**Problem Statement:**Find the linearization of \( y = (1 + 7x)^{-1/2} \) at \( x = 5 \).(Use symbolic notation and fractions where needed.)\[ L(x) = \]**Solution Explanation:**To find the linearization of the function \( y = (1 + 7x)^{-1/2} \) at \( x = 5 \), follow these steps:1. **Identify the function and point of linearization**: \[ f(x) = (1 + 7x)^{-1/2} \] at \( x_0 = 5 \).2. **Calculate \( f(x_0) \)**: Substitute \( x = 5 \) into \( f(x) \): \[ f(5) = (1 + 7 \cdot 5)^{-1/2} = (1 + 35)^{-1/2} = 36^{-1/2} = \frac{1}{6} \]3. **Determine \( f'(x) \)**: Calculate the derivative \( f'(x) \). Using the chain rule: \[ f(x) = (1 + 7x)^{-1/2} \implies f'(x) = -\frac{1}{2} (1 + 7x)^{-3/2} \cdot 7 = -\frac{7}{2} (1 + 7x)^{-3/2} \] Substitute \( x = 5 \) into \( f'(x) \): \[ f'(5) = -\frac{7}{2} (1 + 35)^{-3/2} = -\frac{7}{2} \cdot 36^{-3/2} \] Since \( 36^{3/2} = (36^{1/2})^3 = 6^3 = 216 \): \[ f'(5) = -\frac{7}{2} \cdot \frac{1}{216} = -\frac{7}{432} \]4. **Form the linearization**: The linear approximation or linearization \( L(x) \) around \( x = x_0 \) is given by

Name: **Problem Statement:**Find the linearization of \( y = (1 + 7x)^{-1/2
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: **Problem Statement:**Find the linearization of \( y = (1 + 7x)^{-1/2} \) at \( x = 5 \).(Use symbolic notation and fractions where needed.)\[ L(x) = \]**Solution Explanation:**To find the linearization of the function \( y = (1 + 7x)^{-1/2} \) at \