ANTIDERIVATIVES Definition: reverse operation of finding a derivative

ANTHE Notice that F is called AN antiderivative and not THE antiderivative. This is easily understood by looking at the example above. Because in each case

Theorem 1: If a function has more than one antiderivative, then the antiderivatives differ by a constant.

indefinite integral Let f (x) be a function. The family of all functions that are antiderivatives of f (x) is called the indefinite integral and has the symbol INDEFINITE INTEGRALS

Indefinite Integral Formulas and Properties

Vocabulary: We could calculate the function at a few points and add up slices of width Δx like this (but the answer won't be very accurate): We can make Δx a lot smaller and add up many small slices (answer is getting better): And as the slices approach zero in width, the answer approaches the true answer. We now write dx to mean the Δx slices are approaching zero in width. The area under the curve of a function:

Example 1:

differential equation A differential equation is any equation which contains derivative(s). Solving a differential equation involves finding the original function from which the derivative came. general solution The general solution involves C. particular solution The particular solution uses an initial condition to find the specific value of C. Definition:

Example: general solution: particular solution: INITIAL VALUE PROBLEMS Particular Solutions initial conditions Particular Solutions are obtained from initial conditions placed on the solution that will allow us to determine which solution that we are after.

Example: Find the equation of the curve that passes through (2,6) if its slope is given by dy/dx = 3x 2 at any point x. The curve that has the derivative of 3x 2 is Since we know that the curve passes through (2, 6), we can find out C Therefore, the equation is