**Transcription:**Write the equation of the line normal to g(x) at the point where x = 2.---**Explanation (for educational website):**To find the equation of the line normal to the function \( g(x) \) at a specific point where \( x = 2 \), you should follow these steps:1. **Find the Derivative**: Determine \( g'(x) \), the derivative of the function. This will give you the slope of the tangent line at any point \( x \).2. **Evaluate the Derivative at \( x = 2 \)**: Calculate \( g'(2) \) to find the slope of the tangent line at this point.3. **Find the Slope of the Normal Line**: The slope of the normal line is the negative reciprocal of the slope of the tangent line. So, if the slope of the tangent line is \( m \), the slope of the normal line is \( -1/m \).4. **Determine the Y-Coordinate**: Find the y-coordinate of the point on the curve where \( x = 2 \) by evaluating \( g(2) \).5. **Write the Equation of the Normal Line**: Use the point-slope form of the equation of a line: \[ y - y_1 = m_{\text{normal}}(x - x_1) \] where \( (x_1, y_1) \) is the point \((2, g(2))\), and \( m_{\text{normal}} \) is the slope of the normal line.By following these steps, you can successfully determine the equation of the normal line at \( x = 2 \). The table below provides values of the functions \( f(x) \) and \( g(x) \), along with their first and second derivatives, at different values of \( x \).| \( x \) | \( f(x) \) | \( f'(x) \) | \( f''(x) \) | \( g(x) \) | \( g'(x) \) | \( g''(x) \) ||---------|-----------|-------------|-------------|-----------|-------------|-------------|| 0 | -10 | 1 | 2 | -7 | 3 | -4 || 2 | -4 | 5 | 2 | -1 | 7 | 8 || 5 | 20 | 11 | 2 | 83 | 58 | 26 |### Explanation- **\( x \)**: The independent variable or input value.- **\( f(x) \)**: The output of function \( f \) for each \( x \).- **\( f'(x) \)**: The first derivative of \( f \), representing the rate of change of \( f \) at each \( x \).- **\( f''(x) \)**: The second derivative of \( f \), indicating the curvature or the change of the rate of change of \( f \) at each \( x \).- **\( g(x) \)**: The output of function \( g \) for each \( x \).- **\( g'(x) \)**: The first derivative of \( g \), representing the rate of change of \( g \) at each \( x \).- **\( g''(x) \)**: The second derivative of \( g \), indicating the curvature or the change of the rate of change of \( g \) at each \( x \).

Answered: Image /qna-images/answer/f179f387-838c-4267-bf46-32dfc6cd32b0.jpg

Answered: 1025 f(x) -10 -4 20 f'(x) 1 5 11 f" (x)…

1025 f(x) -10 -4 20 f'(x) 1 5 11 f" (x) 2 2 2 g(x) -7 -1 83 g'(x) 3 7 58 g"(x) -4 8 26

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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