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The Foundation of Integration: Antiderivatives

By Edited May 21, 2015 0 0

Taking a derivative is a fundamental skill in Calculus. If you have a given equation, taking a derivative gives you the function for the slope of the line tangent to your given equation. To find the derivative you must take the:

limxà a [f (x) - f (a)]/[x-a]. This is one of the two forms this limit can take. For more on this, and

how to more quickly accomplish this task, check out Differentiation Shortcuts.

On the other hand, antiderivatives do the inverse of this. Instead of going from X3 to 3X2, you can take the latter and find out its parent function, the former. This process can be used for many, many things such as a physicist who wants to know the position of a particle in which he already knows the velocity, or an engineer who knows the variable rate (the rate at which the flow is) at which water is leaking from a tank, who wants to know the amount leaked over a certain period of time. It could even be used for a biologist who knows the rate at which a bacteria's population is increasing to find out what the size of the population will be at some point in the future. If none of these examples applies to you, you can still use this in Calculus class just to get the grade.

Beginning, let me give you a statement:

"A function, 'F' is called an antiderivative of 'f,' on an interval, 'I,' if F'(x) = f(x) for all x in I."

If you are like most people learning antiderivatives, or many people, in general, who haven't acquired Calculus skills, then you just said "A function 'F' is called an antiderivative of 'f,' on an interval, 'I,' if…" You trailed off thinking what foreign language you need to take to understand the words in the statement. I'll be the first to say math is not easy, but once it is explained, or 'translated' to English, it is much easier to grasp. In English, that statement says: some function, 'F' (the parent function), is called an antiderivative of 'f' (the daughter function), on some interval 'I' (from some number to some other number), if 'F' prime of x (the derivative of 'F') is equal to the daughter function 'f' of x for all numbers (that can be x) in that above stated interval of 'I.'

For example, X4 (we'll call this 'F') is the antiderivative of 4X3(we'll call this 'f') on the open interval (-Â¥ ,Â¥ ), or 'I'. If you take the derivative of X4 you get 4X3 which is equal to 'f,' which is also 4X3. This may be a bit redundant in showing the proof, but that is what it takes in mathematics to prove what you are saying.

General Antiderivatives

To take an antiderivative, you can refer back to the power rule, in Differentiation Shortcuts. In essence, all you must do is increase the power on the variable by one, and divide it all by that same number. The antiderivative of Xn = (Xn+1)/(n+1). An example would be 3X2 = X3. This is because you would increase the power of two, to the power of three, then you would divide by that new power of three, making it 3X3/3. Since three divided by three is one, it leaves X3 behind, and you are done, excepting one thing. On the end of any most general antiderivative, you write a "+C" at the end to show an arbitrary constant. So in all reality, you could have X3 + 3, X3 + 9, or even X3 + 1000. Again, to denote this, you write just "+C."

Specific Antiderivatives

For you to know the 'C' value at the end of the antiderivative, you must be given some information such as what the variable 'X' equals in the antiderivative. Once you are given this specific information, you no longer write "+C" at the end. You will find the actual value that should be placed at the end. Some examples of this include the following:

F'(x)=3X2, where F(0) = 12. You increase the power by one, divide by the new power, and again, we end up with X3. You now add the plus "C." The equation in front of us is now: F(x) = X3 + C. The information above about F(0) will help us in ridding the equation of that constant placeholder, 'C'. You will now substitute in the number zero for every 'X' in the equation, and set X3 + C = 12, just as the information told us to do. You will find out that the specific antiderivative equals X3 + 12. This is the final answer and you are finished with the problem.

As an added side note for science word problems, the antiderivative of the "jerk" function is the "acceleration" function, the antiderivative of the "acceleration" function is the "velocity" function, and the antiderivative of the "velocity" function is the "position" function.

All-in-all, antiderivatives, in my opinion, are easier to work than derivatives are, but that may be from having a firmer grasp on differential equations (any equation that involves the derivative of some function) than you once did when you began. This is the basis for a big part of Calculus, called the integral…So study up, and learn it well!



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