Origin of The Constant
The derivative of any constant function is zero. Once one has found one antiderivative for a function, adding or subtracting any constant C will give us another antiderivative, because . The constant is a way of expressing that every function has an infinite number of different antiderivatives.
For example, suppose one wants to find antiderivatives of . One such antiderivative is . Another one is . A third is . Each of these has derivative, so they are all antiderivatives of .
It turns out that adding and subtracting constants is the only flexibility we have in finding different antiderivatives of the same function. That is, all antiderivatives are the same up to a constant. To express this fact for cos(x), we write:
Replacing C by a number will produce an antiderivative. By writing C instead of a number, however, a compact description of all the possible antiderivatives of cos(x) is obtained. C is called the constant of integration. It is easily determined that all of these functions are indeed antiderivatives of :
Read more about this topic: Constant Of Integration
Famous quotes containing the words origin of, origin and/or constant:
“The origin of storms is not in clouds,
our lightning strikes when the earth rises,
spillways free authentic power:
dead John Brown’s body walking from a tunnel
to break the armored and concluded mind.”
—Muriel Rukeyser (1913–1980)
“In the woods in a winter afternoon one will see as readily the origin of the stained glass window, with which Gothic cathedrals are adorned, in the colors of the western sky seen through the bare and crossing branches of the forest.”
—Ralph Waldo Emerson (1803–1882)
“I should say that the most prominent scientific men of our country, and perhaps of this age, are either serving the arts and not pure science, or are performing faithful but quite subordinate labors in particular departments. They make no steady and systematic approaches to the central fact.... There is wanting constant and accurate observation with enough of theory to direct and discipline it. But, above all, there is wanting genius.”
—Henry David Thoreau (1817–1862)