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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What is the slope
![## Estimating the Slope of a Tangent Line to a Curve
**Estimate the slope (slope = rise/run) of the tangent line to the curve.**
### Graph Details:
The graph shown has the following key features:
- It has a Cartesian coordinate system with the \( x \)-axis (horizontal) and \( y \)-axis (vertical) each marked from 1 to 9 and 0 to 10, respectively.
- The grid lines make it easy to see each unit increase.
- A blue curve represents a function's graph that increases, reaches a maximum point, and then decreases.
- A red dashed line represents the tangent line to the curve at a point of tangency, which seems to be around \( x = 2 \).
### Tangent Line Slope Estimation:
The tangent line intersects the curve at a point where the coordinates can be approximated. To find the slope of the tangent line, we apply the formula for the slope of a line:
\[ \text{slope} = \frac{\text{rise}}{\text{run}} \]
**Instructions for Estimating:**
1. Identify two points on the red dashed line (tangent line). Let's use visible grid points for simplicity. Assume they are \((x_1, y_1)\) and \((x_2, y_2)\).
2. Calculate the rise (change in \( y \)-values) and the run (change in \( x \)-values) between these two points.
### Example Estimation:
- Suppose the two chosen points are approximately \((1, 2)\) and \((3, 7)\).
- **Rise**: Change in \( y \) from point 1 to point 2: \( 7 - 2 = 5 \)
- **Run**: Change in \( x \) from point 1 to point 2: \( 3 - 1 = 2 \)
- Therefore, the slope of the tangent line can be estimated as:
\[ \text{slope} \approx \frac{5}{2} = 2.5 \]
**Note:**
- Different points on the tangent line can slightly affect the estimated slope.
- Enter your estimated slope in the provided box and round your answer if necessary.
---
To ensure accuracy in educational materials, verify the points on the tangent line and compute the slope using precise coordinates. Understanding](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11a64bba-74c0-4fd7-b16b-3d8fff0a6ac2%2F33324f6c-1482-4652-a0c0-6fdcff6502e7%2F8mvpe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Estimating the Slope of a Tangent Line to a Curve
**Estimate the slope (slope = rise/run) of the tangent line to the curve.**
### Graph Details:
The graph shown has the following key features:
- It has a Cartesian coordinate system with the \( x \)-axis (horizontal) and \( y \)-axis (vertical) each marked from 1 to 9 and 0 to 10, respectively.
- The grid lines make it easy to see each unit increase.
- A blue curve represents a function's graph that increases, reaches a maximum point, and then decreases.
- A red dashed line represents the tangent line to the curve at a point of tangency, which seems to be around \( x = 2 \).
### Tangent Line Slope Estimation:
The tangent line intersects the curve at a point where the coordinates can be approximated. To find the slope of the tangent line, we apply the formula for the slope of a line:
\[ \text{slope} = \frac{\text{rise}}{\text{run}} \]
**Instructions for Estimating:**
1. Identify two points on the red dashed line (tangent line). Let's use visible grid points for simplicity. Assume they are \((x_1, y_1)\) and \((x_2, y_2)\).
2. Calculate the rise (change in \( y \)-values) and the run (change in \( x \)-values) between these two points.
### Example Estimation:
- Suppose the two chosen points are approximately \((1, 2)\) and \((3, 7)\).
- **Rise**: Change in \( y \) from point 1 to point 2: \( 7 - 2 = 5 \)
- **Run**: Change in \( x \) from point 1 to point 2: \( 3 - 1 = 2 \)
- Therefore, the slope of the tangent line can be estimated as:
\[ \text{slope} \approx \frac{5}{2} = 2.5 \]
**Note:**
- Different points on the tangent line can slightly affect the estimated slope.
- Enter your estimated slope in the provided box and round your answer if necessary.
---
To ensure accuracy in educational materials, verify the points on the tangent line and compute the slope using precise coordinates. Understanding
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