AP Calculus AB

1
AP Calculus AB

• Antiderivatives,
• Differential Equations,
• and Slope Fields

2
Review

• Consider the equation

Solution
3
Antiderivatives

• What is an inverse operation?

• Examples include

• Addition and subtraction

• Multiplication and division

• Exponents and logarithms

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Antiderivatives

• Differentiation also has an inverse

antidefferentiation
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Antiderivatives

• Consider the function whose derivative is
  given by .
• What is ?

Solution

• We say that is an antiderivative of
  .

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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 .

7
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.

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Differential Equations

• Trying to find y as a function of x

• Can only find indefinite solutions

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Differential Equations

• There are two basic steps to follow

1. Isolate the differential

• Invert both sidesin other words, find
• the antiderivative

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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.

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Differential Equations

• Solve

Solution
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Slope Fields

• Consider the following
• HippoCampus

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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.

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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

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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 .