Estimate the slope (slope = rise/run) of the tangent line to the curve. 10 9 8- 7- -6- 5- -1 0 N. -3 4 5 6 7 8 X to 9 Q N What is your estimate of the slope? slope (Round to the nearest integer.)
Estimate the slope (slope = rise/run) of the tangent line to the curve. 10 9 8- 7- -6- 5- -1 0 N. -3 4 5 6 7 8 X to 9 Q N What is your estimate of the slope? slope (Round to the nearest integer.)
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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![### Estimating the Slope of a Tangent Line
#### Instructions
Estimate the slope (slope = rise/run) of the tangent line to the curve.
#### Visual Representation
On the left side, there is a graph with the following components:
- **Axes**: The horizontal axis is labeled as "x" and the vertical axis is labeled as "y".
- **Grid**: The graph has a grid that helps in estimating values.
- **Curve**: There is a blue curve that rises and falls within the graph boundaries.
- **Tangent Line**: A dashed magenta line represents the tangent line to the curve at a specific point.
- **Point of Tangency**: There is a point where the tangent line touches the blue curve.
The curve starts at the point (0, 4) and rises to a peak, then falls below the x-axis. The tangent line crosses the y-axis at 7 and goes through another point, creating a right triangle that helps in estimating the slope.
#### Task
What is your estimate of the slope?
**Estimated slope ≈** [Input Box] (Round to the nearest integer.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcdac7294-121d-461c-b626-bf307098b471%2Faa04ff07-ed6f-4746-83fb-e17ea51da898%2F7yrze2r_processed.png&w=3840&q=75)
Transcribed Image Text:### Estimating the Slope of a Tangent Line
#### Instructions
Estimate the slope (slope = rise/run) of the tangent line to the curve.
#### Visual Representation
On the left side, there is a graph with the following components:
- **Axes**: The horizontal axis is labeled as "x" and the vertical axis is labeled as "y".
- **Grid**: The graph has a grid that helps in estimating values.
- **Curve**: There is a blue curve that rises and falls within the graph boundaries.
- **Tangent Line**: A dashed magenta line represents the tangent line to the curve at a specific point.
- **Point of Tangency**: There is a point where the tangent line touches the blue curve.
The curve starts at the point (0, 4) and rises to a peak, then falls below the x-axis. The tangent line crosses the y-axis at 7 and goes through another point, creating a right triangle that helps in estimating the slope.
#### Task
What is your estimate of the slope?
**Estimated slope ≈** [Input Box] (Round to the nearest integer.)
![### Estimating the Slope of the Tangent Line
To understand how to estimate the slope of a tangent line, consider the curve in the graph displayed. Your objective is to estimate the slope at the given point, which is \((-4.5, -1.985)\).
The approximate method steps include:
1. **Identify the Point**: Locate the given point \((-4.5, -1.985)\) on the graph. This is where you are interested in finding the slope of the tangent line.
2. **Plot the Tangent Line**: Visualize or sketch a straight line that only touches the curve at this particular point and does not intersect it nearby.
3. **Calculate the Slope**: Use the rise over run method to estimate the slope of this tangent line. Specifically, count the change in y (vertical) and change in x (horizontal) between two clear points on this tangent line.
### Example Graph Analysis
- **Graph Details**:
- The x-axis ranges from \(-10\) to \(10\).
- The y-axis ranges from \(-10\) to \(10\).
- The curve itself appears to oscillate, suggesting a function with local maxima and minima.
- The tangent line (which must be estimated) at the point \((-4.5, -1.985)\) will provide a visual representation of the instantaneous rate of change of the curve at that point.
### Estimated Calculation
From the visual estimate of the tangent line:
- Identify two points on the tangent line that allow for clear determination of rise (Δy) and run (Δx).
- For instance, if the tangent visibly rises 2 units for every 5 units it runs to the right, the slope \( m \) can be estimated as \( \frac{2}{5} = 0.4 \), noting that this should be rounded to the nearest tenth.
**Fill in the Blank**:
The slope of the tangent line is \[ \_\_\_ \].
(Type an integer or decimal rounded to the nearest tenth as needed).
\[
\boxed{m \approx}
\]
This approximate slope represents the rate at which the y-value (function output) is changing at exactly \((-4.5, -1.985)\).
Feel free to use graphing tools or tracing paper to help in more accurately estimating the tangent line and its slope.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcdac7294-121d-461c-b626-bf307098b471%2Faa04ff07-ed6f-4746-83fb-e17ea51da898%2Fozcgmg_processed.png&w=3840&q=75)
Transcribed Image Text:### Estimating the Slope of the Tangent Line
To understand how to estimate the slope of a tangent line, consider the curve in the graph displayed. Your objective is to estimate the slope at the given point, which is \((-4.5, -1.985)\).
The approximate method steps include:
1. **Identify the Point**: Locate the given point \((-4.5, -1.985)\) on the graph. This is where you are interested in finding the slope of the tangent line.
2. **Plot the Tangent Line**: Visualize or sketch a straight line that only touches the curve at this particular point and does not intersect it nearby.
3. **Calculate the Slope**: Use the rise over run method to estimate the slope of this tangent line. Specifically, count the change in y (vertical) and change in x (horizontal) between two clear points on this tangent line.
### Example Graph Analysis
- **Graph Details**:
- The x-axis ranges from \(-10\) to \(10\).
- The y-axis ranges from \(-10\) to \(10\).
- The curve itself appears to oscillate, suggesting a function with local maxima and minima.
- The tangent line (which must be estimated) at the point \((-4.5, -1.985)\) will provide a visual representation of the instantaneous rate of change of the curve at that point.
### Estimated Calculation
From the visual estimate of the tangent line:
- Identify two points on the tangent line that allow for clear determination of rise (Δy) and run (Δx).
- For instance, if the tangent visibly rises 2 units for every 5 units it runs to the right, the slope \( m \) can be estimated as \( \frac{2}{5} = 0.4 \), noting that this should be rounded to the nearest tenth.
**Fill in the Blank**:
The slope of the tangent line is \[ \_\_\_ \].
(Type an integer or decimal rounded to the nearest tenth as needed).
\[
\boxed{m \approx}
\]
This approximate slope represents the rate at which the y-value (function output) is changing at exactly \((-4.5, -1.985)\).
Feel free to use graphing tools or tracing paper to help in more accurately estimating the tangent line and its slope.
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