# Calc06_1 - 6.1: Antiderivatives and Slope Fields Greg...

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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.

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