LECTURE SEVEN
November 24, 2021
Overview
Functions of Several Variables
Applications in Science and
Engineering
Introduction
Applications of Derivatives in Maths
Rate of Change of a Quantity
Increasing and Decreasing Functions
Tangent and Normal To a Curve
Maxima and Minima
Approximation or Finding Approximate Value
Problem examples
Introduction
I There are various applications of
derivatives not only in maths and
real life but also in other fields like
science, engineering, physics, etc.
I Here, you will learn the use of
derivatives with respect to
mathematical concepts and in
real-life scenarios.
Introduction
Derivatives have various important
applications in Mathematics such as:
(1) Rate of Change of a Quantity
(2) Increasing and Decreasing
Functions
(3) Tangent and Normal to a
Curve
(4) Minimum and Maximum
Values
(5) Approximations
Applications of Derivatives in Maths
I The derivative is defined as the
rate of change of one quantity
with respect to another.
I The concept of derivatives has
been used in small scale and large
scale. The concept of derivatives
used in many ways such as change
of temperature or rate of change
of shapes and sizes of an object
depending on the conditions.
Rate of Change of a Quantity
I This is the general and most
important application of derivative.
For example, to check the rate of
change of the volume of a cube
with respect to its decreasing sides.
Increasing and Decreasing Functions
I To find that a given function is
increasing or decreasing or
constant, say in a graph, we use
derivatives.
Tangent and Normal To a Curve
I Tangent is the line that touches
the curve at a point and doesn’t
cross it, whereas normal is the
perpendicular to that tangent.
Maxima and Minima
I To calculate the highest and
lowest point of the curve in a
graph or to know its turning point,
the derivative function is used..
Approximation or Finding Approximate Value
I To find a very small change or
variation of a quantity, we can use
derivatives to give the approximate
value of it.
Problem examples
I The utility function of the portfolio
return of a production industry is
given by
U (WT ) = U (Rp ), where Rp = e x +2y .
If x and y are numbers of the two
products sold, by using the second
order Taylor expansion, express the
portfolio return before making any
transaction.
