Antiderivatives

We will also learn some applications. Think About It f '(x) Suppose this is the graph of the derivative of a function What do we know about the original function? Critical numbers Where it is increasing, decreasing What do we not know? In this chapter we will learn how to determine the original function when given the derivative. We will also learn some applications.

Anti-Derivatives Derivatives give us the rate of change of a function What if we know the rate of change … Can we find the original function? If f '(x) = f(x) Then f(x) is an antiderivative of f’(x) Example – let f(x) = 12x2 Then f '(x) = 24x So f(x) = 12x2 is the antiderivative of f’(x) = 24x

Finding An Antiderivative Given f(x) = 12x3 What is the antiderivative, f(x)? Use the power rule backwards Recall that for f(x) = xn … f '(x) = n • x n – 1 That is … Multiply the expression by the exponent Decrease exponent by 1 Now do opposite (in opposite order) Increase exponent by 1 Divide expression by new exponent

Family of Antiderivatives Consider a family of parabolas f(x) = x2 + n which differ only by value of n Note that f '(x) is the same for each version of f Now go the other way … The antiderivative of 2x must be different for each of the original functions So when we take an antiderivative We specify F(x) + C Where C is an arbitrary constant This indicates that multiple antiderivatives could exist from one derivative

Indefinite Integral The family of antiderivatives of a function f indicated by The symbol is a stylized S to indicate summation

Indefinite Integral The indefinite integral is a family of functions The + C represents an arbitrary constant The constant of integration

Properties of Indefinite Integrals The power rule The integral of a sum (difference) is the sum (difference) of the integrals

Properties of Indefinite Integrals The derivative of the indefinite integral is the original function A constant can be factored out of the integral

Try It Out Determine the indefinite integrals as specified below