Unit 6 Lesson 1 Antiderivatives and Indefinite Integration
- Slides: 32
Unit 6 Lesson 1 Antiderivatives and Indefinite Integration AP Calculus Mrs. Mongold
Definition of Antiderivative • A function F is an antiderivative of f on an interval I if F’(x) = f(x) for all x in I.
Definition of Antiderivative • A function F is an antiderivative of f on an interval I if F’(x) = f(x) for all x in I. • Note F is called an antiderivative not the antiderivative because any constant C, yields the same derivative – F(x) = x 3 -2 and F(x) = x 3 + 12 all produce F’(x) = 3 x 2
More Definitions • Constant of integration: the C in any antiderivative equation • General antiderivative: the family of functions represented by an antiderivative • General Solution: the equation of the antiderivative • Differential Equation: an equation that involved x and y derivatives of y.
Example 1 • Find the general solution to the differential equation y’=2
Example 1 • Find the general solution to the differential equation y’=2 • First find the antiderivative
Example 1 • Find the general solution to the differential equation y’=2 • First find the antiderivative y = 2 x
Example 1 • Find the general solution to the differential equation y’=2 • First find the antiderivative y = 2 x Then get the general solution by adding a constant
Example 1 • Find the general solution to the differential equation y’=2 • First find the antiderivative y = 2 x Then get the general solution by adding a constant y = 2 x + C
Notation for Antiderivatives • Most convenient to write your equation in differential form dy = f(x) dx
Another Definition • Antidifferentiation: the operation of finding all solutions
Another Definition •
Another Definition • Constant of integration Variable of integration An antiderivative of f(x) Integrand
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Important Note • If you are trying to integrate something and you are having trouble consider using some type of math rule to rewrite the problem to make it easier, you can also use your rules to split the function up to make it easier to integrate! • Think outside the box
Initial Conditions and Particular Solutions • Particular Solution: when you are given enough information to find the C value. • Initial condition: information stating a point on the curve.
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Homework • In Packet 4. 1 / 2 -50 Even