Estimation and Limits -- Tangents

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

In this module we look at two problems that are mathematically identical but might at first look different.

• Given information describing the location of an object as it travels, determine the object's velocity at various times.

• Given a curve, determine the slope of the tangent to the curve at a various points along the curve.

• Information about an object's location as it travels is often obtained by actual measurements of the object's location at different times.

• Information about a curve is often obtained by direct measurement.

• In very rare cases, usually in textbooks but sometimes when we are dealing with a theoretical situation rather than actual measurements, data is given to us in the form of a function

  y = f(t)

  that tells us, for example, the location f(t) of an object at time t.

• Geometric curves can often be described algebraically -- for example, a circle of radius R centered at the origin can be described by the equation

  x^2 + y^2 = R^2

We begin by looking at typical data for a falling object. The data below and in your CAS window was collected by dropping a pillow on the sonar range-finder device called a Motion Detector that is often used to collect distance data.

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

If your workstation is equipped with a TI-CBL or other data collection device you can work through this module with real data that you collect. Click on the icon below for more about data collection.

However the information is obtained it is often presented as a table like the table above or as a graph. For example, the graph below presents the same data as is shown in the table above.

This module can be used in either of two ways --

• as a discovery module in which you discover how to estimate the slope of a tangent to a curve.

• as a reference module describing how to estimate the slope of a tangent to a curve.