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
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Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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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.

\[
\boxed{\phantom{00.00}} \quad \text{percent per unit change in } x
\]
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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