6. Consider the function \( f(x)=\frac{x}{x-3} \). Its graph is shown below. a. Find the exact value of the slope of the secant to the graph between \( x=-1 \) and \( x=2 \). b. Find the exact value of the slope of the secant to the graph between \( x=-1 \) and \( x=1 \). c. Find the exact value of the slope of the secant to the graph between \( x=-1 \) and \( x=0 \). d. Use the definition of the derivative to find the exact value of the slope of the tangent at \( x=-1 \). Note that you don't have to use the formula with \( x \) in it, you can immediately use \( x=-1 \).

To find the slope of the secant line between two points on the graph of the function \( f(x)=\frac{x}{x-3} \), we can use the formula for the slope of a secant line, which is given by \( m = \frac{f(b) - f(a)}{b - a} \). a. For the interval between \( x = -1 \) and \( x = 2 \): - Calculate \( f(-1) = \frac{-1}{-1-3} = \frac{-1}{-4} = \frac{1}{4} \). - Calculate \( f(2) = \frac{2}{2-3} = \frac{2}{-1} = -2 \). - Slope \( m = \frac{-2 - \frac{1}{4}}{2 - (-1)} = \frac{-2 - \frac{1}{4}}{3} = \frac{-\frac{8}{4} - \frac{1}{4}}{3} = \frac{-\frac{9}{4}}{3} = -\frac{3}{4} \). b. For the interval between \( x = -1 \) and \( x = 1 \): - Calculate \( f(1) = \frac{1}{1-3} = \frac{1}{-2} = -\frac{1}{2} \). - Slope \( m = \frac{-\frac{1}{2} - \frac{1}{4}}{1 - (-1)} = \frac{-\frac{1}{2} - \frac{1}{4}}{2} = \frac{-\frac{2}{4} - \frac{1}{4}}{2} = \frac{-\frac{3}{4}}{2} = -\frac{3}{8} \). c. For the interval between \( x = -1 \) and \( x = 0 \): - Calculate \( f(0) = \frac{0}{0 - 3} = 0 \). - Slope \( m = \frac{0 - \frac{1}{4}}{0 - (-1)} = \frac{-\frac{1}{4}}{1} = -\frac{1}{4} \). d. To find the slope of the tangent at \( x = -1 \), we use the definition of the derivative: \[ f'(-1) = \lim_{h \to 0} \frac{f(-1 + h) - f(-1)}{h} = \lim_{h \to 0} \frac{\frac{-1 + h}{-1 + h - 3} - \frac{1}{4}}{h}. \] Substituting \( f(-1) = \frac{1}{4} \) and solving the limit gives \( f'(-1) = -\frac{3}{4} \). So to sum up: - Slope between \( x=-1 \) and \( x=2 \): \(-\frac{3}{4}\) - Slope between \( x=-1 \) and \( x=1 \): \(-\frac{3}{8}\) - Slope between \( x=-1 \) and \( x=0 \): \(-\frac{1}{4}\) - Tangent slope at \( x=-1 \): \(-\frac{3}{4}\)