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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.

Definition

- The anti-derivative of f(x) is the function F(x)

such that F' (x) f(x).

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

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.

Find the anti-derivative of

- x13
- x-5
- vx
- 1/x3

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.

Question?

- How are integrals and derivatives related to each

other? - How are limits and derivatives related to each

other?

But.

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

Power Rule for Integration

- These integrals are called indefinite integrals,

because the constant term is yet to be determined.

Properties of Integrals

- Constant multiple property
- Sum/difference property

Evaluate

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.

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.

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?

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