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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**Problem #37 (Modified):**
Find the equation of the tangent line at the point \( x = 1 \) for the function:
\[ y = x^4 + \frac{1}{x} - 3e^x \]
---
### Explanation:
To find the equation of the tangent line, we need to:
1. Calculate the derivative of the given function \( y = x^4 + \frac{1}{x} - 3e^x \) to find the slope of the tangent line at \( x = 1 \).
2. Evaluate the original function to find the y-coordinate at \( x = 1 \).
3. Use the point-slope form of a line to write the equation of the tangent line.
### Steps:
1. Differentiate the function to find \( y' \).
2. Compute \( y' \) at \( x = 1 \) for the slope.
3. Calculate \( y \) at \( x = 1 \) for the point.
4. Apply the point-slope formula: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point on the curve.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b79b02b-7faf-4a60-85a8-80947491ab86%2F013131ec-cba4-403e-b4e5-f0d4bacf1338%2Fzhr8oo3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Problem #37 (Modified):**
Find the equation of the tangent line at the point \( x = 1 \) for the function:
\[ y = x^4 + \frac{1}{x} - 3e^x \]
---
### Explanation:
To find the equation of the tangent line, we need to:
1. Calculate the derivative of the given function \( y = x^4 + \frac{1}{x} - 3e^x \) to find the slope of the tangent line at \( x = 1 \).
2. Evaluate the original function to find the y-coordinate at \( x = 1 \).
3. Use the point-slope form of a line to write the equation of the tangent line.
### Steps:
1. Differentiate the function to find \( y' \).
2. Compute \( y' \) at \( x = 1 \) for the slope.
3. Calculate \( y \) at \( x = 1 \) for the point.
4. Apply the point-slope formula: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point on the curve.
---
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