1. Differential equations and Slope Fields Greg Kelly, Hanford High School, Richland, Washington

2. First, a little review: Consider: then: or It doesn’t matter whether the constant was 3 or -5, since when we take the derivative the constant disappears. However, when we try to reverse the operation: Given:findWe don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.

3. If we have some more information we can find C. Given: and when, find the equation for. This is called an initial value problem. We need the initial values to find the constant. An equation containing a derivative is called a differential equation. It becomes an initial value problem when you are given the initial condition and asked to find the original equation.

4. Initial value problems and differential equations can be illustrated with a slope field. Slope fields are mostly used as a learning tool and are mostly done on a computer or graphing calculator, but a recent AP test asked students to draw a simple one by hand.

5. Draw a segment with slope of 2. Draw a segment with slope of 0. Draw a segment with slope of 4. 000 010 00 00 2 3 10 2 112 204 0 -2 0-4

6. If you know an initial condition, such as (1,-2), you can sketch the curve. By following the slope field, you get a rough picture of what the curve looks like. In this case, it is a parabola.

7. Separable Differential Equations A separable differential equation can be expressed as the product of a function of x and a function of y. Example: Multiply both sides by dx and divide both sides by y 2 to separate the variables. (Assume y 2 is never zero.)

8. Separable Differential Equations A separable differential equation can be expressed as the product of a function of x and a function of y. Example:

9. Example 9: Separable differential equation Combined constants of integration

10. Example 9: We now have y as an implicit function of x. We can find y as an explicit function of x by taking the tangent of both sides. Notice that we can not factor out the constant C, because the distributive property does not work with tangent.

11. 6.4 Exponential Growth and Decay Greg Kelly, Hanford High School, Richland, Washington

12. The number of rabbits in a population increases at a rate that is proportional to the number of rabbits present (at least for awhile.) So does any population of living creatures. Other things that increase or decrease at a rate proportional to the amount present include radioactive material and money in an interest-bearing account. If the rate of change is proportional to the amount present, the change can be modeled by:

13. Rate of change is proportional to the amount present. Divide both sides by y. Integrate both sides.

14. Exponentiate both sides. When multiplying like bases, add exponents. So added exponents can be written as multiplication.

15. Exponentiate both sides. When multiplying like bases, add exponents. So added exponents can be written as multiplication. Since is a constant, let.

16. At,. This is the solution to our original initial value problem.

17. Exponential Change: If the constant k is positive then the equation represents growth. If k is negative then the equation represents decay.

18. Continuously Compounded Interest If money is invested in a fixed-interest account where the interest is added to the account k times per year, the amount present after t years is: If the money is added back more frequently, you will make a little more money. The best you can do is if the interest is added continuously.

19. Of course, the bank does not employ some clerk to continuously calculate your interest with an adding machine. We could calculate: but we won’t learn how to find this limit until chapter 8. Since the interest is proportional to the amount present, the equation becomes: Continuously Compounded Interest: You may also use: which is the same thing.