Let's solve each part step by step: --- ### **a. Sketch a graph of the elevation \( y \) as a function of distance downriver \( x \).** **Table Data:** | Distances (miles) | 0 | 0.56 | 0.92 | 1.19 | 1.30 | 1.39 | 1.57 | 1.74 | 1.98 | 2.18 | 2.41 | 2.64 | 3.24 | |-------------------|------|------|------|------|------|------|------|------|------|------|------|------|------| | Elevation (feet) | 1577 | 1512 | 1448 | 1384 | 1319 | 1255 | 1191 | 1126 | 1062 | 998 | 933 | 869 | 805 | To **sketch the graph**, plot each pair \((x, y)\) on the axes. The points should show a decreasing trend since elevation drops as you move downriver. --- ### **b. Using the midpoint of each interval and the slope on the interval to create a table of the derivative relationship.** #### **Step 1: Find the midpoint of each interval.** The midpoint for each interval is the average of the two \( x \)-values: \[ \text{Midpoint} = \frac{x_1 + x_2}{2} \] #### **Step 2: Find the slope (rate of change) for each interval.** The slope (approximate derivative) is: \[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} \] #### **Step 3: Fill out the table.** Let's calculate the first few as an example, then show the rest in a table: | Interval | \( x_1 \) | \( x_2 \) | \( y_1 \) | \( y_2 \) | Midpoint (\( x \)) | Slope (\( \frac{\Delta y}{\Delta x} \)) | |----------|-----------|-----------|-----------|-----------|--------------------|-----------------------------------------| | 1 | 0 | 0.56 | 1577 | 1512 | 0.28 | \( \frac{1512-1577}{0.56-0} = \frac{-65}{0.56} \approx -116.1 \) | | 2 | 0.56 | 0.92 | 1512 | 1448 | 0.74 | \( \frac{1448-1512}{0.92-0.56} = \frac{-64}{0.36} \approx -177.8 \) | | 3 | 0.92 | 1.19 | 1448 | 1384 | 1.055 | \( \frac{1384-1448}{1.19-0.92} = \frac{-64}{0.27} \approx -237.0 \) | | 4 | 1.19 | 1.30 | 1384 | 1319 | 1.245 | \( \frac{1319-1384}{1.30-1.19} = \frac{-65}{0.11} \approx -590.9 \) | | ... | ... | ... | ... | ... | ... | ... | Let's do the rest quickly: | \( x_1 \) | \( x_2 \) | \( y_1 \) | \( y_2 \) | Midpoint (\( x \)) | Slope (\( \frac{\Delta y}{\Delta x} \)) | |-----------|-----------|-----------|-----------|--------------------|-----------------------------------------| | 1.30 | 1.39 | 1319 | 1255 | 1.345 | \( (1255-1319)/(1.39-1.30) = -64/0.09 \approx -711.1 \) | | 1.39 | 1.57 | 1255 | 1191 | 1.48 | \( (1191-1255)/(1.57-1.39) = -64/0.18 \approx -355.6 \) | | 1.57 | 1.74 | 1191 | 1126 | 1.655 | \( (1126-1191)/(1.74-1.57) = -65/0.17 \approx -382.4 \) | | 1.74 | 1.98 | 1126 | 1062 | 1.86 | \( (1062-1126)/(1.98-1.74) = -64/0.24 \approx -266.7 \) | | 1.98 | 2.18 | 1062 | 998 | 2.08 | \( (998-1062)/(2.18-1.98) = -64/0.20 \approx -320.0 \) | | 2.18 | 2.41 | 998 | 933 | 2.295 | \( (933-998)/(2.41-2.18) = -65/0.23 \approx -282.6 \) | | 2.41 | 2.64 | 933 | 869 | 2.525 | \( (869-933)/(2.64-2.41) = -64/0.23 \approx -278.3 \) | | 2.64 | 3.24 | 869 | 805 | 2.94 | \( (805-869)/(3.24-2.64) = -64/0.60 \approx -106.7 \) | **Unit of measure:** The derivative \( y' \) is the change in elevation per change in distance, so its unit is **feet per mile**. --- ### **c. Sketch a graph of the rate of change of elevation (\( y' \)) as a function of distance downriver (\( x \)).** Plot the midpoints of each interval on the \( x \)-axis, and the corresponding slopes (calculated above) on the \( y \)-axis. The graph should reflect how the rate of elevation drop changes along the route. --- ## **Summary Table for (b):** | Midpoint Distance (miles) | Slope (feet per 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 | -266.7 | | 2.08 | -320.0 | | 2.295 | -282.6 | | 2.525 | -278.3 | | 2.94 | -106.7 | --- ### **Final Answers** - **(a):** Plot the given data points (\( x, y \)) on a graph. - **(b):** Use the table above for midpoints and slopes. **Unit:** feet per mile. - **(c):** Plot the midpoints from (b) on \( x \)-axis, slopes on \( y \)-axis. This shows how quickly elevation drops at various points. If you need the graphs sketched or want a digital plot, let me know!

