Find the percentage rate of change of f at the given value of x. (Round your answer to two decimal places.) x + 1 f(x) ; x = 2 X + x + 1 percent per unit change in x
Find the percentage rate of change of f at the given value of x. (Round your answer to two decimal places.) x + 1 f(x) ; x = 2 X + x + 1 percent per unit change in x
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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![**Calculus Lesson: Finding the Percentage Rate of Change**
**Problem:**
Find the percentage rate of change of \( f \) at the given value of \( x \). (Round your answer to two decimal places.)
\[ f(x) = \frac{x+1}{x^3 + x + 1}; \quad x = 2 \]
**Solution:**
1. **Differentiate \( f(x) \) with respect to \( x \).**
Given \( f(x) = \frac{x+1}{x^3 + x + 1} \), we use the quotient rule for differentiation:
\[
\text{If } f(x) = \frac{g(x)}{h(x)}, \text{ then } f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}
\]
2. **Evaluate \( f'(x) \) at \( x = 2 \).**
3. **Calculate the rate of change as a percentage.**
Percentage rate of change = \(\left( \frac{f'(2)}{f(2)} \right) \times 100 \)
**Graph/Diagram Explanation:**
The problem statement includes an equation and a particular value of \( x \). There is no accompanying graph or diagram in this problem.
**Result Field:**
After solving the differential equation and evaluating the expression, enter the result in the provided field.
\[
\boxed{\phantom{00.00}} \quad \text{percent per unit change in } x
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22542f52-fc18-4f9b-90f3-86db01220e87%2F93e15b3d-77c2-40be-a720-befe79e605ce%2Fzpaed4_processed.png&w=3840&q=75)
Transcribed Image Text:**Calculus Lesson: Finding the Percentage Rate of Change**
**Problem:**
Find the percentage rate of change of \( f \) at the given value of \( x \). (Round your answer to two decimal places.)
\[ f(x) = \frac{x+1}{x^3 + x + 1}; \quad x = 2 \]
**Solution:**
1. **Differentiate \( f(x) \) with respect to \( x \).**
Given \( f(x) = \frac{x+1}{x^3 + x + 1} \), we use the quotient rule for differentiation:
\[
\text{If } f(x) = \frac{g(x)}{h(x)}, \text{ then } f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}
\]
2. **Evaluate \( f'(x) \) at \( x = 2 \).**
3. **Calculate the rate of change as a percentage.**
Percentage rate of change = \(\left( \frac{f'(2)}{f(2)} \right) \times 100 \)
**Graph/Diagram Explanation:**
The problem statement includes an equation and a particular value of \( x \). There is no accompanying graph or diagram in this problem.
**Result Field:**
After solving the differential equation and evaluating the expression, enter the result in the provided field.
\[
\boxed{\phantom{00.00}} \quad \text{percent per unit change in } x
\]
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