Objectives:
1. to be able to find the rate of change from two points.
2. to be able to find the rate of change using a graph.
Definitions:
Rate of change: change in distance divided by change in time
Rate of change is a specific form of slope. So, if one can find the slope of a line, one should have little trouble fnding the rate of change. Just like slope, rate of change is written as a fraction so that one can observe the relationship between distance and time. This relationship is very hard to determine when one uses decimals instead of fractions to represent rate of change.
The equation for rate of change is analagous to that of slope. Instead of subtracting x and y-values, here we subtract distance and time values. In order to do that however, we treat distance as a y-coordinant and time as an x-coordinant. So now our ordered pairs look like (time, distance).
Now that we have ordered pairs, we modify the slope formula(which we learned on day 2) with the numerator being distance2 - distance1 and the denominator being time2 - time1. Problems would work like this.
Example1: A jet is parked at the terminal. Two hours later it has traveled 600 miles. What is the rate of change of the jet?
We need two ordered pairs of the form (time, distance). So, the first ordered pair comes from the plane being at the terminal. Here its time is zero because it hasn't left and its distance is zero because it hasn't moved. That gives us our first orded pair of (0,0). The second ordered pairs comes from the information from 2 hours later. Here time is 2 and distance is 600. This gives us the ordered pair (2,600). Now, apply the formula and find rate of change.
The formula is: (distance2- distance1)/(time2 - time1).
Substitution yields the expression: (600-0)/(2-0). Then use order of operations to get the fraction 600/2 which is 300 mph. Remember that rate of change will always have units involved.
Example 2: The Smith family drives through Europe on vacation. Four hours into their trip they had traveled 280 kilometers. Six hours later, they had traveled a total of 1000 kilometers. What is the rate of change?
Again find the two ordered pairs. The first is at the four hour point of the trip which gives us the point (4,280). The second is at the ten hour point of trip which gives us the point (10, 1000). Now apply the formula to get: (1000-280)/(10-4). Now subtract to get 720/6 which reduces to 120 km/h.
Rate of change is not always linear. This is where it is different from slope. Think about driving a car to illustrate this. The car starts and accelerates to a given speed, then it can slow down or accelerate depending upon the speed limit of the road. The car might have to stop at a stop sign or at a stop light. So, when graphing rate of change, one rarely gets a linear relationship. It is usually jagged. It could look something like the following where the x-axis shows time and the y-axis shows distance.
The graph above represents a rate of change and it is definitely not linear. It does however have four different parts that appear linear. We can now use the graph to find the five points in time that have been reprented. In a table it would look like the following:
| Time | 1 | 2 | 3 | 4 | 5 |
| Distance | 3 | 8 | 8 | 10 | 14 |
Now that we have the points, we can find the four different rates of change. The four different rates of change happen between the different times. So we can find a rate of change between hour 1 and hour 2, hour 2 and hour 3, hour 3 and hour 4, or hour 4 and hour 5.
Example3: The rate of change between hour 1 and hour 2.
Now that we have the points, it becomes a simple substitution. We initially get, (8 - 3)/(2 - 1). This reduces to 5/1 which is just 5 units per hour.
On your Own: Find the rates of change for the remaining portions of the graph.
Did you get 0, 2 and 4 units per hour? If not, check your work and find you mistake(s).
Now, you should be able to find a rate of change from a graph or from given points.