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AP Calculus AB

- Antiderivatives,
- Differential Equations,
- and Slope Fields

Review

- Consider the equation

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

- Consider the function whose derivative is

given by . - What is ?

Solution

- We say that is an antiderivative of

.

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

- 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

- 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

- Invert both sidesin 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

Solution

Slope Fields

- Consider the following
- HippoCampus

Slope Fields

- A slope field shows the general flow of a

differential equations 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 simplicitys sake

well use Slope Fields - Rather than solving the differential equation,

well 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. - Construct a slope field for .

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