AP Calculus AB
and Slope Fields
Consider the equation
What is an inverse operation?
Addition and subtraction
Multiplication and division
Exponents and logariths
Differentiation also has an inverse…
Consider the function whose derivative is given by .
What is ?
We say that is an antiderivative of .
Notice that we say is an antiderivative and not the antiderivative. Why?
Since is an antiderivative of , we can
say that .
If and , find
Recall the earlier equation .
This is called a differential equation and could also be written as .
We can think of solving a differential equation as being similar to solving any other equation.
Trying to find y as a function of x
Can only find indefinite solutions
There are two basic steps to follow:
1. Isolate the differential
Invert both sides…in other words, find
Since we are only finding indefinite solutions, we must indicate the ambiguity of the constant.
Normally, this is done through using a letter to represent any constant. Generally, we use C.
Consider the following:
A slope field shows the general “flow” of a differential equation’s solution.