2.15 Antiderivatives and Initial Value Problems

2.15 Antiderivatives and Initial Value Problems

Summary

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:

The definition of an antiderivative.
Any two antiderivatives of a function on an interval differ by a constant.
The definition of an indefinite integral.
How to find indefinite integrals.
How to solve introductory first-order and second-order differential equations.
How to recognize an Initial Value Problem (IVP).
How to solve an Initial Value Problem (IVP) of first or second order.

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

Examples

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

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Videos

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Quiz

Exercises

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.

Resources on the Web

Information on Newton
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"

Information on Leibniz
Biographical data from St. Andrew's University's Web site
Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics"

Calculus Applications
Project Intermath

Antiderivatives
Visual Calculus

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2.14 Linear Approximations

Table of Contents

2.16 Velocity and Acceleration

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