Instantaneous Rate of Change -Mark Sanders

 Mark Sanders

Scenario

Hunters have begun to hunt in an area heavily populated by deer. After several months, additional hunters arrive. Before the hunters arrived, there were 1100 deer in the area. Over the course of a year, the deer population drastically declines.

What is the instantaneous rate of change for the deer population during the fourth month?

Time(Months)	Deer Population
0	1100
1	1087
2	1058
3	1027
4	978
5	902
6	832
7	741
8	602
9	402
10	286
11	103
12	68

Average Rate of Change

Rate of Change of secant lines:

(1, 1087) to (4,978): (978-1087)/(4-1)= -36.33

(2, 1058) to (4,978): (978-1058)/(4-2)= -40

(3, 1027) to (4,978): (978-1027)/(4-3)=  -49

(4, 978) to (6, 832): (832-978)/(6-4)= -73

(4, 978) to (8, 602): (602-978)/(8-4)= -94

(4, 978) to (10, 286): (286-978)/(10-4)= -115.333333

The rate of change for each secant line is found by calculating the slope between two points. Starting at the ordered pair (4,978), I found the slope/average rate of change (ARC) for three secant lines. The secants lines I used all began at (4,978) and ended at one of the following points: (6,832), (8,602), (10,286). Also, several of the secant line I used began at the points (1,1087), (2,1058), (3,1027) and ended at (4,978).  By finding the slopes of each of the lines, I found the ARC for each of the respective intervals on which the secant lines belonged.

I noticed that the smaller the interval for each secant line, the closer the ARC is to the IRC, which I calculated in the next section. As the ARC is more closely focused on a section around the point (4,978) on either side, the ARC is closer to the IRC as the IRC only looks at one ordered pair. Therefore, the larger the interval/secant line, the larger and less focused the ARC will be then the IRC at that one particular ordered pair as there will be additional data involved. The smaller the interval/secant line is, the more focused the ARC will be on that one particular point and the closer the ARC will be to the IRC.

Rate of Change of Tangent Line (Instantaneous Rate of Change)

The tangent line cuts through the ordered pair (4,978). By taking another ordered pair on the tangent line, I can find the instantaneous rate of range at the fourth month by finding the slope of the line.

(Y2-Y1)/(X2-X1)= Slope or Instantaneous Rate of Change of tangent line.

(4,978) and (10,608) are both ordered pairs on the tangent line.

(608-978)/(10-4)= -61.67

-61.67 is the slope of the tangent line. This number is also the instantaneous rate of change.

Instantaneous Rate of Change/ Conclusion 

The instantaneous rate of change (IRC) at the point (4,978) is -61.67. The IRC is a measure of the rate of change at that particular ordered pair. This means that during the fourth month of the year, the deer population saw a total decrease/negative rate of change of approximately 62 deer. The deer population saw 62 deer killed by hunters in that year.

The value from part G is the IRC for the point (4,978) as it is equal to the slope  of the tangent line. I found the slope of the tangent line coming off at the particular ordered pair of (4,978). The tangent line only touches this point on the graph. Because of this reason, if I were to take the slope of the tangent line, it would only be considering the slope/IRC for that particular ordered pair as none of the other ordered pairs are influencing the tangent line directly. Also, the rate of change for each of the secant lines from part E get closer to the slope of the tangent line as the secant lines move closer to the original ordered pair.