Learning Resources

Example 1

An object in free fall from the top of a building can be described by the function . If a marble is dropped from the top of the CN Tower, 450 meters above the ground, how fast is the marble traveling at 5 seconds?

Solution

To find the instantaneous rate at 5 seconds, we require the slope of the tangent at that point to the graph of the function. Using technology we have the following graph with an approximate tangent drawn at the point where x = 5  

Note: To obtain the graph above, the window used included only a domain with positive values since time is considered to be a positive quantity.

Since we need to analyze the graph near the point where x = 5 , the following table from the TI-83 will be helpful in determining the various slopes for the secants.  

If we use the points (5.1,127.45) and (5,122.5) for a secant we have the following slope

Therefore the average rate is 49.5 m/s

To obtain a better result use values that are closer again to x = 5 . Use the points (5.01,122.99) and (5,122.5) and calculate the slope.

Therefore the average rate is 49 m/s. From this we can conclude that the slope of the tangent line is 49 m/s or the marble is falling at the rate of 49 m/ sec at 5 seconds.