# Calculus

**Calculus**, branch of mathematics dealing with calculating rates of change (differential calculus) and determining functions from information about their rate of change (integral calculus).

Differential calculus is used to calculate accelerations, velocities, slopes of curves, and maximum and minimum values, based on experimental or theoretical relationships expressed as continuous equations. If a relationship can be expressed as *y* = *f*(*x*), where the response *y* is a continuous function of *x*, then the average rate of change can be found by determining the change in

*y*(Δ*y* =*y*_{2}−*y*_{1} = *f*(*x*_{2})−*f*(*x*_{1}))

and dividing it by the change in

*x* (Δ*x* =*x*_{2}−*x*_{1}).

The average speed traveling in a car, 55 mph (88.5 kmph), or the average fuel efficiency during a trip, 30 mi/gal (12.8 km/l), are examples. If we take the limit of the expression Δ*y*/Δ*x* as Δ*x* =*x*_{2}−*x*_{1} gets smaller and smaller, we arrive at the instantaneous rate of change, which is the derivative of the function *y* =*f*(*x*) expressed as *f*'(*x*)=*dy*/*dx*:

“the derivative of the function *y* =*f*(*x*) with respect to *x*.” The symbol *dy* and *dx* are referred to as the differential of *y* and the differential of *x* respectively. In the automobile example above, although the average speed was 55 mph (88.5 kmph) over the length of a trip, each moment the traveler glanced at the speedometer different instantaneous speeds (rates) were observed.

When the derivative of a function *f*'(*x*) is known, the function itself can be determined. This process of finding the *antiderivative* is done with integral calculus. Just as differential calculus is concerned with changes and differences, integral calculus is concerned with summations. Rearranging the expression above, *dy* =*f*'(*x*)*dx*. Summing each side of this equation over a range gives the expression:

read “*y* is the integral of *f* prime of *x* with respect to *x*.” The elongated s stands for “sum” and is the symbol for integration. The range of summation indicated is from *x* = *a* to *x* = *b* and takes place over tiny intervals of *x*, *dx*. Integration is used to find lengths of curves, areas bounded by curves, volumes enclosed by surfaces, and centers of gravity of attracting bodies.

Many 17th-century mathematicians, astronomers, and physicists contributed to the development of calculus, predominant developers were Isaac Newton and Gottfried Wilhelm Leibniz.

*See also:* Newton, Sir Isaac; Leibniz, Gottfried Wilhelm von.

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