Let's address each part step by step: --- ### **a. Sketch a graph of the elevation (y) as a function of distance downriver (x).** - **X-axis:** Distance (miles) - **Y-axis:** Elevation (feet) Let's list the points to plot: ``` (0, 1577), (0.56, 1512), (0.92, 1448), (1.19, 1384), (1.30, 1319), (1.39, 1255), (1.57, 1191), (1.74, 1126), (1.98, 1062), (2.18, 998), (2.41, 933), (2.64, 869), (3.24, 805) ``` **Graph Sketch:** You would plot each point and connect them smoothly (the graph will be a decreasing curve). --- ### **b. Using the midpoint of each interval and the slope, create a table for the derivative.** **Derivative unit:** \[ \frac{\text{change in elevation (feet)}}{\text{change in distance (miles)}} = \frac{\text{feet}}{\text{mile}} \] This is the rate of change of elevation with respect to distance. #### **Steps:** 1. Compute the midpoint of each interval: \[ \text{Midpoint} = \frac{x_1 + x_2}{2} \] 2. Compute the slope (difference in y over difference in x) for each interval: \[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} \] #### **Table Construction:** Let's do the first few as an example: | Interval | Midpoint (miles) | Slope (feet/mile) | |--------------|------------------|--------------------------------------------| | 0–0.56 | (0 + 0.56)/2 = 0.28 | (1512 - 1577)/(0.56 - 0) = -65/0.56 ≈ **-116.1** | | 0.56–0.92 | (0.56 + 0.92)/2 = 0.74 | (1448 - 1512)/(0.92 - 0.56) = -64/0.36 ≈ **-177.8** | | 0.92–1.19 | (0.92 + 1.19)/2 = 1.055 | (1384 - 1448)/(1.19 - 0.92) = -64/0.27 ≈ **-237.0** | | 1.19–1.30 | (1.19 + 1.30)/2 = 1.245 | (1319 - 1384)/(1.30 - 1.19) = -65/0.11 ≈ **-590.9** | | 1.30–1.39 | (1.30 + 1.39)/2 = 1.345 | (1255 - 1319)/(1.39 - 1.30) = -64/0.09 ≈ **-711.1** | | 1.39–1.57 | (1.39 + 1.57)/2 = 1.48 | (1191 - 1255)/(1.57 - 1.39) = -64/0.18 ≈ **-355.6** | | ... | ... | ... | Continue this process for all intervals. --- ### **c. Sketch a graph of the rate of change of elevation (y) as a function of distance downriver (x).** - **X-axis:** Midpoint of each interval (miles) - **Y-axis:** Slope (feet per mile) Plot the midpoints (from part b) on the x-axis and the slopes on the y-axis. This graph will show how steeply the river drops at each section. --- ### **Summary Table for Part (b):** | Midpoint (miles) | Slope (feet/mile) | |------------------|------------------| | 0.28 | -116.1 | | 0.74 | -177.8 | | 1.055 | -237.0 | | 1.245 | -590.9 | | 1.345 | -711.1 | | 1.48 | -355.6 | | 1.655 | -382.4 | | 1.86 | -282.6 | | 2.08 | -320.0 | | 2.295 | -282.6 | | 2.525 | -278.8 | | 2.94 | -104.6 | --- If you need the full table or a specific graph, let me know!