AP Calculus AB: AP Calculus AB Antiderivatives, Differential Equations, and Slope Fields
Review: Solution Review Consider the equation Find
Antiderivatives: Antiderivatives What is an inverse operation? Examples include: Addition and subtraction Multiplication and division Exponents and logarithms
Antiderivatives: Antiderivatives Differentiation also has an inverse… antidefferentiation
Antiderivatives: Antiderivatives Consider the function whose derivative is given by . What is ? Solution We say that is an antiderivative of .
Antiderivatives: 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 .
Differential Equations: 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.
Differential Equations: Differential Equations Trying to find y as a function of x Can only find indefinite solutions
Differential Equations: Differential Equations There are two basic steps to follow: 1. Isolate the differential Invert both sides…in other words, find the antiderivative
Differential Equations: 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: Solution Differential Equations Solve
Slope Fields: Slope Fields Consider the following: HippoCampus
Slope Fields: 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.
Slope Fields: 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
Slope Fields: 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 .