Chapter 2
Models of Growth: Rates of Change





2.1 Rates of Change

Problems

  1. Draw a line with each of the indicated slopes. Start each part by making a grid like the one at the right on your paper — or click on the figure to get a page you can print with four such grids.

    1. slope  `-2//3`

    2. slope `3//5`

    3. slope  `-3//4`

    4. slope  `-2`

 

For each of the lines in the `x`,`y`-plane in Problems 2-11:

    1. Find an equation of the line.
    2. Sketch the line. (If you need graph paper, click on the image at the right to get a page from which you can print your own.)
    3. Choose two `x` values, `x_1` and `x_2`, and calculate `Delta ytext(/)Delta x`. Then choose another pair of `x` values and calculate `Delta ytext(/)Delta x`.
  1. The line with slope `1.5` through the point `text[(]-1, 2.3text[)]`.

  2. The line through the points `text[(]2.1, 1.7text[)]` and `text[(]-1.5, 4.2text[)]`.

  3. The line with slope `1text(/)2` through the point `text[(]5, -2text[)]`.

  4. The line through the points `text[(]3, -7text[)]` and `text[(]1, -3text[)]`.

  5. The line with slope `-1.7` through the point `text[(]1.5, -3.2text[)]`.

  6. The line through the points `text[(]1.3, -0.7text[)]` and `text[(]1.5, -3.2text[)]`.

  7. The line through the point `text[(]12,14text[)]` that is parallel to the `x`-axis.

  8. The line through the point `text[(]5,4text[)]` that is parallel to the line `4x+5y=3.`

  9. The line through the point `text[(]5,4text[)]` that is parallel to the line through the points `text[(]-1,6text[)]` and `text[(]5,5text[)]`.

  10. The line through the point `text[(]-9,11text[)]` that is parallel to the line `2x-5y+7=0`.

  1. A common way to check the accuracy of your speedometer is to drive one or more measured miles, holding the speedometer at a constant 60 MPH and checking your watch at the start and end of the measured distance.

    1. Explain the method. How does your watch tell you whether the speedometer is accurate?

    2. Why shouldn't you use your odometer to measure the mile(s)?

    1. Make a table of the average rate of change of `y=x^2` over each of the following intervals: `[0,1]`, `[1,2]`, `[2,3]`, `[3,4]`.

    2. What pattern do you see? Explain this pattern by algebra.

    1. Make a table of the average rate of change of `y=x^3` over each of the following intervals: `[0,1]`, `[1,2]`, `[2,3]`, `[3,4]`.

    2. Can you find a pattern? Use algebra to confirm (or find) a pattern.
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