Let's go through each part step-by-step with detailed calculations, explanations, and final answers, without using tables. --- ### **Part (a): Sketch a graph of elevation \( y \) as a function of distance \( x \).** **Given Data:** | \( x \) (miles) | \( y \) (feet) | |----------------|----------------| | 0.00 | 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 | **Approach:** - Plot these points with \( x \) on the horizontal axis and \( y \) on the vertical axis. - The points will show a generally decreasing trend, indicating the elevation drops as you move downstream. - Connect the points smoothly to visualize how elevation decreases along the route. **Final note:** Since this is a rough sketch, just plot the points and connect them with a smooth curve to see the trend. --- ### **Part (b): Calculate the approximate derivative (rate of change of elevation) at midpoints of each interval.** **Step 1: Find the midpoints of each interval** For a segment between \( x_1 \) and \( x_2 \), the midpoint \( x_m \): \[ x_m = \frac{x_1 + x_2}{2} \] **Step 2: Calculate the slope between each pair of points** The slope (approximate derivative) between two points: \[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} \] --- **Calculations:** 1. For the segment from \( x=0.00 \) to \( x=0.56 \): - \( y_1 = 1577 \), \( y_2=1512 \) - Slope: \( \frac{1512 - 1577}{0.56 - 0.00} = \frac{-65}{0.56} \approx -116.07 \) feet per mile - Midpoint \( x_m = \frac{0+0.56}{2} = 0.28 \) miles --- 2. Between \( x=0.56 \) and \( x=0.92 \): - \( y_1=1512 \), \( y_2=1448 \) - Slope: \( \frac{1448 - 1512}{0.92 - 0.56} = \frac{-64}{0.36} \approx -177.78 \) - Midpoint: \( \frac{0.56 + 0.92}{2} = 0.74 \) miles --- 3. Between \( x=0.92 \) and \( x=1.19 \): - \( y_1=1448 \), \( y_2=1384 \) - Slope: \( \frac{-64}{0.27} \approx -237.04 \) - Midpoint: \( 1.055 \) miles --- 4. Between \( x=1.19 \) and \( x=1.30 \): - \( y_1=1384 \), \( y_2=1319 \) - Slope: \( \frac{-65}{0.11} \approx -590.91 \) - Midpoint: \( 1.245 \) miles --- 5. Between \( x=1.30 \) and \( x=1.39 \): - \( y_1=1319 \), \( y_2=1255 \) - Slope: \( \frac{-64}{0.09} \approx -711.11 \) - Midpoint: \( 1.345 \) miles --- 6. Between \( x=1.39 \) and \( x=1.57 \): - \( y_1=1255 \), \( y_2=1191 \) - Slope: \( \frac{-64}{0.18} \approx -355.56 \) - Midpoint: \( 1.48 \) miles --- 7. Between \( x=1.57 \) and \( x=1.74 \): - \( y_1=1191 \), \( y_2=1126 \) - Slope: \( \frac{-65}{0.17} \approx -382.35 \) - Midpoint: \( 1.655 \) miles --- 8. Between \( x=1.74 \) and \( x=1.98 \): - \( y_1=1126 \), \( y_2=1062 \) - Slope: \( \frac{-64}{0.24} \approx -266.67 \) - Midpoint: \( 1.86 \) miles --- 9. Between \( x=1.98 \) and \( x=2.18 \): - \( y_1=1062 \), \( y_2=998 \) - Slope: \( \frac{-64}{0.20} = -320.00 \) - Midpoint: \( 2.08 \) miles --- 10. Between \( x=2.18 \) and \( x=2.41 \): - \( y_1=998 \), \( y_2=933 \) - Slope: \( \frac{-65}{0.23} \approx -282.61 \) - Midpoint: \( 2.295 \) miles --- 11. Between \( x=2.41 \) and \( x=2.64 \): - \( y_1=933 \), \( y_2=869 \) - Slope: \( \frac{-64}{0.23} \approx -278.26 \) - Midpoint: \( 2.525 \) miles --- 12. Between \( x=2.64 \) and \( x=3.24 \): - \( y_1=869 \), \( y_2=805 \) - Slope: \( \frac{-64}{0.60} \approx -106.67 \) - Midpoint: \( 2.94 \) miles --- ### **Final Result for Part (b):** The approximate derivatives (rate of change of elevation) at midpoints are: - 0.28 miles: **-116.07 ft/mile** - 0.74 miles: **-177.78 ft/mile** - 1.055 miles: **-237.04 ft/mile** - 1.245 miles: **-590.91 ft/mile** - 1.345 miles: **-711.11 ft/mile** - 1.48 miles: **-355.56 ft/mile** - 1.655 miles: **-382.35 ft/mile** - 1.86 miles: **-266.67 ft/mile** - 2.08 miles: **-320.00 ft/mile** - 2.295 miles: **-282.61 ft/mile** - 2.525 miles: **-278.26 ft/mile** - 2.94 miles: **-106.67 ft/mile** --- ### **Part (c): Sketch a graph of the rate of change of elevation \( y' \) versus distance \( x \).** - Plot the midpoints on the \( x \)-axis. - Plot the corresponding slope values on the \( y \)-axis. - Connect these points smoothly to observe how the rate of elevation change varies along the route. --- ### **Summary:** - **(a)** Graph shows elevation decreasing from 1577 ft to 805 ft over about 3.24 miles. - **(b)** The approximate rate of elevation change (feet per mile) varies from about \(-711\) to \(-107\) ft/mile, indicating the steepest drop occurs around 1.3 miles. - **(c)** The graph of \( y' \) versus \( x \) illustrates how the slope (rate of elevation change) increases sharply near 1.3 miles and then decreases again. **Note:** The units for the derivative are **feet per mile**.

Certainly! Here's a detailed step-by-step calculation for part (b), including all the work and final results. --- ### **Part (b): Find the approximate rate of change of elevation (\( y' \)) at midpoints between each pair of points** **Given Data:** | \( x \) (miles) | \( y \) (feet) | |----------------|----------------| | 0.00 | 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 | --- ### **Step 1: Calculate the midpoints of each interval** For each interval between two points \((x_1, y_1)\) and \((x_2, y_2)\): \[ x_{\text{mid}} = \frac{x_1 + x_2}{2} \] --- ### **Step 2: Calculate the slope (approximate derivative) between each pair** The slope: \[ y' \approx \frac{y_2 - y_1}{x_2 - x_1} \] --- ### **Calculations for each interval:** --- #### **Interval 1:** between \( x=0.00 \) and \( x=0.56 \) - \( y_1 = 1577 \), \( y_2=1512 \) - \(\Delta y = 1512 - 1577 = -65\) - \(\Delta x = 0.56 - 0.00 = 0.56\) \[ \text{slope} = \frac{-65}{0.56} \approx -116.07 \text{ ft/mile} \] - Midpoint: \[ x_{\text{mid}} = \frac{0.00 + 0.56}{2} = 0.28 \text{ miles} \] --- #### **Interval 2:** between \( 0.56 \) and \( 0.92 \) - \( y_1=1512 \), \( y_2=1448 \) - \(\Delta y = -64\) - \(\Delta x= 0.92 - 0.56= 0.36\) \[ \text{slope} = \frac{-64}{0.36} \approx -177.78 \text{ ft/mile} \] - Midpoint: \[ x_{\text{mid}} = \frac{0.56 + 0.92}{2} = 0.74 \text{ miles} \] --- #### **Interval 3:** between \( 0.92 \) and \( 1.19 \) - \( y_1=1448 \), \( y_2=1384 \) - \(\Delta y = -64\) - \(\Delta x= 1.19 - 0.92= 0.27\) \[ \text{slope} = \frac{-64}{0.27} \approx -237.04 \text{ ft/mile} \] - Midpoint: \[ x_{\text{mid}}= \frac{0.92 + 1.19}{2} = 1.055 \text{ miles} \] --- #### **Interval 4:** between \( 1.19 \) and \( 1.30 \) - \( y_1=1384 \), \( y_2=1319 \) - \(\Delta y= -65\) - \(\Delta x= 0.11\) \[ \text{slope} = \frac{-65}{0.11} \approx -590.91 \text{ ft/mile} \] - Midpoint: \[ x_{\text{mid}}= \frac{1.19 + 1.30}{2} = 1.245 \text{ miles} \] --- #### **Interval 5:** between \( 1.30 \) and \( 1.39 \) - \( y_1=1319 \), \( y_2=1255 \) - \(\Delta y= -64\) - \(\Delta x= 0.09\) \[ \text{slope} = \frac{-64}{0.09} \approx -711.11 \text{ ft/mile} \] - Midpoint: \[ x_{\text{mid}}= \frac{1.30 + 1.39}{2} = 1.345 \text{ miles} \] --- #### **Interval 6:** between \( 1.39 \) and \( 1.57 \) - \( y_1=1255 \), \( y_2=1191 \) - \(\Delta y= -64\) - \(\Delta x= 0.18\) \[ \text{slope} = \frac{-64}{0.18} \approx -355.56 \text{ ft/mile} \] - Midpoint: \[ x_{\text{mid}}= \frac{1.39 + 1.57}{2} = 1.48 \text{ miles} \] --- #### **Interval 7:** between \( 1.57 \) and \( 1.74 \) - \( y_1=1191 \), \( y_2=1126 \) - \(\Delta y= -65\) - \(\Delta x= 0.17\) \[ \text{slope} = \frac{-65}{0.17} \approx -382.35 \text{ ft/mile} \] - Midpoint: \[ x_{\text{mid}}= \frac{1.57 + 1.74}{2} = 1.655 \text{ miles} \] --- #### **Interval 8:** between \( 1.74 \) and \( 1.98 \) - \( y_1=1126 \), \( y_2=1062 \) - \(\Delta y= -64\) - \(\Delta x= 0.24\) \[ \text{slope} = \frac{-64}{0.24} \approx -266.67 \text{ ft/mile} \] - Midpoint: \[ x_{\text{mid}}= \frac{1.74 + 1.98}{2} = 1.86 \text{ miles} \] --- #### **Interval 9:** between \( 1.98 \) and \( 2.18 \) - \( y_1=1062 \), \( y_2=998 \) - \(\Delta y= -64\) - \(\Delta x= 0.20\) \[ \text{slope} = \frac{-64}{0.20} = -320.00 \text{ ft/mile} \] - Midpoint: \[ x_{\text{mid}}= \frac{1.98 + 2.18}{2} = 2.08 \text{ miles} \] --- #### **Interval 10:** between \( 2.18 \) and \( 2.41 \) - \( y_1=998 \), \( y_2=933 \) - \(\Delta y= -65\) - \(\Delta x= 0.23\) \[ \text{slope} = \frac{-65}{0.23} \approx -282.61 \text{ ft/mile} \] - Midpoint: \[ x_{\text{mid}}= \frac{2.18 + 2.41}{2} = 2.295 \text{ miles} \] --- #### **Interval 11:** between \( 2.41 \) and \( 2.64 \) - \( y_1=933 \), \( y_2=869 \) - \(\Delta y= -64\) - \(\Delta x= 0.23\) \[ \text{slope} = \frac{-64}{0.23} \approx -278.26 \text{ ft/mile} \] - Midpoint: \[ x_{\text{mid}}= \frac{2.41 + 2.64}{2} = 2.525 \text{ miles} \] --- #### **Interval 12:** between \( 2.64 \) and \( 3.24 \) - \( y_1=869 \), \( y_2=805 \) - \(\Delta y= -64\) - \(\Delta x= 0.60\) \[ \text{slope} = \frac{-64}{0.60} \approx -106.67 \text{ ft/mile} \] - Midpoint: \[ x_{\text{mid}}= \frac{2.64 + 3.24}{2} = 2.94 \text{ miles} \] --- ### **### Final Answer:** | **Midpoint (miles)** | **Approximate rate of change \( y' \) (feet per mile)** | |----------------------|-----------------------------------------------------------| | 0.28 | **-116.07** | | 0.74 | **-177.78** | | 1.055 | **-237.04** | | 1.245 | **-590.91** | | 1.345 | **-711.11** | | 1.48 | **-355.56** | | 1.655 | **-382.35** | | 1.86 | **-266.67** | | 2.08 | **-320.00** | | 2.295 | **-282.61** | | 2.525 | **-278.26** | | 2.94 | **-106.67** | --- This detailed calculation provides the approximate slope (rate of change) at each midpoint, with units in **feet per mile**.