Transcript Antiderivatives, Differential Equations, and

AP Calculus AB
Antiderivatives,
Differential Equations,
and Slope Fields
Review
• Consider the equation
dy
• Find
 2x
dx
yx
Solution
2
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
• We say that
5
Solution
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 .
• If

f x , we can
Gx   x  3 and H x   x  2, find
5
g x  and hx  .
5
Differential Equations
dy
• Recall the earlier equation
 2 x.
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