In fact, the complete statement of the integral involve another argument: C: For example, in traditional notation, it’s quite accepted to write an integration statement like this: This is a point that is somewhat obscured by traditional mathematical notation, which allows you to write down statements that are not completely explicit. The value of C sets, implicitly, the lower end of the range over which the accumulation is going to occur. So, as long as you are not concerned about additive constants, the anti-derivative of the derivative of a function gives you back the original function. So the problem of indefiniteness of the antiderivative amounts just to an additive constant - the anti-derivative of the derivative of a function will be the function give or take an additive constant: In fact, they are identical except for an additive constant.
Given a function \(f(x)\), the derivative with respect to \(x\) is written \(df/dx\) and the anti-derivative is written \(\int f(x) dx\).Īll of the functions that have the same derivative are similar. To start, it helps to review the traditional mathematical notation, so that it can be compared side-by-side with the computer notation. Or, rather, there are two answers that work out to be different faces of the same thing. The answer to this question is by no means a philosophical mystery. When you “undo” the derivative of any of df1, df2, or df3, what should the answer be? Should you get \(f_1\) or \(f_2\) or \(f_3\) or some other function? It appears that the antiderivative is, to some extent, indefinite. Consider the following functions, each of which is different: This global or distributed property of the anti-derivative is what makes anti-derivatives a bit more complicated than derivatives, but not much more so.Īt the core of the problem is that there is more than one way to “undo” a derivative. If you’ve ever stood on a hill, you know that you can tell the local slope without being able to see the whole hill just feel what’s under your feet.Īn anti-derivative undoes a derivative, but what does it mean to “undo” a local property? The answer is that an anti-derivative (or, in other words, an integral) tells you about some global or distributed properties of a function: not just the value at a point, but the value accumulated over a whole range of points. This will be confusing at first, but you’ll soon get a feeling for what’s going on.Īs you know, a derivative tells you a local property of a function: how the function changes when one of the inputs is changed by a small amount. The function that’s produced by the process is generally called an “integral.” The terms “indefinite integral” and “definite integral” are often used to distinguish between the function produced by anti-differentiation and the value of that function when evaluated at specific inputs. “Integration” is a shorter and nicer term than “anti-differentiation,” and is the more commonly used term. This is often called “integrating” a function. But it often happens that you are working with a function that describes the derivative of some unknown function, and you wish to find the unknown function. One undoes the other, so there is little point except to illustrate in a textbook how differentiation and anti-differentiation are related to one another. It’s rarely the case that you will want to anti-differentiate a function that you have just differentiated. 9.2.1 Example: Diving from the high board.
6.3 Functions with nonlinear parameters.
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