This week in AP Calc we discussed the applications of differential equations especially when they pertain to exponential growth. The law of exponential change is as follows:
dy/dx = k * y (k is the growth constant)
When you solve this equation for y it becomes in terms of the growth constant and the equation will end up:
ln(y) = kt + C therefore, --------> y = e^ln(y)
(because you are taking the anti-derivative and do not know the constant)
y = e^ln(y) = e^(k * t) + C = e^(k * t) * e^(C)
Because the A value will be equal to e^(C), you can substitute it in giving you the equation for exponential change solved for y:
y(t) = Ae^(k * t)
And, because y(0) = A, you can deduce that the differential equation will be:
y(t) = y(0)e^(k * t)
which is also true for radioactive decay, however, since the number would be consistently decreasing the equation would look like this:
y(t) = y(0)e^(-k * t)
So, for example, say a group of bacteria starts with 500 bacteria and after 3 hours there are 8,000 bacteria (implying that the exponential change is positive since the bacteria are growing).
We know that y(0) = 500 because we started with 500 bacteria
We also know that (t) = 3 hours because that is the time duration
And, we know that y(t) = 8,000 because after the duration of time that is the amount of bacteria present.
So, we can transform y(t) = y(0)e^(k * t) into 8,000 = 500e^(k * 3).
To find the growth constant we must solve for (k).
8,000/500 = e^(k * 3) ---> 80/5 = e^(k * 3) ---> ln(80/5) = 3k ---> k = (ln(80/5))/3
Once the model is developed, it is easy to find when a population of bacteria will reach a certain number of bacteria or how many will be present after a certain amount of time. The most crucial part is finding the correct model which was challenging at first, but once we went through how to get the exponential change and radioactive decay models in class, it made a lot of sense. One thing I always have to remind myself is to not forget the importance of the (C) value when you are taking the anti-derivative because it comes into play when developing the model for exponential change.
Click this link for more information or watch the video below.
dy/dx = k * y (k is the growth constant)
When you solve this equation for y it becomes in terms of the growth constant and the equation will end up:
ln(y) = kt + C therefore, --------> y = e^ln(y)
(because you are taking the anti-derivative and do not know the constant)
y = e^ln(y) = e^(k * t) + C = e^(k * t) * e^(C)
Because the A value will be equal to e^(C), you can substitute it in giving you the equation for exponential change solved for y:
y(t) = Ae^(k * t)
And, because y(0) = A, you can deduce that the differential equation will be:
y(t) = y(0)e^(k * t)
which is also true for radioactive decay, however, since the number would be consistently decreasing the equation would look like this:
y(t) = y(0)e^(-k * t)
So, for example, say a group of bacteria starts with 500 bacteria and after 3 hours there are 8,000 bacteria (implying that the exponential change is positive since the bacteria are growing).
We know that y(0) = 500 because we started with 500 bacteria
We also know that (t) = 3 hours because that is the time duration
And, we know that y(t) = 8,000 because after the duration of time that is the amount of bacteria present.
So, we can transform y(t) = y(0)e^(k * t) into 8,000 = 500e^(k * 3).
To find the growth constant we must solve for (k).
8,000/500 = e^(k * 3) ---> 80/5 = e^(k * 3) ---> ln(80/5) = 3k ---> k = (ln(80/5))/3
Once the model is developed, it is easy to find when a population of bacteria will reach a certain number of bacteria or how many will be present after a certain amount of time. The most crucial part is finding the correct model which was challenging at first, but once we went through how to get the exponential change and radioactive decay models in class, it made a lot of sense. One thing I always have to remind myself is to not forget the importance of the (C) value when you are taking the anti-derivative because it comes into play when developing the model for exponential change.
Click this link for more information or watch the video below.