|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.
, where F'(x) = f(x) and a is any constant.
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.
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
on your TI-89.
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
and confirm it by entering the integral on your TI-89.
Checking Indefinite Integrals
Because you can check each indefinite integral's result by finding the derivative of the result. For example, can be verified by taking the derivative of the result: . Because the result of the differentiation is the original function, the integration is confirmed.
The Generalized Rule
The generalized version of this rule is , where and C is a constant. Recall that the derivative of a constant is 0, so for any constant C.
The indefinite integral
may be illustrated by graphing the family of curves that are represented by
for different values of C. For example,
. Letting C take the values -240, -200, -160, -120, -80, -40, 0, 40, 80, the family of curves is shown below in a [0, 50] x [-50, 100] window where the particular curve associated with
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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