Calculus

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

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

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 whose derivative is given
by .
• What is ?
F
  4
5x
x
f 
 
x
F
 
x
F  
x
f
Solution
• We say that is an antiderivative of .
  5
F x x


6. 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 .
 
x
F
 
x
F  
x
f
   
x
f
x
F 
'
  3
5

 x
x
G   2
5

 x
x
H
 
x
g  
x
h

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.
dx
dy
x
2

xdx
dy 2


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. Solution
Differential Equations
• Solve
dx
dy
x
2

C
x
y 
 2

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 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
xdx
dy 2



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.
• Construct a slope field for .
y
x
dx
dy

