Answers for Chapter 14 Sample Problems

2. The slope of a straight line between 2 points, is the (difference in the y's) divided by (the difference in the x's), sometimes called the rise/run. In the dot graph below, for the slope of the line between (3,3) and (0,1), we get (3-1)/(3-0) = 2/3 .

3.The two problems below were originally done by hand by a 6th grader.

You can see that he found, not surprisingly, that east-central Illinois is quite flat and the slope is 0! There are other nearby spots not as flat as the part he chose.  You could do this near your house or school.
Both of these figures were computer-enhanced, mainly for readability purposes. But it was done from a topographic map.

One of my calculus students was talking about the downward slope of her driveway into their garage, which made it very difficult to maneuver in the winter time. I'll bring up the slope of a roof (really 1/2 the slope), the gradient of a road (given in %), the pitch of a screw, etc.
5. Estimating slopes

6. Zooming in on a curve at a point, then finding the slope of the tangent at that point.
The table below shows various x-coordinates and the slope of the tangent, for the curve y = x2
x-coordinate     slope of tangent
        1                         2
        2                         4
        3                         6
Looking at the pattern, the slope of the tangent is 2x, which is the derivative of x2.
For the horizontal line say, y = 7 the slope is 0, for the straight line y = x, the slope is 1, so the derivative of x is 1. If you zoom in on points on the curve x3, you'll get 3x2. Putting these in the table below:
      function           derivative
        7                         0
        x                         1
        x2                       2x
        x3                       3x2
Look for a pattern in the table above. Can you predict the derivative of x4? of xn ? of 7x3 ? of 5x3 + 7x2 ?

7. Derivatives as done in textbooks.
8. Tickertape.
9. The slope of the tangent to the curve y = x3 and beyond!
10. Rectangles of constant perimeter of 20

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