1. AP Calculus AB
Antiderivatives,
Differential Equations,
and Slope Fields

2. Review
2
• Consider the equation y x
dy
• Find 2x Solution
dx

3. Antiderivatives
• What is an inverse operation?
• Examples include:
Addition and subtraction
Multiplication and division
Exponents and logarithms

4. Antiderivatives
• Differentiation also has an inverse…
antidefferentiation

5. 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 .

6. 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 .

7. 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.

8. Differential Equations
• Trying to find y as a function of x
• Can only find indefinite solutions

9. Differential Equations
• There are two basic steps to follow:
1. Isolate the differential
2. Invert both sides…in other words, find
the antiderivative

10. 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.

11. Differential Equations
• Solve dy
2x
dx
y x2 C Solution

12. Slope Fields
• Consider the following:
HippoCampus

13. 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.

14. 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

15. 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