AntiDerivatives

1
Anti-Derivatives

• Is it possible, knowing the derivative of a
  function, to work backwards and determine the
  function?
• This process of converting a derivative back to
  the original function is called finding the
  anti-derivative, or anti-differentiation.

2
Definition

• The anti-derivative of f(x) is the function F(x)
  such that F' (x) f(x).

3
Find the anti-derivative of

• f(x) 0
• f(x) c (constant)
• f(x) c (a constant)
• f(x) cx (linear)
• f(x) x
• f(x) x2/2
• f(x) x2
• f(x) x3/3
• f(x) x3
• f(x) x4/4

4
Power Rule for Anti-Derivatives

• The anti-derivative of xn is xn 1
• n
  1
• where n ? -1.
• Notice how this is related to the Power Rule for
  derivatives.

5
Find the anti-derivative of

• x13
• x-5
• vx
• 1/x3

6
New Name

• The symbol used to indicate finding an
  anti-derivative is called the integral and is
  denoted as
• The process of evaluating an integral is called
  integration.

7
Question?

• How are integrals and derivatives related to each
  other?
• How are limits and derivatives related to each
  other?

8
But.

• Dx x2 2x
• Dx (x2 4) 2x
• Dx (x2 - 11) 2x
• So what is the anti-derivative of 2x???

9
Power Rule for Integration

• These integrals are called indefinite integrals,
  because the constant term is yet to be determined.

10
Properties of Integrals

• Constant multiple property
• Sum/difference property

11
Evaluate
12
Determining the C value

• Find the equation of the curve passing
• through (2, -3) if the slope of the curve is
• given by dy/dx 2x3 6x 5.
• Use the integral to convert dy/dx to y f(x).
• Substitute f(2) -3 into the equation to
  determine the value of C.

13
Marginal Cost to Cost Function

• A companys marginal cost is
• M 21x4/3 6x1/2 50, where x is the
• number of units produced and fixed costs
• are 3000. Find the cost function.

14
Rate of change to actual value

• A 20,000 art collection is increases in
• value at a rate of 300vt dollars per year
• after t years. What is the value of the
• collection after 25 years?