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Average Rate of Change - Problem 2

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Calculating the average rate of change is much like calculating average velocity (in fact, finding average velocity is simply a special case of calculating the average rate of change). In order to determine the average rate of change, divide the difference between the quantities by the difference between the times at which those quantities are measured. The quantities can be determined by plugging in values of time into a given function. For instance, if the function is f(t)=4t^2+4, and you want to know the average rate of change from t=1 to t=3, evaluate f at each of these times. f(1)=4(1)^2+4=8, and f(3)=4(3)^2+4=40. Then, to find the rate of change, divide the differences between these points by the time interval: the average rate of change = [f(3)-f(1)]/(3-1) = (40-8)/2 = 32/2 = 16. So, the average rate of change would be 16. Remember that rate of change is the change of some quantity per unit of time.

I have another average rate of change problem. The amount A of t in milligrams of pain reliever in a patients system after t minutes is given by function A(t) equals 8t times e to the (minus t divided by 50). I want to compute the average rates of change of A(t) on the interval between zero and 50.

So remember the formula for average rate of change, it’s A(50) minus A(0). The final quantity minus the initial quantity over the change in time; 50 minus 0. So A (50) is going to be 8 times 50, e to the -50/50. Minus A(0) is 8 times 0, e to the -0/50, this is just going to be 0, all over 50.

Now this is just zero. So I’m left with 8 times 50, times something, over 50. The 50s cancel and I have 8 times e to the -1. I can approximate that on my calculator. 8e to the -1 is about 2.943. But what are the units? Remember the units of A are in milligrams and the units of time are in minutes. So this is going to be in milligrams per minute. This means that the average rate of increase of the medicine in the blood stream is 2.943 milligrams per minute. It’s going up.

Now let’s calculate the average rate of change on this interval between 50 and 150. So it’s A(150) minus A(50) over 150 minus 50. So A(150) is 8 times 150e to the -150/50. A(50) is 8 times 50e to the -50/50. This will turn out nice. 150 minus 50 is 100 and this simplifies.

You have a factor of 50 here and here, and also down here. And you have a factor of 2 here, here and also down here. This denominator, let me just rewrite it, is 2 times 50. So the 50s cancel, leaves a 3 here, and the 2s cancel leaving a 4 here and here. So we just have 1 in the denominator. This is 12 times e to the -3, minus 4 times e to the -1.

I’ll need my calculator to approximate this. 12e to the -3 minus 4e to the -1 equals, and I get -0.874. Just like before the units are milligrams per minute. This is really important. We have a negative answer here and that means that the amount of pain reliever in the system is decreasing at this point. Decreasing at the rate of 0.874 milligrams per minute.

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