Use Newton's method to approximate a root of the equation cos(x² + 4) = x³ as follows. Let 1 = 2 be the initial approximation. The second approximation 2 is Submit Question
Use Newton's method to approximate a root of the equation cos(x² + 4) = x³ as follows. Let 1 = 2 be the initial approximation. The second approximation 2 is Submit Question
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter1: Trigonometry
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![### Using Newton's Method for Root Approximation
In this exercise, we will use Newton's method to approximate a root of the equation \(\cos(x^2 + 4) = x^3\). We'll follow the steps outlined below and begin with an initial approximation:
#### Given Equation:
\[ \cos(x^2 + 4) = x^3 \]
#### Step-by-Step Instructions:
1. **Initial Approximation:**
Let \( x_1 = 2 \) be the initial approximation.
2. **Newton's Method Formula:**
Newton's method is given by the iterative formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
3. **Define the Function \( f(x) \):**
Rearrange the given equation to define \( f(x) \):
\[ f(x) = \cos(x^2 + 4) - x^3 \]
4. **Compute the Derivative \( f'(x) \):**
\[ f'(x) = -\sin(x^2 + 4) \cdot 2x - 3x^2 \]
5. **Second Approximation:**
Using \( x_1 = 2 \), compute the second approximation \( x_2 \):
\[ x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} \]
Substitute and solve for \( x_2 \):
\[ x_2 = 2 - \frac{\cos(2^2 + 4) - 2^3}{-\sin(2^2 + 4) \cdot 2(2) - 3(2^2)} \]
#### Question Box:
Please compute the value of \( x_2 \) and enter your second approximation below:
\[ \text{The second approximation } x_2 \text{ is } \boxed{\phantom{0}} \]
#### Submission:
- Click the **Submit Question** button to save your response.
**Note:** Ensure your calculations are accurate to receive full credit for this exercise.
### Explanation of Graphs and Diagrams
There are no graphs or diagrams associated with this particular exercise.
By following these steps carefully, you can successfully approximate the root of the given equation using Newton's method.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf909276-6ad3-4144-b98c-235c5a32e436%2F04e0baf2-769c-4419-8701-66f1446df323%2Fyonsa49_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Using Newton's Method for Root Approximation
In this exercise, we will use Newton's method to approximate a root of the equation \(\cos(x^2 + 4) = x^3\). We'll follow the steps outlined below and begin with an initial approximation:
#### Given Equation:
\[ \cos(x^2 + 4) = x^3 \]
#### Step-by-Step Instructions:
1. **Initial Approximation:**
Let \( x_1 = 2 \) be the initial approximation.
2. **Newton's Method Formula:**
Newton's method is given by the iterative formula:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]
3. **Define the Function \( f(x) \):**
Rearrange the given equation to define \( f(x) \):
\[ f(x) = \cos(x^2 + 4) - x^3 \]
4. **Compute the Derivative \( f'(x) \):**
\[ f'(x) = -\sin(x^2 + 4) \cdot 2x - 3x^2 \]
5. **Second Approximation:**
Using \( x_1 = 2 \), compute the second approximation \( x_2 \):
\[ x_2 = x_1 - \frac{f(x_1)}{f'(x_1)} \]
Substitute and solve for \( x_2 \):
\[ x_2 = 2 - \frac{\cos(2^2 + 4) - 2^3}{-\sin(2^2 + 4) \cdot 2(2) - 3(2^2)} \]
#### Question Box:
Please compute the value of \( x_2 \) and enter your second approximation below:
\[ \text{The second approximation } x_2 \text{ is } \boxed{\phantom{0}} \]
#### Submission:
- Click the **Submit Question** button to save your response.
**Note:** Ensure your calculations are accurate to receive full credit for this exercise.
### Explanation of Graphs and Diagrams
There are no graphs or diagrams associated with this particular exercise.
By following these steps carefully, you can successfully approximate the root of the given equation using Newton's method.
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