Complete the following steps for the given function, interval, and value of \( n \). a. Sketch the graph of the function on the given interval. b. Calculate \( \Delta x \) and the grid points \( x_{0}, x_{1}, \ldots, x_{n} \). c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles. d. Calculate the midpoint Riemann sum. \( f(x)=\frac{1}{x} \) on \( [1,6] ; n=5 \)

Let's go through the steps one by one for the function \( f(x) = \frac{1}{x} \) on the interval \([1, 6]\) with \( n = 5 \). ### Step a: Sketch the graph of the function on the given interval The function \( f(x) = \frac{1}{x} \) is a hyperbola that decreases as \( x \) increases. On the interval \([1, 6]\), the function values are: - \( f(1) = 1 \) - \( f(2) = 0.5 \) - \( f(3) \approx 0.333 \) - \( f(4) = 0.25 \) - \( f(5) = 0.2 \) - \( f(6) \approx 0.167 \) The graph will start at the point (1, 1) and decrease to (6, 0.167). ### Step b: Calculate \( \Delta x \) and the grid points \( x_{0}, x_{1}, \ldots, x_{n} \) 1. **Calculate \( \Delta x \)**: \[ \Delta x = \frac{b - a}{n} = \frac{6 - 1}{5} = 1 \] 2. **Calculate the grid points**: - \( x_0 = 1 \) - \( x_1 = x_0 + \Delta x = 1 + 1 = 2 \) - \( x_2 = x_1 + \Delta x = 2 + 1 = 3 \) - \( x_3 = x_2 + \Delta x = 3 + 1 = 4 \) - \( x_4 = x_3 + \Delta x = 4 + 1 = 5 \) - \( x_5 = x_4 + \Delta x = 5 + 1 = 6 \) Thus, the grid points are: - \( x_0 = 1 \) - \( x_1 = 2 \) - \( x_2 = 3 \) - \( x_3 = 4 \) - \( x_4 = 5 \) - \( x_5 = 6 \) ### Step c: Illustrate the midpoint Riemann sum by sketching the appropriate rectangles The midpoints for each subinterval are: - Midpoint for \([1, 2]\): \( m_0 = \frac{1 + 2}{2} = 1.5 \) - Midpoint for \([2, 3]\): \( m_1 = \frac{2 + 3}{2} = 2.5 \) - Midpoint for \([3, 4]\): \( m_2 = \frac{3 + 4}{2} = 3.5 \) - Midpoint for \([4, 5]\): \( m_3 = \frac{4 + 5}{2} = 4.5 \) - Midpoint for \([5, 6]\): \( m_4 = \frac{5 + 6}{2} = 5.5 \) The heights of the rectangles will be: - \( f(m_0) = f(1.5) = \frac{1}{1.5} \approx 0.667 \) - \( f(m_1) = f(2.5) = \frac{1}{2.5} = 0.4 \) - \( f(m_2) = f(3.5) = \frac{1}{3.5} \approx 0.286 \) - \( f(m_3) = f(4.5) = \frac{1}{4.5} \approx 0.222 \) - \( f(m_4) = f(5.5) = \frac{1}{5.5} \approx 0.182 \) ### Step d: Calculate the midpoint Riemann sum The midpoint Riemann sum \( M_n \) is given by: \[ M_n = \Delta x \cdot \left( f(m_0) + f(m_1) + f(m_2) + f(m_3) + f(m_4) \right) \] Substituting the values: \[ M_n = 1 \cdot \left( f(1.5) + f(2.5) + f(3.5) + f(4.5) + f(5.5) \right) \] \[ M_n = 1 \cdot \left( 0.667 + 0.4 + 0.286 + 0.222 + 0.182 \right) \] \[ M_n = 0.667 + 0.4 + 0.286 + 0.222 + 0.182 \] Calculating the sum: \[ M_n \approx 1.757 \] Thus, the midpoint Riemann sum is approximately \( 1.757 \).