(1 point) Approximate cos(4.6) using a quadratic approximat First note that cos(4.6) cos(3/2). Let f(x) = cos(x). Then, f'(x) = -sinx and f"(x) = -cosx Let a = 3x/2. Then f' (3/2) = -sin(3pi/2) and f" (3л/2) = -cos(3pi/2) Q(x), the quadratic approximation to cos(x) at a = 3/2 is Q(x) = cos(3pi/2) Use Q(x) to approximate cos(4.6). cos(4.6) cos(3pi/2)
(1 point) Approximate cos(4.6) using a quadratic approximat First note that cos(4.6) cos(3/2). Let f(x) = cos(x). Then, f'(x) = -sinx and f"(x) = -cosx Let a = 3x/2. Then f' (3/2) = -sin(3pi/2) and f" (3л/2) = -cos(3pi/2) Q(x), the quadratic approximation to cos(x) at a = 3/2 is Q(x) = cos(3pi/2) Use Q(x) to approximate cos(4.6). cos(4.6) cos(3pi/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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![**Quadratic Approximation of Cosine Function:**
To approximate \( \cos(4.6) \) using a quadratic approximation, follow these steps:
1. **Identify the Closest Known Value:**
First, note that \( \cos(4.6) \approx \cos(3\pi/2) \).
2. **Function and Derivatives:**
- Let \( f(x) = \cos(x) \). Then,
- \( f'(x) = -\sin(x) \)
- \( f''(x) = -\cos(x) \)
3. **Evaluate at \( a = 3\pi/2 \):**
- \( f(3\pi/2) = \cos(3\pi/2) \)
- \( f'(3\pi/2) = -\sin(3\pi/2) \)
- \( f''(3\pi/2) = -\cos(3\pi/2) \)
4. **Quadratic Approximation:**
The quadratic approximation \( Q(x) \) to \( \cos(x) \) at \( a = 3\pi/2 \) is:
\[
Q(x) = \cos(3\pi/2)
\]
5. **Approximate the Desired Value:**
- Use \( Q(x) \) to approximate \( \cos(4.6) \).
- Thus, \(\cos(4.6) \approx \cos(3\pi/2)\).
This method provides an approximate value of the cosine function near \( 4.6 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F58cffe43-d701-4c21-9740-08fe8d98ee79%2Fabd1291f-b677-4379-b354-0d0be7e1faf9%2Fcdw2u49_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Quadratic Approximation of Cosine Function:**
To approximate \( \cos(4.6) \) using a quadratic approximation, follow these steps:
1. **Identify the Closest Known Value:**
First, note that \( \cos(4.6) \approx \cos(3\pi/2) \).
2. **Function and Derivatives:**
- Let \( f(x) = \cos(x) \). Then,
- \( f'(x) = -\sin(x) \)
- \( f''(x) = -\cos(x) \)
3. **Evaluate at \( a = 3\pi/2 \):**
- \( f(3\pi/2) = \cos(3\pi/2) \)
- \( f'(3\pi/2) = -\sin(3\pi/2) \)
- \( f''(3\pi/2) = -\cos(3\pi/2) \)
4. **Quadratic Approximation:**
The quadratic approximation \( Q(x) \) to \( \cos(x) \) at \( a = 3\pi/2 \) is:
\[
Q(x) = \cos(3\pi/2)
\]
5. **Approximate the Desired Value:**
- Use \( Q(x) \) to approximate \( \cos(4.6) \).
- Thus, \(\cos(4.6) \approx \cos(3\pi/2)\).
This method provides an approximate value of the cosine function near \( 4.6 \).
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