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February 6, 2011 12:01 PM   Subscribe

Mathfilter: Antiderivative explained?

Could someone give a clear, concise definition/description of what the relationship is between derivative and an antiderivative at a very basic level without going into any sort of proofs. Also what is the antiderivative itself? Real world examples? Google offers a plethora of information, of course, but I am after an extremely basic this-is-what-is explanation for a math novice.

Thank you all!
posted by mooselini to Education (14 answers total) 4 users marked this as a favorite
 
The antiderivative of function f(x) is the function, when differentiated, gives f(x).

Example: f(x) = 2x

The antiderivative is x^2, since you get 2x when when you differentiate x^2.
posted by Wet Spot at 12:06 PM on February 6, 2011


If a function g is the derivative of a function f (g = f'), then f is the antiderivative of g.
posted by mr_roboto at 12:06 PM on February 6, 2011


Best answer: If you already know calculus and this is just a vocabulary question, the antiderivative is pretty much equivalent to the integral. Otherwise, here's a really dumbed-down, non-rigorous explanation:

The derivative gives the slope at any point. It's like someone telling you which way to turn at each intersection. The antiderivative tries to recover the original path from those directions. I say "tries to" because you also need a starting point, which is why formally we have that "constant of integration" C.

See the fundamental theorem(s) of calculus for a real answer.
posted by d. z. wang at 12:11 PM on February 6, 2011


Response by poster: Thank you, I guess I should have been more clear: I would like to understand antiderivative more conceptually. Instead of representing it as a f of (x) definition, could someone explain the concept behind antiderivative? Thank you all once more!
posted by mooselini at 12:13 PM on February 6, 2011


Instead of representing it as a f of (x) definition, could someone explain the concept behind antiderivative?

Can you give us your current understanding of the derivative in terms that would be appropriate?
posted by mr_roboto at 12:16 PM on February 6, 2011


Best answer: It's a little bit hard because you don't typically use the antiderivative directly itself - it's combined with the fundamental theorem of calculus to let you solve problems. But, if you remember high school physics: we'll use ' to denote the derivative with respect to time.

For an object accelerating from a starting velocity of 0,

x = (1/2)at^2
x' = at
x'' = a

We get this by taking the derivative of the position at each step. Lets say we start with x'' = a and want to work back up to an expression for x itself. We take the antiderivative of x'', and we get x' = at + C (the C is important because it implies we don't know anything about the starting velocity if all we start with is the acceleration). Then we take the antiderivative of x' and get x = (1/2)at^2 + C, where the C implies that only knowing the velocity doesn't give us any information about the starting position. So the derivative takes you down the chain, the antiderivative brings you back up.
posted by devilsbrigade at 12:26 PM on February 6, 2011


Is it sort of like saying that derivative:antiderivative::multiplication:division ?
posted by entropone at 12:33 PM on February 6, 2011 [1 favorite]


Best answer: With some detail removed (for example, I'm implicitly taking the derivative of everything with respect to time below):

The derivative of something is its rate of change. For example, the derivative of your position is your velocity (how fast your position is changing), and the derivative of your velocity is your acceleration (how fast your velocity is changing). You can see how this is useful; for example, you can take snapshots of a car's position and find out later how fast it was going.

Taking the antiderivative is simply performing this process in reverse; the antiderivative of acceleration is velocity and the antiderivative of velocity is position. Again, you can see that this will be useful too; for example, given the acceleration of a car, you can find out how fast it'll be going after some period of time.
posted by dfan at 12:35 PM on February 6, 2011 [4 favorites]


The antiderivative is the integral. The integral is "the area under the curve."
posted by exphysicist345 at 12:35 PM on February 6, 2011


Best answer: Instead of representing it as a f of (x) definition, could someone explain the concept behind antiderivative?

Think of differentiation as a process, one that acts on a function. Other mathematical operations are also processes; some act on functions, others act on numbers.

Mathematical operations all have inverses. An inverse operation is one that undoes the first operation. For example, if you have the number pi/6, and you perform the operation "take the sine of" on this number, you will end up with 1/2. The operation "take the sine of" is a process, one that turns pi/6 into 1/2. The inverse operation is "take the arc sine of"; that process turns 1/2 into pi/6. Or, equivalently, the arc sine function undoes the sine function.

Antidifferentiation is the inverse of differentiation. If you take the derivative, and then the antiderivative, you're back to where you started.

tl,dr: The antiderivative is like differentiating and then hitting ^Z (undo).
posted by Wet Spot at 12:53 PM on February 6, 2011 [1 favorite]


Best answer: Mathematical operations all have inverses.

Almost

The operation "take the sine of" is a process, one that turns pi/6 into 1/2.

But it also, for example, takes p/6 + 2pi to 1/2. (Jargon: sine is not one-to-one/injective)

The inverse operation is "take the arc sine of"; that process turns 1/2 into pi/6. Or, equivalently, the arc sine function undoes the sine function.

That's one choice of the inverse for sine. Another is "take the arc sine of" + 2pi.

Similarly, there can be several functions which, when differentiated, give f. They all differ from each other by a constant factor (the "C" in F(x)+C).

tl,dr: The antiderivative is like differentiating and then hitting ^Z (undo), if undo occasionally gave us something slightly different to what we had before...
posted by Omission at 2:47 PM on February 6, 2011



For those that are interested in a more precise explanation of inverses: functions map specific elements between two sets which may or may not be the same. Let's call them A and B. For every element a in A, f(a) is some element in B. Many different elements in A might map to the same element in B, and there may be elements in B that nothing from A maps to. The term 'inverse' is slightly misleading. There's a notion called a 'preimage': the preimage of a set of elements in B are all those elements in A that the function takes to those elements in B (confusing, I know). So if you take the preimage of f(x), you may not get just one thing back - in the case of differentiation, if f(x) is the differentiation function, you'll get a set of functions back that all differ by an added number - this is why we write the '+ C' when we compute an indefinite integral.

posted by devilsbrigade at 3:28 PM on February 6, 2011


Area under the curve, where f(x) is the curve, and the anti-derivative F(x) is the area under the curve.
posted by JesseBikman at 5:13 PM on February 6, 2011


Best answer: Analogy time*! Imagine you have a building with three flights of stairs, and each flight of stairs has its own landing. The bottom landing is labeled f'(x). The middle landing is labeled f(x). The top landing is labeled F(x). You start at f(x). You derive and go down the stairs to f'(x). So how do you return to the f(x) landing? Well, you need to "undo" what you just did, and integrate, or go up the stairs. So now you are at the f(x) landing and want to go up to the F(x) landing. All you have to do is integrate, or go up the stairs to reach the F(x) landing. To return to the f(x) landing you need to derive or go down the stairs.

The fundamental theorem of calculus says that you can always travel between the landings by deriving aka going down or integrating aka going up.

*excluding "+C"
posted by oceano at 6:05 PM on February 6, 2011


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