Calc06_1 - 6.1 Antiderivatives and Slope Fields Greg Kelly...

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6.1: Antiderivatives and Slope Fields Greg Kelly, Hanford High School, Richland, Washington
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First, a little review: Consider: 2 3 y x = + then: 2 y x = 2 y x = 2 5 y x = - or It 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: 2 y x = find y 2 y x C = + 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.
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If we have some more information we can find C. Given: and when , find the equation for . 2 y x = y 4 y = 1 x = 2 y x C = + 2 4 1 C = + 3 C = 2 3 y x = + 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 condition and asked to find the original equation.
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Initial value problems and differential equations can be illustrated with a slope field.
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