For the given function, (a) find the slope of the tangent line to the graph at the given point; (b) find the equation of the tangent line. \( g(x)=3 \sqrt{x} \) at \( x=1 \)

To solve the problem, we will follow these steps: 1. **Find the slope of the tangent line** at the given point by calculating the derivative of the function \( g(x) \) and evaluating it at \( x = 1 \). 2. **Find the equation of the tangent line** using the point-slope form of the equation of a line. ### Step 1: Find the slope of the tangent line The function is given as: \[ g(x) = 3 \sqrt{x} \] To find the slope of the tangent line at \( x = 1 \), we need to calculate the derivative \( g'(x) \). Let's calculate the derivative of \( g(x) \). The derivative of \( g(x) \) is: \[ g'(x) = \frac{d}{dx}(3 \sqrt{x}) = 3 \cdot \frac{1}{2\sqrt{x}} = \frac{3}{2\sqrt{x}} \] Now, we will evaluate the derivative at \( x = 1 \): \[ g'(1) = \frac{3}{2\sqrt{1}} = \frac{3}{2} \] So, the slope of the tangent line at \( x = 1 \) is \( \frac{3}{2} \). ### Step 2: Find the equation of the tangent line To find the equation of the tangent line, we will use the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope, and \( (x_1, y_1) \) is the point on the line. First, we need to find the value of \( g(1) \): \[ g(1) = 3 \sqrt{1} = 3 \] Thus, the point on the graph at \( x = 1 \) is \( (1, 3) \). Now we can substitute \( m = \frac{3}{2} \), \( x_1 = 1 \), and \( y_1 = 3 \) into the point-slope form: \[ y - 3 = \frac{3}{2}(x - 1) \] Now, we can simplify this equation to find the equation of the tangent line: \[ y - 3 = \frac{3}{2}x - \frac{3}{2} \] \[ y = \frac{3}{2}x - \frac{3}{2} + 3 \] \[ y = \frac{3}{2}x + \frac{3}{2} \] ### Final Answers (a) The slope of the tangent line at \( x = 1 \) is \( \frac{3}{2} \). (b) The equation of the tangent line is: \[ y = \frac{3}{2}x + \frac{3}{2} \]