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, sincewhen we take the derivative the constant disappears.However, when we try to reverse the operation:Given:2yx′=findy2yxC=+We don’t know what theconstant is, so we put “C” inthe answer to remind us thatthere might have been aconstant.→
If we have some more information we can find C.Given:andwhen, find the equation for.2yx′=y4y=1x=2yxC=+241C=+3C=23yx=+This is called an initial valueproblem.We need the initialvalues to find the constant.An equation containing a derivative is called a differentialequation.It becomes an initial value problem when youare given the initial conditionand asked to find the originalequation.→
Initial value problems and differential equations can beillustrated with a slope field.