(vii) 5.001 = mpQ (viii) 5.0001 MpQ = (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at P(5, -3). m = (c) Using the slope from part (b), find an equation of the tangent line to the curve at P(5, -3).
(vii) 5.001 = mpQ (viii) 5.0001 MpQ = (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at P(5, -3). m = (c) Using the slope from part (b), find an equation of the tangent line to the curve at P(5, -3).
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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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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![### Calculus: Finding the Slope of the Secant Line
The point \( P(5, -3) \) lies on the curve \( y = \frac{3}{4 - x} \).
**Problem Statement:**
(a) If \( Q \) is the point \(\left( x, \frac{3}{4 - x} \right)\), find the slope of the secant line \( PQ \) (correct to six decimal places) for the following values of \( x \).
### Individual Tasks:
1. For \( x = 4.9 \):
\( m_{PQ} = \) [____]
2. For \( x = 4.99 \):
\( m_{PQ} = \) [____]
3. For \( x = 4.999 \):
\( m_{PQ} = \) [____]
4. For \( x = 4.9999 \):
\( m_{PQ} = \) [____]
5. For \( x = 5.1 \):
\( m_{PQ} = \) [____]
6. For \( x = 5.01 \):
\( m_{PQ} = \) [____]
7. For \( x = 5.001 \):
\( m_{PQ} = \) [____]
### Explanation:
To find the slope of the secant line \( PQ \), we use the slope formula:
\[ m_{PQ} = \frac{y_2 - y_1}{x_2 - x_1} \]
where \( P(5, -3) \) and \( Q \) \(\left( x, \frac{3}{4 - x} \right)\).
\[ m_{PQ} = \frac{\left( \frac{3}{4 - x} \right) - (-3)}{x - 5} \]
This expression needs to be simplified and evaluated for each given \( x \).
Please compute the value of \( m_{PQ} \) for each specified \( x \) value to complete the table.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00067d91-5c79-4b53-8e03-40299a52da99%2F0a4ba16b-f0dc-459b-9ecd-030b67dc824f%2Fxxqsty_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus: Finding the Slope of the Secant Line
The point \( P(5, -3) \) lies on the curve \( y = \frac{3}{4 - x} \).
**Problem Statement:**
(a) If \( Q \) is the point \(\left( x, \frac{3}{4 - x} \right)\), find the slope of the secant line \( PQ \) (correct to six decimal places) for the following values of \( x \).
### Individual Tasks:
1. For \( x = 4.9 \):
\( m_{PQ} = \) [____]
2. For \( x = 4.99 \):
\( m_{PQ} = \) [____]
3. For \( x = 4.999 \):
\( m_{PQ} = \) [____]
4. For \( x = 4.9999 \):
\( m_{PQ} = \) [____]
5. For \( x = 5.1 \):
\( m_{PQ} = \) [____]
6. For \( x = 5.01 \):
\( m_{PQ} = \) [____]
7. For \( x = 5.001 \):
\( m_{PQ} = \) [____]
### Explanation:
To find the slope of the secant line \( PQ \), we use the slope formula:
\[ m_{PQ} = \frac{y_2 - y_1}{x_2 - x_1} \]
where \( P(5, -3) \) and \( Q \) \(\left( x, \frac{3}{4 - x} \right)\).
\[ m_{PQ} = \frac{\left( \frac{3}{4 - x} \right) - (-3)}{x - 5} \]
This expression needs to be simplified and evaluated for each given \( x \).
Please compute the value of \( m_{PQ} \) for each specified \( x \) value to complete the table.
![### Calculating the Slope and Equation of the Tangent Line
#### Instructions:
(vii) Calculate the value of \( m_{PQ} \) when \( P = 5.001 \):
\[ m_{PQ} = \_\_\_\_\_\_\_\_\_ \]
(viii) Calculate the value of \( m_{PQ} \) when \( P = 5.0001 \):
\[ m_{PQ} = \_\_\_\_\_\_\_\_\_ \]
(b) Using the results of parts (a)(vii) and (a)(viii), estimate the value of the slope of the tangent line to the curve at \( P(5, -3) \):
\[ m = \_\_\_\_\_\_\_\_\_ \]
(c) Using the slope from part (b), determine the equation of the tangent line to the curve at \( P(5, -3) \):
\[ \_\_\_\_\_\_\_\_\_ \]
### Explanation:
**Part (vii) and (viii):**
Calculate the values of \( m_{PQ} \) for the specific given points. The slope \( m_{PQ} \) typically refers to the slope of the secant line between points \( P \) and \( Q \) on a curve.
**Part (b):**
Using the values derived in parts (vii) and (viii), you can estimate the slope of the tangent line at the point \( P(5, -3) \).
**Part (c):**
Utilize the slope obtained in part (b) to determine the equation of the tangent line at the given point using the point-slope form of the line equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00067d91-5c79-4b53-8e03-40299a52da99%2F0a4ba16b-f0dc-459b-9ecd-030b67dc824f%2F2cqug5_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating the Slope and Equation of the Tangent Line
#### Instructions:
(vii) Calculate the value of \( m_{PQ} \) when \( P = 5.001 \):
\[ m_{PQ} = \_\_\_\_\_\_\_\_\_ \]
(viii) Calculate the value of \( m_{PQ} \) when \( P = 5.0001 \):
\[ m_{PQ} = \_\_\_\_\_\_\_\_\_ \]
(b) Using the results of parts (a)(vii) and (a)(viii), estimate the value of the slope of the tangent line to the curve at \( P(5, -3) \):
\[ m = \_\_\_\_\_\_\_\_\_ \]
(c) Using the slope from part (b), determine the equation of the tangent line to the curve at \( P(5, -3) \):
\[ \_\_\_\_\_\_\_\_\_ \]
### Explanation:
**Part (vii) and (viii):**
Calculate the values of \( m_{PQ} \) for the specific given points. The slope \( m_{PQ} \) typically refers to the slope of the secant line between points \( P \) and \( Q \) on a curve.
**Part (b):**
Using the values derived in parts (vii) and (viii), you can estimate the slope of the tangent line at the point \( P(5, -3) \).
**Part (c):**
Utilize the slope obtained in part (b) to determine the equation of the tangent line at the given point using the point-slope form of the line equation.
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