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AP Calculus AB Antiderivatives, Differential Equations, and Slope Fields
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Solution Review Consider the equation Find
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Antiderivatives What is an inverse operation? Examples include: Addition and subtraction Multiplication and division Exponents and logarithms
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Antiderivatives Differentiation also has an inverse… antidefferentiation
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Antiderivatives Consider the function whose derivative is given by. What is ? Solution We say that is an antiderivative of.
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Antiderivatives Notice that we say is an antiderivative and not the antiderivative. Why? Since is an antiderivative of, we can say that. If and, find and.
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Differential Equations 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.
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Differential Equations Trying to find y as a function of x Can only find indefinite solutions
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Differential Equations There are two basic steps to follow: 1. Isolate the differential 2.Invert both sides…in other words, find the antiderivative
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Differential Equations 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.
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Solution Differential Equations Solve
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Slope Fields Consider the following: HippoCampus
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Slope Fields A slope field shows the general “flow” of a differential equation’s solution. Often, slope fields are used in lieu of actually solving differential equations.
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Slope Fields To construct a slope field, start with a differential equation. For simplicity’s sake we’ll use Slope FieldsSlope Fields Rather than solving the differential equation, we’ll construct a slope field Pick points in the coordinate plane Plug in the x and y values The result is the slope of the tangent line at that point
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Slope Fields Notice that since there is no y in our equation, horizontal rows all contain parallel segments. The same would be true for vertical columns if there were no x. Construct a slope field for.
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