Module 18 - Antiderivatives as Indefinite Integrals and Differential Equations | ||||||||||||||||||||||||||||||
Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Lesson 4 | Self-Test | ||||||||||||||||||||||||||||||
Lesson 18.1: Antiderivatives as Indefinite Integrals | ||||||||||||||||||||||||||||||
This lesson exlores the relationship between antiderivatives and indefinite integrals and discusses families of curves. Mathematics can be discovered using the TI-89, as illustrated in Module 2 and Module 10. There is a sense of ownership and interest that is acquired with inductive learning. Review the discovery-learning process which was described in Module 2 and Module 10 and is shown below.
Defining Indefinite Integrals
Recall that an antiderivative of a function f is a function F whose derivative is
The Fundamental theorem gives a relationship between an antiderivative F and the function f.
A modified notation is used to signify the antiderivatives of f. The new notation is called an indefinite integral and the antiderivatives of f are written as
Using the Integral Key The integral key, which is used to find definite integrals, can also be used to find indefinite integrals by simply omitting the limits of integration.
Exploring
Examine the antiderivative of each of the following functions that have the form xn and look for a pattern that will lead you to a general rule for finding
Notice that the arbitrary constant C is not part of the result given by the TI-89.
Describing the Pattern and Predicting
18.1.1 Describe the pattern you found when evaluating the indefinite integrals above and use it to predict
18.1.2 Confirm your prediction of
Extending the Examples Extend the exploration of examples by predicting the following indefinite integrals.
18.1.3 Confirm your predictions by entering the integrals on your TI-89. Generalizing the Pattern
18.1.4 Predict a general rule for
Checking Indefinite Integrals
Because
The Generalized Rule
The generalized version of this rule is
Illustrating
The indefinite integral
![]() Each curve in the family can be obtained by choosing a different value of C and vertically translating the curve corresponding to C = 0. |
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