If f(x) = 3x² − 5x+7, find f'(-5). | Use this to find the equation of the tangent line to the parabola y - 3x² - 5x+7 at the point (-5,107) . The equation of this tangent line can be written in the form y=mx+b where m is: and where b is: Graph both f(x) and the tangent line on Desmos to verify.
If f(x) = 3x² − 5x+7, find f'(-5). | Use this to find the equation of the tangent line to the parabola y - 3x² - 5x+7 at the point (-5,107) . The equation of this tangent line can be written in the form y=mx+b where m is: and where b is: Graph both f(x) and the tangent line on Desmos to verify.
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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![### Example Problem: Finding the Derivative and Tangent Line
**1. Given Function and Derivative Calculation**
Given the function \( f(x) = 3x^2 - 5x + 7 \):
1. Find \( f'(-5) \). \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)
**2. Equation of the Tangent Line**
Use the derivative found to determine the tangent line to the parabola \( y = 3x^2 - 5x + 7 \) at the point \((-5, 107)\). The equation of this tangent line can be written in the form \( y = mx + b \):
- where \( m \) is: \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)
- and where \( b \) is: \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)
**Graphing**
Graph both \( f(x) \) and the tangent line on Desmos to verify the results.
---
**3. Second Example Problem: Function and Derivative Involving Product Rule**
Given:
\[ f(x) = x^2 h(x) \]
\[ h(-1) = 4 \]
\[ h'(-1) = 7 \]
Calculate \( f'(-1) \). \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)
*Hint: Use the product rule and the power rule.*
**Additional Instructions:**
For the given problem, utilize:
- **The Power Rule**: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
- **The Product Rule**: If \( f(x) = u(x)v(x) \), then \( f'(x) = u'(x)v(x) + u(x)v'(x) \).
---
In this exercise, we aim to strengthen your understanding of calculus concepts including the derivative of a function and the application of derivatives to find tangent lines. Graphical verification through software such as Desmos will solidify your conceptual grasp.
### Note:
In an educational setting, steps to reach the solution are encouraged to be detailed for thorough understanding. Please follow through derivative rules and algebraic manipulation step-by-step.
Feel free to seek assistance if any step is unclear, and always verify your results practically](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdccbde07-1885-4681-9cbb-26aa67baf026%2F2297fc0b-33e3-4cd1-88cf-e6e85a7394e7%2Fdztky5_processed.png&w=3840&q=75)
Transcribed Image Text:### Example Problem: Finding the Derivative and Tangent Line
**1. Given Function and Derivative Calculation**
Given the function \( f(x) = 3x^2 - 5x + 7 \):
1. Find \( f'(-5) \). \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)
**2. Equation of the Tangent Line**
Use the derivative found to determine the tangent line to the parabola \( y = 3x^2 - 5x + 7 \) at the point \((-5, 107)\). The equation of this tangent line can be written in the form \( y = mx + b \):
- where \( m \) is: \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)
- and where \( b \) is: \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)
**Graphing**
Graph both \( f(x) \) and the tangent line on Desmos to verify the results.
---
**3. Second Example Problem: Function and Derivative Involving Product Rule**
Given:
\[ f(x) = x^2 h(x) \]
\[ h(-1) = 4 \]
\[ h'(-1) = 7 \]
Calculate \( f'(-1) \). \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)
*Hint: Use the product rule and the power rule.*
**Additional Instructions:**
For the given problem, utilize:
- **The Power Rule**: If \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).
- **The Product Rule**: If \( f(x) = u(x)v(x) \), then \( f'(x) = u'(x)v(x) + u(x)v'(x) \).
---
In this exercise, we aim to strengthen your understanding of calculus concepts including the derivative of a function and the application of derivatives to find tangent lines. Graphical verification through software such as Desmos will solidify your conceptual grasp.
### Note:
In an educational setting, steps to reach the solution are encouraged to be detailed for thorough understanding. Please follow through derivative rules and algebraic manipulation step-by-step.
Feel free to seek assistance if any step is unclear, and always verify your results practically
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