# differentation, intigraton

# differentation, intigraton

## 1 Answers

## Differentiation andintegration

Independently, Newton and Leibniz established simple rules for finding the formula for the slope of the tangent to a curve at any point on it, given only a formula for the curve. The rate of change of a function*f*(denoted by*f*') is known as itsderivative. Finding the formula of thederivativefunction is calleddifferentiation, and the rules for doing so form the basis of differential calculus. Depending on the context, derivatives may be interpreted as slopes of tangent lines, velocities of moving particles, or other quantities, and therein lies the great power of the differential calculus.

An important application of differential calculus is graphing a curve given itsequation*y*=*f*(*x*). This involves, in particular, finding local maximum and minimum points on thegraph, as well as changes in inflection (convex to concave, or vice versa). When examining a function used in amathematical model, such geometric notions have physical interpretations that allow a scientist or engineer to quickly gain a feeling for the behaviour of a physical system.

The other great discovery of Newton and Leibniz was that finding the derivatives of functions was, in a precise sense, theinverseof the problem of finding areas under curves—a principle now known as thefundamental theorem of calculus. Specifically, Newton discovered that if there exists a function*F*(*t*) that denotes the area under the curve*y*=*f*(*x*) from, say, 0 to*t*, then this function’s derivative will equal the original curve over that interval,*F*'(*t*)=*f*(*t*). Hence, to find the area under the curve*y*=*x*2from 0 to*t*, it is enough to find a function*F*so that*F*'(*t*)=*t*2. The differential calculus shows that the most general such function is*x*3/3+*C*, where*C*is an arbitrary constant. This is called the(indefinite)integralof the function*y*=*x*2, and it is written as ?*x*2*d**x*. The initial symbol ? is an elongated S, which stands for sum, and*d**x*indicates an infinitely small increment of the variable, or axis, over which the function is being summed. Leibniz introduced this because he thought ofintegrationas finding the area under a curve by a summation of the areas of infinitely manyinfinitesimallythin rectangles between the*x*-axis and the curve. Newton and Leibniz discovered that integrating*f*(*x*) is equivalent to solving adifferential equation—i.e., finding a function*F*(*t*) so that*F*'(*t*)=*f*(*t*). In physical terms, solving this equation can be interpreted as finding the distance*F*(*t*) traveled by an object whose velocity has a given expression*f*(*t*).

The branch of the calculus concerned with calculating integrals is the integral calculus, and among its many applications are finding work done by physical systems and calculating pressure behind a dam at a given depth.

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Rinkoo Gupta

AskIITians Faculty