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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![**Transcription for Educational Website:**
---
**Problem Statement:**
Find an equation of the tangent line to the given curve at the specified point.
\[ y = \frac{4x}{x^3 + 3} \] at the point (1, 1).
**Solution:**
Given the function:
\[ y = \frac{3}{12} x^n + \frac{1}{2} \]
---
The problem involves determining the equation of the tangent line to a curve defined by a rational function. The approximate derivative at the specified point can be used to find the slope of the tangent line. The solution proceeds by calculating the derivative of the function, evaluating it at \(x = 1\), and using the point-slope form of a line to write the equation of the tangent.
Note: The second part of the solution appears to involve simplifying terms, but additional context is required for complete clarity.
No graphs or diagrams are present in the image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e8b0010-4aed-4d57-b920-96d0bf9eca5d%2F2f5175d0-c1c6-483c-886a-8b9450577b03%2F6og7q1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
---
**Problem Statement:**
Find an equation of the tangent line to the given curve at the specified point.
\[ y = \frac{4x}{x^3 + 3} \] at the point (1, 1).
**Solution:**
Given the function:
\[ y = \frac{3}{12} x^n + \frac{1}{2} \]
---
The problem involves determining the equation of the tangent line to a curve defined by a rational function. The approximate derivative at the specified point can be used to find the slope of the tangent line. The solution proceeds by calculating the derivative of the function, evaluating it at \(x = 1\), and using the point-slope form of a line to write the equation of the tangent.
Note: The second part of the solution appears to involve simplifying terms, but additional context is required for complete clarity.
No graphs or diagrams are present in the image.
![### Evaluating the Derivative of \( y = (x^2 + 4)(x^3 + 6) \) in Two Methods
#### Method 1: Using the Product Rule
The first approach involves applying the Product Rule. The derivative is:
\[ y' = 5x^4 + 12x^2 + 12x \]
This expression is marked as correct.
#### Method 2: Expanding the Expression Before Differentiation
The second method involves expanding the expression before performing the differentiation:
\[ y' = 20x^3 + 24x + 4 \]
This result is indicated as incorrect.
#### Discussion
(c) Do your answers agree?
Through these methods, one can analyze the process of finding derivatives in different ways, understanding how the product rule can sometimes yield different forms and interpretations based on the step-by-step approach used.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e8b0010-4aed-4d57-b920-96d0bf9eca5d%2F2f5175d0-c1c6-483c-886a-8b9450577b03%2Fvodwf0l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Evaluating the Derivative of \( y = (x^2 + 4)(x^3 + 6) \) in Two Methods
#### Method 1: Using the Product Rule
The first approach involves applying the Product Rule. The derivative is:
\[ y' = 5x^4 + 12x^2 + 12x \]
This expression is marked as correct.
#### Method 2: Expanding the Expression Before Differentiation
The second method involves expanding the expression before performing the differentiation:
\[ y' = 20x^3 + 24x + 4 \]
This result is indicated as incorrect.
#### Discussion
(c) Do your answers agree?
Through these methods, one can analyze the process of finding derivatives in different ways, understanding how the product rule can sometimes yield different forms and interpretations based on the step-by-step approach used.
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