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Physicist: If you’ve taken calculus, then at some point you learned that to find the area under a function (generally written ) you need to find the anti-derivative of that function. It comes back (in a roundabout way) to the fact that the derivative of a function is the slope of that function or the “rate of change”.
This theorem is so important and widely used that it’s called the “fundamental theorem of calculus”, and it ties together the (opposite of the derivative) so tightly that the two words are essentially interchangeable.
: Say you’ve got a function f(x), and the area under f(x) (up to some value x) is given by A(x).
Then the statement “the area, A, is given by the anti-derivative of f” is equivalent to “the derivative of A is given by f”.
In other words, the rate at which the area increases (as you slide x to the right) is given by the height, f(x).
For a constant function the area is given by A=cx, and the rate of increase (the amount that the area increases if x increases by 1) is c.
Whether or not the function moves around makes no difference.
From moment-to-moment the rate of increase is always equal to the height (the value of f).For example, if the height of the function were 3, then, for a moment, the area under the function is increasing by 3 for every 1 unit of distance you slide to the right.Keep in mind that the function can move up and down as much as it wants.As far as the function “knows”, at any particular moment it may as well be constant (dotted line in picture above).So if the height of the function (which is just the function) is the rate at which the area changes, then f is the derivative of the area: A’=f.But that’s exactly the same as saying that the area is the anti-derivative of the function.