Find the slope and the equation of the tangent line to the graph of the function at the given value of x. f(x) = x* – 25x2 + 144; x = – 1
Find the slope and the equation of the tangent line to the graph of the function at the given value of x. f(x) = x* – 25x2 + 144; x = – 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Concept explainers
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
![Transcription for Educational Website:
---
**Problem Statement:**
Find the slope and the equation of the tangent line to the graph of the function at the given value of x.
**Given Function:**
\[ f(x) = x^4 - 25x^2 + 144 \]
**Value of x:**
\[ x = -1 \]
---
**Solution Steps:**
1. Calculate the first derivative of the function \( f(x) \) to find the slope of the tangent line.
2. Evaluate the derivative at \( x = -1 \) to find the slope.
3. Use the point-slope form to write the equation of the tangent line.
**Output Requirements:**
- The slope of the tangent line is: \(\_\_\_\).
*(Simplify your answer.)*
- The equation of the tangent line is: \( y = \_\_\_\).
---
*Note: The boxes are placeholders for the numerical values obtained from the calculations.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ccf583b-3141-45e1-ad26-60f361ecbb8d%2Fd3d7d100-88f2-41ac-bb2e-367c2944b19d%2Fhtolxym_processed.png&w=3840&q=75)
Transcribed Image Text:Transcription for Educational Website:
---
**Problem Statement:**
Find the slope and the equation of the tangent line to the graph of the function at the given value of x.
**Given Function:**
\[ f(x) = x^4 - 25x^2 + 144 \]
**Value of x:**
\[ x = -1 \]
---
**Solution Steps:**
1. Calculate the first derivative of the function \( f(x) \) to find the slope of the tangent line.
2. Evaluate the derivative at \( x = -1 \) to find the slope.
3. Use the point-slope form to write the equation of the tangent line.
**Output Requirements:**
- The slope of the tangent line is: \(\_\_\_\).
*(Simplify your answer.)*
- The equation of the tangent line is: \( y = \_\_\_\).
---
*Note: The boxes are placeholders for the numerical values obtained from the calculations.*
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