Liam Instant rate of change blog

Introduction: You are sitting in a bath tub watching the water swirl down the drain and you begin to wonder how much water flows out of the drain at a given time. In order to figure out the instantaneous rate of change, you make a table of the amount of water in the bathtub from the time you pull the plug until it is completely drained. In order to find the Instantaneous rate of change of the water draining from the tub at t=20 (seconds), you must use the table and chart below.

Average Rate of Change: 

 1.         (20, 1111) to (30, 900)

               (900-1111)/(30-20) = -21.1

2.          (10, 1344) to (20, 1111)

               (1111-1344)/(20-10) = -23.3

3.          (20, 1111) to (50, 544)

               (544-1111)/(50-20) = -18.9

I used the slope formula to find the average rate of change between the points on my graph. Since I wanted to know what happened at t=20, I used the point (20,1111) and 3 other points around t=20. I calculated the ARC of the secant line between these points by calculating the slopes. I noticed that the secant lines got closer to 20 in slope as I used points to the right and left of t=20.

Instantaneous Rate of Change:

Based on the Secant line, I chose the points (20, 1111) and (40,711) to calculate the slope of the Secant.

(20,1111) to (40,711)

(711-1111)/ (40-20) = -20

To calculate the slope, (711-1111)/ (40-20). The slope between these points is -20.  This means that after the tub has been emptying for exactly 20 seconds, the rate of change -20 liters of water per second at that exact period in time. On my graph at the point t=20, the slope of the line at t=20 has a rate of change of -20.

I know this is correct because at that point of my graph, the slopes around it are getting closer and closer together, until they reach -20. It is safe to say that as I pick two points closer and closer to t=20, they will have slopes that are closer and closer to -20, and for this reason I can assume that the correct IRC is -20 liters per second. -23.3, -21.1, and -18.9 are all very close slopes to -20 and they are all slopes of points to the right and left of t=20 on my graph.