Imagine this much bacteria in a Petri dish

1. Imagine this much bacteria in a Petri dish Now this amount of the same bacteria Assuming that each bacterium would reproduce at the same rate, which dish will have a larger rate of growth? Answer: The second one simply because there are more of them. So if we presume that the rate of growth is given by: y = amount of bacteria, t = time And that the population produces at a rate proportional to itself with the proportion represented by the constant k

2. So if we presume that the rate of growth is given by: y = amount of bacteria, t = time And that the population produces at a rate proportional to itself with the proportion represented by the constant k Then Use your newfound skills for solving differential equations to solve for y here:

3. Since the initial amount is at t = 0 In this case, y0 is the initial amount So our equation for this type of growth would be…

4. Exponential Change: Remember too that we’ve just shown that is the solution to the differential equation If the constant k is positive then the equation represents growth. If k is negative then the equation represents decay. There is a similar growth equation used in finance that you may remember from pre-calc…and we’ll talk about that soon

5. One straight-forward application of this is Half-life is the period of time it takes for a substance undergoing decay to decrease by half. In this case, think of y0 as the initial amount of a substance undergoing decay. To find its half life…

6. Compounded Interest If money is invested in a fixed-interest account where the total interest r (which is a % written as a decimal) is broken into k equal portions and added to the account k times per year, the amount present after t years is: # times per year over t years % added each time 100% (initial investment) Initial investment If the interest is broken down more and added back more frequently (k is larger), you will make a little more money. We can add as many times as we want which means we can make k as large as… 

7. The larger k gets, the more times per year we compound the interest. If we can theoretically compound an infinite number of times, we say that the interest is compounded continuously We could calculate: Using an old limit from pre-calc

8. The larger k gets, the more times per year we compound the interest. If we can theoretically compound an infinite number of times, we say that the interest is compounded continuously We could calculate: which turns out to be: Continuously Compounded Interest: Just like the exponential growth model we just saw, the interest is directly proportional to the amount present. You may also use: Remember PERT? Same equation, different letters

9. Newton’s Law of Cooling Coffee left in a cup will cool to the temperature of the surrounding air. The rate of cooling is proportional to the difference in temperature between the liquid and the air. (The colder the air, the faster the coffee cools) This would give us the differential equation: where is the temperature of the surrounding medium, which is a constant.

10. Newton’s Law of Cooling Coffee left in a cup will cool to the temperature of the surrounding air. The rate of cooling is proportional to the difference in temperature between the liquid and the air. (The colder the air, the faster the coffee cools) If we solve the differential equation, we get: Newton’s Law of Cooling Don’t be afraid of the size of this equation. It really is not that different from the first exponential growth/decay equation. Don’t forget also that TS and T0 are constants. Just look at this comparison…

11. Newton’s Law of Cooling Coffee left in a cup will cool to the temperature of the surrounding air. The rate of cooling is proportional to the difference in temperature between the liquid and the air. (The colder the air, the faster the coffee cools) If we solve the differential equation, we get: Newton’s Law of Cooling It’s just a matter of sorting through the constants

12. The End