AP Calculus AB

Antiderivatives, Differential Equations,

and Slope Fields

Solution

Review

• Consider the equation

2xy

2xdy

dx• Find

Antiderivatives

• What is an inverse operation?

• Examples include:

Addition and subtractionMultiplication and divisionExponents and logarithms

Antiderivatives

• Differentiation also has an inverse…

antidefferentiation

Antiderivatives

• Consider the function whose derivative is given by .

• What is ?

F 45xxf

xF

xF xf

Solution

• We say that is an antiderivative of .

5F x x

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 .

xF

xF xf xfxF '

35 xxG 25 xxH

xg xh

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.

dx

dyx2

xdxdy 2

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.

Solution

Differential Equations

• Solve dx

dyx2

Cxy 2

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 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

xdxdy 2

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 .yx

dx

dy