Use a second degree Taylor polynomial to approximate the two solutions to cos(x) = 3/4 in the interval (-"/2, "/2).
Use a second degree Taylor polynomial to approximate the two solutions to cos(x) = 3/4 in the interval (-"/2, "/2).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Taylor Polynomials! I would appreciate any assistance! The more explanation the better.
![**Problem Statement:**
Use a second-degree Taylor polynomial to approximate the two solutions to \(\cos(x) = \frac{3}{4}\) in the interval \((- \pi/2, \pi/2)\).
**Explanation:**
This problem involves finding approximate solutions for the equation \(\cos(x) = \frac{3}{4}\) using a second-degree Taylor polynomial. The task is restricted to finding the solutions within the interval from \(-\pi/2\) to \(\pi/2\).
*Taylor Polynomial Approach:*
1. **Taylor Polynomial**:
- A Taylor polynomial of degree 2 for a function \(f(x)\) about a point \(a\) is given by:
\[
f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2
\]
2. **Cosine Function Approximation**:
- To approximate \(\cos(x)\) near a point \(a\), apply the Taylor polynomial formula. For the cosine function, this would generally start from specific values of interest, such as \(x=0\).
3. **Interval Consideration**:
- Focus is on solutions within \((- \pi/2, \pi/2)\), which restricts the domain to ensure the solutions found are contained in this specified range.
This educational example demonstrates the use of polynomial approximations, such as Taylor series, to solve trigonometric equations numerically in specific intervals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7679a1-29bd-41dc-b06e-05edfd912e91%2F9777b0f6-f8a1-47a4-9146-e0fc67dd9a5f%2Fygln71d_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use a second-degree Taylor polynomial to approximate the two solutions to \(\cos(x) = \frac{3}{4}\) in the interval \((- \pi/2, \pi/2)\).
**Explanation:**
This problem involves finding approximate solutions for the equation \(\cos(x) = \frac{3}{4}\) using a second-degree Taylor polynomial. The task is restricted to finding the solutions within the interval from \(-\pi/2\) to \(\pi/2\).
*Taylor Polynomial Approach:*
1. **Taylor Polynomial**:
- A Taylor polynomial of degree 2 for a function \(f(x)\) about a point \(a\) is given by:
\[
f(x) \approx f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2
\]
2. **Cosine Function Approximation**:
- To approximate \(\cos(x)\) near a point \(a\), apply the Taylor polynomial formula. For the cosine function, this would generally start from specific values of interest, such as \(x=0\).
3. **Interval Consideration**:
- Focus is on solutions within \((- \pi/2, \pi/2)\), which restricts the domain to ensure the solutions found are contained in this specified range.
This educational example demonstrates the use of polynomial approximations, such as Taylor series, to solve trigonometric equations numerically in specific intervals.
Expert Solution
Step 1
Given that,
The function
We want to find two solutions of using second-degree Taylor polynomial
Take
We need to find Taylor's polynomial at x =0.
The Taylor series approximation is given by formula,
Step 2
f'(0) =-sin(0) =0
Thus the second-degree Taylors approximation is,
Step by step
Solved in 3 steps
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