Suppose f(x) is a continuous function. Then a function F(x) such that F'(x) = f(x) is called:
A.) the indefinite integral of f
B.) the antiderivative of f
C.) an antiderivative of f
D.) a definite integral of f
E.) All of the above
Surely your textbook has a shaded box explaining that C is the right choice.
E.) All of the above
Well, if we were to choose only one, it would definitely be "All of the above." But I have to say, these mathematical terms can be a bit dramatic. It's like they couldn't make up their mind and wanted to cover all their bases. So, yes, F(x) is the indefinite integral of f, the antiderivative of f, and an antiderivative of f. Talk about an identity crisis!
C.) an antiderivative of f
To answer this question, we need to understand the concept of antiderivatives and indefinite integrals.
An antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x). In other words, it is the "reverse" operation of taking derivatives. So, an antiderivative is a function that, when differentiated, gives us the original function as its derivative.
Now, when we talk about the indefinite integral of a function f(x), we are specifically referring to finding antiderivatives. The indefinite integral is represented by the symbol ∫f(x)dx, where ∫ represents the integral and dx indicates the variable of integration (usually x).
Therefore, the correct answer to the question is C.) an antiderivative of f. Option E.) "All of the above" is not correct because a definite integral, as indicated by option D.), involves finding the area under a curve between specific limits of integration.
In summary, an antiderivative and an indefinite integral refer to the same concept of finding a function whose derivative is equal to a given function.