AP Calculus AB: Antiderivatives, Differential Equations, and Slope Fields
Delve into the world of antiderivatives, differential equations, and slope fields in AP Calculus AB. Explore inverse operations, solving differential equations, and understanding antiderivatives as you deepen your calculus knowledge.
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AP Calculus AB Antiderivatives, Differential Equations, and Slope Fields
Review y = 2 x Consider the equation dy dx= 2x Find Solution
Antiderivatives What is an inverse operation? Examples include: Addition and subtraction Multiplication and division Exponents and logarithms
Antiderivatives Differentiation also has an inverse antidefferentiation
Antiderivatives F Consider the function whose derivative is given by . ( ) 5x x f = 4 ( ) x ( ) = 5 F x x F Solution What is ? ( ) x ( ) x f F We say that is an antiderivative of .
Antiderivatives ( ) x F Notice that we say is an antiderivative and not the antiderivative. Why? ( ) x F ( ) ( ) x f x F = ' ( ) 3 = x x G H ( ) x g ( ) x h ( ) x f Since is an antiderivative of , we can say that . ( ) x = x 5 5+ 2 If and , find and .
Differential Equations dy = Recall the earlier equation . 2 x dx This is called a differential equation and could also be written as . dy = 2 xdx We can think of solving a differential equation as being similar to solving any other equation.
Differential Equations Trying to find y as a function of x Can only find indefinite solutions
Differential Equations There are two basic steps to follow: 1. Isolate the differential 2. Invert both sides in other words, find the antiderivative
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.
Differential Equations dy Solve = 2 x dx = + 2 y x C Solution
Slope Fields Consider the following: HippoCampus
Slope Fields A slope fieldshows the general flow of a differential equation s solution. Often, slope fields are used in lieu of actually solving differential equations.
Slope Fields To construct a slope field, start with a differential equation. For simplicity s sake we ll use Slope 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 = 2 dy xdx
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. dy = + Construct a slope field for . y x dx
