The instantaneous rate of change is the rate of change of a function at a certain time. If given the function values before, during, and after the required time, the instantaneous rate of change can be estimated. While estimates of the instantaneous rate of change can be found using values and times, an exact calculation requires using the derivative function. This rate of change is not the same as the average rate of change.
We want to talk about instantaneous rate of change, instantaneous rate of change is a lot like instantaneous velocity only it's a little more general. Let's take a look at a problem. A barrel of maple syrup is tapped at t=0. The total amount f of t of syrup that is poured out at time t is given by a table and here I have values 2, 4 and 6 for time and this is the amount of syrup that's leaked out by that time. t is equals to 35.83 gallons of leaked out, t equals 4, 54.03 and t equals 6, 68.11.
Now the idea behind instantaneous rate of change is the same as the idea behind instantaneous velocity I want to take average rates of change over a shorter and shorter increments of time. Here the increment of time is 2 seconds, so if I take an average rate of change over this increment from t equals 2 and t equals 4 I get 18.2 gallons over 2 minutes and that gives me 9.1 gallons per minute. So that's an average rate of change of the amount of syrup that's leaked out.
But if I do the same thing over this time interval fromt equals 4 to t equals 6 I get a different answer 14.08 that's the change in the amount of syrup that's leaked out over 2 minutes. I get 7.04 gallons per minute so a different value. Again this is a pretty large increment of time. I want to take smaller and smaller increments and see what values these average rates approach. Now here my increment of time delta t is 0.2 seconds, t equals 3.8, 4, 4.2 and I've calculated average rates here 54.03-52.45 that's 1.58 divided by the increment of time 0.2 and I get 7.90.
I make the same calculation from 4 to 4.2 and I get 7.70 these two values are getting very close together and that's the idea that as the increment of time gets smaller and smaller the average rates of change get closer to each other. Finally when my increment is as small as 0.02 right that's the difference between these values here and these values here. My average rate is the same to the nearest tenth 7.8 gallons per minute. So the idea behind average rate of change is as delta t approaches 0 that's the increment of time that you're averaging over if that approaches zero, the average rate of change approaches the instantaneous rate of change. And so in our example t equals 4 the instantaneous rate of change is this value that was approached 7.8 gallons per minute.