Introduction to Integrals Unit 4 Day 1

Do Now  Write a function for which dy / dx = 2 x.  Can you think of more than one?

Derivative = 2 x  Each function pictured has the same derivative, 2 x.  Notice that at any x, the tangents to the graphs are all parallel, which implies _____________________

Antiderivatives  A function F is an antiderivative of f on an interval I if F '( x ) = f ( x ) for all x in I.  Thm. 4.1: Representation of Antiderivatives  If F is an antiderivative of f, then G is an antiderivative of f if and only if G ( x ) = F ( x ) + C, where C is a constant.  C is called the constant of integration.  The family of functions represented by G is called the general antiderivative of f.

Differentials and Antiderivatives  A differential equation in x and y is an equation that involves x, y, and derivatives of y.  Ex.: G '( x ) = 2 x  G ( x ) = x 2 + C is the general solution of the differential equation.

Ex. 1: Solving a Differential Equation  Find the general solution to the differential equation. a) y ' = 6 x 2 – 2 x b) dy / dx = sin x

Notation for Antiderivatives  dy / dx = f ( x ) in differential form: ___________  The operation of solving this equation is called antidifferentiation (or indefinite integration ) and is denoted by an integral sign:  General solution:   “the antiderivative (aka integral) of f with respect to x”

General Power Rule for Integration  What is the antiderivative of x 3 ?  What is the antiderivative of x n ?

Ex. 2: Powers and Constant Multiples  Find the antiderivative. a) b) c) d)

Ex. 3: Polynomials  Find the antiderivative. a) b) c)