April 11, 2007

Scribe Post


In class, we learned how to draw slope fields.

Let's review from our previous calculus class:

The order of a differential equation is the highest-order derivative.

y'=y is a first order differential equation.
y"=-y is a second order differential equation.

The solution to a differential equation is any function that fits the eqatuion.

For example, to find the differential equation to y'=y you find that you have to find a function that is its own derivative. We know that y=ex is a solution.

When we think about derivatives, we think about slopes! For every input x, f''(x) is the slope of f(x) at (x,y). This is a first order differential equation: when it describes the slopes at a specific point.

Example: y'=y is a differential equation. At any point (x,y), the slope of the solution curve is the same as the y coordinate. If we use the point (1,2), the slope of the curve is 2. Whenever the y coordinate is y=3, the slope will be 3.

We were given a handout with some examples. We did this in class and the solutions are in the slide post.

1) y'=y; you would use y=ex because it is its own derivative.

2) y'=2x; the parent function is x2

3) y'=-x/y;
x2 + y2 = k
2x + 2yy' = 0
y'= -x/y

*parent function is a circle

4) y'=x+y

For homework, we need to create a new program to create slope fields. You can find how to create it in 9.2 or you can ask a calculus buddy to send it to your calculator! Also do all the odd questions!

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