20 50 -45 16 -40 -35 12- 30 C. 25- -20- 15 10 2 4 8. 10 50 45 40 4- 35 2- 30 d. 25 2. 10 -2. 20 4. 15 -0 10 -8. -10 10 2

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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The image displays four graphs labeled (a), (b), (c), and (d).

**Graph (a):**  
- **Type:** Line Graph
- **Description:** This graph has a linear progression. It starts at the point (0,4) on the y-axis and ends at (10,20). The graph line ascends steadily from left to right. Both axes are labeled from 0 to 20 for y and 0 to 10 for x.

**Graph (b):**  
- **Type:** Horizontal Line Graph
- **Description:** This graph displays a constant value. The line starts at (-10,10) and extends horizontally to (10,10). The y-coordinate stays constant at 10 while the x-axis runs from -10 to 10. Both axes are labeled from -10 to 10.

**Graph (c):**  
- **Type:** Line Graph
- **Description:** This graph begins at the point (0,-5) and ascends sharply to (4,50), where it transitions into a horizontal line extending to (8,50). The x-axis ranges from 0 to 10, and the y-axis spans -5 to 55.

**Graph (d):**  
- **Type:** Triangle Graph
- **Description:** It begins at (0,-5), ascends to a peak at (5,55), and then descends back to the y-coordinate of -5 at (10,5). The x-axis ranges from 0 to 10, while the y-axis ranges from -5 to 55.

Each graph demonstrates different mathematical functions and trends, useful for understanding the representation of linear, constant, and non-linear data patterns.
Transcribed Image Text:The image displays four graphs labeled (a), (b), (c), and (d). **Graph (a):** - **Type:** Line Graph - **Description:** This graph has a linear progression. It starts at the point (0,4) on the y-axis and ends at (10,20). The graph line ascends steadily from left to right. Both axes are labeled from 0 to 20 for y and 0 to 10 for x. **Graph (b):** - **Type:** Horizontal Line Graph - **Description:** This graph displays a constant value. The line starts at (-10,10) and extends horizontally to (10,10). The y-coordinate stays constant at 10 while the x-axis runs from -10 to 10. Both axes are labeled from -10 to 10. **Graph (c):** - **Type:** Line Graph - **Description:** This graph begins at the point (0,-5) and ascends sharply to (4,50), where it transitions into a horizontal line extending to (8,50). The x-axis ranges from 0 to 10, and the y-axis spans -5 to 55. **Graph (d):** - **Type:** Triangle Graph - **Description:** It begins at (0,-5), ascends to a peak at (5,55), and then descends back to the y-coordinate of -5 at (10,5). The x-axis ranges from 0 to 10, while the y-axis ranges from -5 to 55. Each graph demonstrates different mathematical functions and trends, useful for understanding the representation of linear, constant, and non-linear data patterns.
### Matching Velocity Graphs to Position Graphs

Learn how to match a velocity graph to its corresponding position graph by analyzing the changes in velocity and position over time.

#### Graph Description:

**Velocity Graphs (Blue):**

1. **Graph 1 (Top Left):**
   - The velocity \( v \) remains constant at 10 units from time \( t = 0 \) to \( t = 4 \).
   - After \( t = 4 \), the velocity drops to 0 at \( t = 6 \) and remains 0 till \( t = 10 \).

2. **Graph 2 (Bottom Left):**
   - The velocity \( v \) decreases linearly from 4 units at \( t = 0 \) to -4 units at \( t = 10 \).

**Position Graphs (Red):**

1. **Graph a (Top Right):**
   - The position \( f \) increases from 0, peaks around \( t = 5 \) at a value slightly above 12, then decreases back to 0 at \( t = 10 \).
   - This suggests a parabolic shape, indicating a change in direction at the peak.

2. **Graph b (Bottom Right):**
   - The position \( f \) begins at 10 at \( t = 0 \) and increases in a curved manner, reaching around 100 at \( t = 10 \).
   - This indicates continuous acceleration.

#### Analysis:

- **Matching Velocity to Position Graph:**

  - **Graph 1 (Velocity) matches with Graph b (Position)**
    - Initially constant velocity and then the velocity becomes zero, suggesting steady movement initially and then stopping, which aligns with the continuous increase in position seen in Graph b.

  - **Graph 2 (Velocity) matches with Graph a (Position)**
    - The linear decrease in velocity indicates deceleration, reaching zero and then negative values, suggesting a return to the starting point, matching the parabolic trajectory of Graph a.

By understanding these graphs, students can learn to interpret how velocity affects position over time, an important concept in physics and calculus.
Transcribed Image Text:### Matching Velocity Graphs to Position Graphs Learn how to match a velocity graph to its corresponding position graph by analyzing the changes in velocity and position over time. #### Graph Description: **Velocity Graphs (Blue):** 1. **Graph 1 (Top Left):** - The velocity \( v \) remains constant at 10 units from time \( t = 0 \) to \( t = 4 \). - After \( t = 4 \), the velocity drops to 0 at \( t = 6 \) and remains 0 till \( t = 10 \). 2. **Graph 2 (Bottom Left):** - The velocity \( v \) decreases linearly from 4 units at \( t = 0 \) to -4 units at \( t = 10 \). **Position Graphs (Red):** 1. **Graph a (Top Right):** - The position \( f \) increases from 0, peaks around \( t = 5 \) at a value slightly above 12, then decreases back to 0 at \( t = 10 \). - This suggests a parabolic shape, indicating a change in direction at the peak. 2. **Graph b (Bottom Right):** - The position \( f \) begins at 10 at \( t = 0 \) and increases in a curved manner, reaching around 100 at \( t = 10 \). - This indicates continuous acceleration. #### Analysis: - **Matching Velocity to Position Graph:** - **Graph 1 (Velocity) matches with Graph b (Position)** - Initially constant velocity and then the velocity becomes zero, suggesting steady movement initially and then stopping, which aligns with the continuous increase in position seen in Graph b. - **Graph 2 (Velocity) matches with Graph a (Position)** - The linear decrease in velocity indicates deceleration, reaching zero and then negative values, suggesting a return to the starting point, matching the parabolic trajectory of Graph a. By understanding these graphs, students can learn to interpret how velocity affects position over time, an important concept in physics and calculus.
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