Estimation and Limits -- Tangents -- Reference

The graph below shows data collected when we dropped a pillow on a range-finding device. The table below the graph shows the same data. This data is also available in the CAS window that you opened when you entered this module. You can work through this module using this data or you can collect your own data using the TI-CBL.

Missing Graphic

Time  Height 

0.00     4.18803
0.02     4.18083
0.04     4.17722
0.06     4.15202
0.08     4.11241

0.10     4.07639
0.12     4.02958
0.14     3.96836
0.16     3.89634
0.18     3.81352

0.20     3.71629
0.22     3.61186
0.24     3.49662
0.26     3.36699
0.28     3.23015

0.30     3.0897
0.32     2.94566
0.34     2.77281
0.36     2.60716
0.38     2.43071

0.40     2.24706
0.42     2.0526
0.44     1.85454
0.46     1.64928
0.48     1.44762

Think of two problems at the same time.

Find the velocity of the pillow at the time t = 0.20
The data above came from from measurements made of the height of a pillow as it fell. They might, just as well, have been measurements made of a curved piece of metal at various points. Think of them in this way and find the slope of the tangent to this curve at the point t = 0.20

The figure below shows the tangent that we seek (in red).

Missing Graphic

One way to estimate the velocity of our falling pillow at time t = 0.20 is by computing the change in the pillow's height from time t = 0.20 to time t = 0.22 and then dividing this by the length of time this change took.

3.61186 - 3.71629
----------------- = -5.2215
       0.02

Notice the answer is negative because the pillow is falling and its height is decreasing.

Can you think of any other way to estimate the velocity of the pillow at time t = 0.20?

answer

Use your CAS window to draw a graph showing (estimated) velocity as a function of time.

Now we look at the second problem.

The data above came from from measurements made of the height of a pillow as it fell. They might, just as well, have been measurements made of a curved piece of metal at various points. Think of them in this way and find the slope of the tangent to this curve at the point t = 0.20

One way to estimate the slope of the tangent is by looking at the secant line going through the two points (0.20, 3.71629) and (0.22, 0.361186). The figure below shows this secant line (in green). Notice how close it is to the tangent (in red). Thus, its slope will be close to the slope of the tangent. The computations that we make for the slope of the this secant line are exactly the same as the computations we made earlier for our first estimate of the velocity of our falling pillow.

Missing graphic

More generally, we can estimate either the slope of the tangent to a curve y = f(t) at the point t = a or the velocity at time t = a of a moving object whiose location is given by the function y = f(t) by computing

f(a + h) - f(a)
---------------
        h

where h is a relatively small number. The exact answer is the limit of these approximations.

                           f(a + h) - f(a)
Slope of tangent =  Lim    ---------------
                  h --> 0          h


                   f(a + h) - f(a)
Velocity =  Lim    ---------------
          h --> 0          h

Check Your Understanding

For each of the following functions, f(t) think about two problems.

The function describes the location f(t) of a moving abject at time t and we want to find its velocity at the indicated time.
The function describes a curve and we want to find the slope of the tangent to the curve at the indicated point.

Find several estimates for the two answers (they are the same) and then try to find the exact answer.

f(t) = t^2 at the time (point) t = 2.
answer
f(t) = t^2 at the time (point) t = 3.
answer
f(t) = t^2 at the time (point) t = a.
answer
f(t) = 100 - 16t^2 at the time (point) t = 1.
answer
f(t) = 100 - 16t^2 at the time (point) t = 2.
answer
f(t) = 100 - 16t^2 at the time (point) t = a.
answer