**Problem Statement:**Find the equation of the tangent line to the graph of \( f(x) = 7 \sin(x) + 6 \cos(x) \) at \( x = \frac{\pi}{2} \) in the form \( y = mx + b \).**Solution:**To find the equation of the tangent line, we need to determine the slope of the tangent line at \( x = \frac{\pi}{2} \) and the point on the graph \( (x, f(x)) \).1. **Calculate \( f(x) \) at \( x = \frac{\pi}{2} \):** \[ f\left(\frac{\pi}{2}\right) = 7 \sin\left(\frac{\pi}{2}\right) + 6 \cos\left(\frac{\pi}{2}\right) \] \[ f\left(\frac{\pi}{2}\right) = 7(1) + 6(0) = 7 \] Thus, the point on the graph is \( \left(\frac{\pi}{2}, 7\right) \).2. **Calculate the derivative \( f'(x) \):** \[ f'(x) = \frac{d}{dx}[7 \sin(x) + 6 \cos(x)] = 7 \cos(x) - 6 \sin(x) \]3. **Determine the slope at \( x = \frac{\pi}{2} \):** \[ f'\left(\frac{\pi}{2}\right) = 7 \cos\left(\frac{\pi}{2}\right) - 6 \sin\left(\frac{\pi}{2}\right) \] \[ f'\left(\frac{\pi}{2}\right) = 7 (0) - 6 (1) = -6 \]4. **Equation of the tangent line:** Using the point-slope form of a line \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = \left(\frac{\pi}{2}, 7\right) \) and \( m = -6 \): \[ y - 7 = -6 \left(x

Name: **Problem Statement:**Find the equation of the tangent line to the gr
Uploaded: 2024-03-20T06:47:22.000Z
Channel: Bartleby
Description: **Problem Statement:**Find the equation of the tangent line to the graph of \( f(x) = 7 \sin(x) + 6 \cos(x) \) at \( x = \frac{\pi}{2} \) in the form \( y = mx + b \).**Solution:**To find the equation of the tangent line, we need to determine the sl