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 4: Linearization for Estimating Roots**
**Objective:**
Use linearization to give an estimate for the fourth root of 264, denoted as \(\sqrt[4]{264}\).
**Approach:**
Linearization involves approximating a complex function with a linear function (tangent line) that is easier to evaluate. This is particularly useful for estimating values close to known points.
1. **Identify the Function and Point of Tangency:**
- Function: \( f(x) = x^{1/4} \)
- Choose a nearby point where the fourth root is simple to calculate, such as \( x = 256 \), since \( \sqrt[4]{256} = 4 \).
2. **Calculate the Derivative:**
- Derivative of \( f(x) \): \( f'(x) = \frac{1}{4}x^{-3/4} \)
3. **Evaluate the Derivative at the Chosen Point:**
- \( f'(256) = \frac{1}{4}(256)^{-3/4} \)
- Simplifying gives \( f'(256) = \frac{1}{4 \times 64} = \frac{1}{256} \)
4. **Construct the Linear Approximation:**
- Formula: \( L(x) = f(a) + f'(a)(x - a) \)
- Here, \( L(x) = 4 + \frac{1}{256}(x - 256) \)
5. **Estimate \( \sqrt[4]{264} \) Using Linear Approximation:**
- Substitute \( x = 264 \) into the linear equation.
- \( L(264) = 4 + \frac{1}{256}(264 - 256) \)
- \( L(264) = 4 + \frac{1}{256} \times 8 \)
- \( L(264) = 4 + \frac{8}{256} = 4 + \frac{1}{32} \)
- Result: \( L(264) \approx 4.03125 \)
**Conclusion:**
The linear approximation suggests that \(\sqrt[4]{264} \approx 4.03125\). This method provides a quick way of estimating values using calculus and is useful for small deviations from known](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf00542e-b5b8-4896-9f9f-a2b46551c658%2F60eedac1-7e7d-4cda-8798-56c9ddb2dda1%2Fvtfzkq4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 4: Linearization for Estimating Roots**
**Objective:**
Use linearization to give an estimate for the fourth root of 264, denoted as \(\sqrt[4]{264}\).
**Approach:**
Linearization involves approximating a complex function with a linear function (tangent line) that is easier to evaluate. This is particularly useful for estimating values close to known points.
1. **Identify the Function and Point of Tangency:**
- Function: \( f(x) = x^{1/4} \)
- Choose a nearby point where the fourth root is simple to calculate, such as \( x = 256 \), since \( \sqrt[4]{256} = 4 \).
2. **Calculate the Derivative:**
- Derivative of \( f(x) \): \( f'(x) = \frac{1}{4}x^{-3/4} \)
3. **Evaluate the Derivative at the Chosen Point:**
- \( f'(256) = \frac{1}{4}(256)^{-3/4} \)
- Simplifying gives \( f'(256) = \frac{1}{4 \times 64} = \frac{1}{256} \)
4. **Construct the Linear Approximation:**
- Formula: \( L(x) = f(a) + f'(a)(x - a) \)
- Here, \( L(x) = 4 + \frac{1}{256}(x - 256) \)
5. **Estimate \( \sqrt[4]{264} \) Using Linear Approximation:**
- Substitute \( x = 264 \) into the linear equation.
- \( L(264) = 4 + \frac{1}{256}(264 - 256) \)
- \( L(264) = 4 + \frac{1}{256} \times 8 \)
- \( L(264) = 4 + \frac{8}{256} = 4 + \frac{1}{32} \)
- Result: \( L(264) \approx 4.03125 \)
**Conclusion:**
The linear approximation suggests that \(\sqrt[4]{264} \approx 4.03125\). This method provides a quick way of estimating values using calculus and is useful for small deviations from known
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