6.1: Antiderivativesand Slope FieldsGreg Kelly, Hanford High School, Richland, Washington
First, a little review:Consider:23yx=+then:2yx′=2yx′=25yx=-orIt 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:2yx′=findy2yxC=+We don’t know what the constant is, so we put “C” in the answer to remind us that there might have been a constant.→
If we have some more information we can find C.Given: and when , find the equation for .2yx′=y4y=1x=2yxC=+241C=+3C=23yx=+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 conditionand asked to find the original equation.→
Initial value problems and differential equations can be illustrated with a slope field.