## Sample Problems from Chapter 14

Chapter 14: Slopes and The Derivative

1. Slope, rate and ratios.

2. Linear graphs and their slopes.

3.Alec's slopes of mountains and stairs.
Use a topographic map to find the slope of your area of the country or the slope of a mountain and the slope of a ramp or the slope of the steps in your house.
4. Stories from graphs.
5. Estimating slopes:

6. Using the computer program Derive to zoom in on a curve at a point, then find the slope of the tangent at that point.
When you zoom in at the point, the curve approaches a straight line! Increase the number grid points so you can find the slope more easily.

On the right, going down, we zoom in on the curve y = x2 at the point (1,1).

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The second graph shows the same curve, again centered at (1,1), but the scale is X: 0.2, Y: 0.2 .
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`The third graph ends up 0.02 of a unit apart. Notice what happens. In this small portion `
`of the graph, the curve looks like a straight line. The slope of this line is essentially `
`the slope of the curve or the slope of the tangent to the curve at the point (1,1). `
`Using the dots on the screen, we can find the slope of this line, which is 2. `
`The slope of the tangent to the curve y = x2   at (1,1) is 2.`

Zoom in on other points like (2,4) and (3,9), and make a table like this:
x-coordinate     slope of tangent
1                         2
2
3

Find a rule for these pairs of numbers. Whatever you get will be the derivative of x2

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On the right we zoom in on the circle x2 + y2 = 25 at the point (3,4).

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Look at the third graph on the right. What's the slope of this 'line'? It looks like -3/4. The point was (3,4). What do you notice?

What would you predict would be the slope of the tangent to the circle at the point (4,3)? at (0,5)? Wow!

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Suppose you don't have a computer to "zoom in" on the curve x2, can you do this? SURE!

Zoom in on other points like (1,2) and (2,4), and some others, and make a table like this:
x-coordinate     slope of tangent
1
2
3                         6

Find a rule for these pairs of numbers.

7. Derivatives as done in most textbooks..not done here.
8. Tickertape -- application of the derivative..not done here.
9. Now let's find the slope of the tangent to the curve y = x3 ...not done here.
10. Rectangles of constant perimeter of 20

Ways young people have solved these kinds of problems
To choose sample problems from other chapters
To order Don's materials
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