The primary objective of differential calculus is to partition something into smaller parts (infinitesimal parts), in order to determine how it changes.

**Applications
of Differential Calculus**

* *

*“Nothing takes place in the world whose meaning is not that of
some maximum or minimum”*

**- Leonhard Euler**

** **

The
primary objective of differential calculus is to partition something into
smaller parts (infinitesimal parts), in order to determine how it changes. For
this reason today’s differential calculus was earlier named as **infinitesimal
calculus**. Applications of differential
calculus to problems** **in physics and
astronomy was contemporary with the origin of science. All through the 18^{th}
century these applications were multiplied until Laplace and Lagrange, towards
the end of the 18^{th} century, had brought the whole range of
the study of forces into the realm of analysis.

The
development of applications of differentiation are also due to Lejeune
Dirichlet, Riemann, von Neumann, Heine, Kronecker, Lipschitz, Christoffel,
Kirchhoff, Beltrami, and many of the leading physicists of the century.

• Differential
calculus has applications in geometry and dynamics.

• Derivatives
of function, representing cost, strength, materials in a process, profit, etc.,
are used to determine the monotonicity of functions and there by to determine the
extreme values of the quantities represented by those functions.

• Derivatives
of a function do find a prominent place in many of the modelling problems in
engineering and sciences.

• Differential
calculus has applications in social sciences and medical sciences too.

Using
just the first two derivatives of a function *f* ( *x*) , in this chapter,
the nature of the function, sketching of curve *y* = *f* ( *x*) , and local extrema (maxima or
minima) of *f* ( *x*) are determined. Further, using certain higher derivatives of *f* ( *x*)
(if they exist), series expansion of *f*
( *x*) about a point are also
discussed.

** **

Upon completion of this chapter, students will be
able to

• apply derivatives to geometrical problems

• use derivatives to physical problems

• identify the nature of curves like monotonicity,
convexity, and concavity

• model real time problems for computing the
extreme values using derivatives

• trace the curves for polynomials and other
functions.

** **

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