Understanding Antiderivatives in Calculus
Learn about antiderivatives in calculus, their importance in various fields, and how to find antiderivatives of functions. Discover the concept of anti-differentiation and examples to deepen your understanding of calculus concepts.
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Presentation Transcript
The integral is of fundamental importance in statistics, the sciences, and engineering. As with the derivative, the integral also arises as a limit, this time of increasingly fine approximations. We use it to calculate quantities ranging from probabilities and averages to energy consumption and the forces against a dam s floodgates. In this session we focus on the integral concept and its use in computing areas of various regions with curved boundaries.
James Stewart, Calculus: Early Transcendentals, 7th Edition, Brooks/ Cole 2012. Howard Anton, Irl C. Bivens and Stephen Davis, Calculus: Early Transcendentals, 9th Edition, John Wiley & Sons, Inc. 2010. Salas, Etgen, and Hille, Calculus: One and Several Variables, 10th Edition, John Wiley & Sons, Inc. 2007.
We have studied how to find the derivative of a function. However, many problems require that we recover a function from its known derivative. More generally, starting with a function ?, we want to find a function ? whose derivative is ?. If such a function ? exists, it is called an antiderivatives of ?. We will see next that antiderivatives are the link connecting the two major elements of calculus: derivatives and definite integrals.
Definition: A function ? is an antiderivative of ? on an interval ? if ? ? = ?(?) for all ? in ?. The process of recovering a function ? ? from its derivative ?(?) is called anti-differentiation. We use capital letters such as ? to represent an antiderivative of a function ?, ? to represent an antiderivative of ? , and so forth. Example: Find an antiderivative for ? ? = ?? .
Solution: We need to think backward here: What function do we know has a derivative equal to ??? It is not difficult to this question if we keep the Power Rule in mind. In fact, if ? ? = ??, then ? ? = ?? = ?(?). But the function ? ? = ?? ? also satisfies ? ? = ?? = ?(?). Therefore both ? and ? are antiderivatives of ? . Indeed, any function of the form ? ? = ??+ ?, where ? is a constant, is an antiderivative of ? ? = ??.
Theorem: If ? is an antiderivative of ? on an interval ?, then the most general antiderivative of ? on ? is ?(?) + ? where ? is an arbitrary constant. Example: Find an antiderivative ? of ? ? = ???that satisfies ?(?) = ?. Solution: Since the derivative of ??is ???, the general antiderivative ? ? = gives all the antiderivatives of ?(?). The condition ? ? = ? determines a specific value for ?. Substituting ? = ? into ?(?) gives ? = ?. So, ? ? = ?? ? is the antiderivative satisfying ? ? = ?.
