2.15 Antiderivatives and Initial Value Problems



The concept of an antiderivative of a function f is introduced as a function F such that F' = f. Thus, finding an antiderivative can be thought of as solving the differential equation F' = f for F. Differential equations are at the heart of modelling motion in dynamic systems. They provide the language in which we can describe the state of a physical system. For example an object in motion may be described in terms of its position, velocity, and acceleration. An equation relating these properties is thus an equation involving a function and its first and second derivatives. Also introduced in this section are Initial-value problems where additional conditions are present that allow a particular solution of a differential equation to be picked out from the general solution.

By the end of your studying, you should know:

On-screen applet instructions: Use the slider to display solutions of the differential equation.


Graph sin2(x), then use the graph to sketch an antiderivative of sin2(x).

Let g'''(x) = sin(x), with the conditions g(0) = 0, g'(0) = 1, g''(0) = –1. Find g(x).

Find a solution to the 4th-order differential equation

What can you say about


See short videos of worked problems for this section.


Take a quiz.


See Exercises for 2.15 Antiderivatives and Initial Value Problems (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

Interesting Application

Nothing yet has been found. Any ideas?

2.14 Linear Approximations Table of Contents 2.16 Velocity and Acceleration

Software requirements: For best results viewing and interacting with this page, get the free software listed here.

Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel