Antiderivatives, Differential Equations, and Slope Fields

AP Calculus AB Antiderivatives, Differential Equations, and Slope Fields Review • Consider the equation yx Solution 2 dy • Find  2x dx 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 • Consider the function by f x   5x 4. • What is F whose derivative is given F x ? F  x   x 5 Solution • We say that . F x is an antiderivative of f x  Antiderivatives • Notice that we say F x is an antiderivative and not the antiderivative. Why? • Since F x is an antiderivative of say that F ' x  f x .    5  f x , we can • If Gx   x  3 and H x   x  2, find 5 g x  and hx . Differential Equations dy • Recall the earlier equation .  2x 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  2x dx y  x2  C Solution Slope Fields • Consider the following: HippoCampus 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 • To construct a slope field, start with a differential equation. For simplicity’s sake we’ll use dy  2 xdx 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 • 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  x  y. • Construct a slope field for dx