### Tangent Line Problem**Problem Statement:**Find the equation of the tangent line to the curve \( y = x + \tan x \) at the point \( (\pi, \pi) \).**Answer Options:**1. \( y = 3x - 2\pi \)2. \( y = 3x + 2\pi \)3. \( y = 2x - \pi \) 4. \( y = 2x + \pi \)**Detailed Explanation:**To find the equation of the tangent line to the curve \( y = x + \tan x \) at the given point, we follow these steps:1. **Find the Derivative:** The first derivative of the function \( y = x + \tan x \) with respect to \( x \) is given by: \[ \frac{dy}{dx} = \frac{d}{dx}(x) + \frac{d}{dx}(\tan x) = 1 + \sec^2 x \]2. **Evaluate the Derivative at \( x = \pi \):** \[ \frac{dy}{dx} \bigg|_{x = \pi} = 1 + \sec^2 (\pi) \] Since \( \sec(\pi) = -1 \), we have \( (\sec(\pi))^2 = 1\): \[ \frac{dy}{dx} \bigg|_{x = \pi} = 1 + 1 = 2 \] Therefore, the slope (m) of the tangent line at \( x = \pi \) is 2.3. **Point-Slope Form of the Tangent Line:** The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] Substituting \( m = 2 \) and the point \( (\pi, \pi) \): \[ y - \pi = 2(x - \pi) \] Simplifying this equation, we get: \[ y - \pi = 2x - 2\pi \] \[ y = 2x - \

Name: ### Tangent Line Problem**Problem Statement:**Find the equation of th
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: ### Tangent Line Problem**Problem Statement:**Find the equation of the tangent line to the curve \( y = x + \tan x \) at the point \( (\pi, \pi) \).**Answer Options:**1. \( y = 3x - 2\pi \)2. \( y = 3x + 2\pi \)3. \( y = 2x - \pi \) 4. \( y = 2x + \