Antiderivatives, differential equations, and slope fields
AP Calculus AB Antiderivatives,Differential Equations, and Slope Fields
Review 2• Consider the equation y x dy• Find 2x Solution 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 F whose derivative is given by f x 5x 4. 5 Solution• What is F x ? F x x• 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 f x , we can say that F x f x. 5 5• If Gx x 3 and H x x 2, find g x and hx .
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• Solve dy 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• Construct a slope field for x y. dx