and the y-axis represents \( f(t) \) values ranging from -3 to 3.
Key points on the graph:
- From \( t = 0 \) to \( t = 1 \), \( f(t) \) increases linearly from \( 0 \) to \( 2 \).
- From \( t = 1 \) to \( t = 3 \), \( f(t) \) decreases linearly from \( 2 \) to \( -1 \).
- From \( t = 3 \) to \( t = 5 \), \( f(t) \) decreases linearly from \( -1 \) to \( -3 \).
- From \( t = 5 \) to \( t = 9 \), \( f(t) \) is constant at \( 0 \).
#### Integration of \( f(t) \) to Find \( F(t) \):
The value of \( F(t) \) can be found by integrating \( f(t) \) and using the initial condition \( F(0) = 3 \).
1. **From \( t = 0 \) to \( t = 2 \):**
- The function \( f(t) = 2t \).
- The area under the curve (integral of \( f(t) \)): \(\int_0^2 2t dt = 2 \).
2. **From \( t = 2 \) to \( t = 3 \):**
- The function \( f(t) = -3(t-1) + 2 \).
- The area under the curve (integral of \( f(t) \)): -1/2.
So, from the cumulative integration:
\[ F(2) = F(0) + \int_0^2 f(t) dt = 3 + 1 \cdot 2 = 5 \]
3
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