maths > differential-calculus

Differential Calculus : Understanding Application Scenarios

what you'll learn...

Overview

The application scenarios of derivative are explained in detail with examples.

»  cause-effect relation in quantities.

»  cause is the "rate-of-change" of effect

eg: Speed is "rate-of-change" of displacement<

cause-effect relation

One of the fundamental aspects of science is to measure and specify quantities. Some examples are

•  length of a pen is $10$$10$cm

•  mass of an object: $20$$20$ gram

•  temperature of water: ${30}^{\circ }$${30}^{\circ}$ Celsius

•  the amount of time taken: $3$$3$ seconds

•  the amount of distance traveled: $20$$20$ meter

•  the speed of a car : $20$$20$meter per second

A pen can be used to write $30$$30$ pages. With $4$$4$ pens, one can write $4×30=120$$4 \times 30 = 120$ pages. Increase in the number of pen causes increase in the number of pages, which is the effect of the cause.

In this "number of pen" is a cause and "number of pages" is an effect.

This is an example of cause and effect relation.

This is a brief on "relations and functions".

Some cause-effect relations are

•  Volume of Paint and painted area

•  Number of tickets sold and the money collected in the sale

•  speed of a car and distance covered in an hour

2 liter of paint is required to paint 3 square meter. If 14 liter paint is available, how much area can be painted?

The answer is "$14×\frac{3}{2}$$14 \times \frac{3}{2}$"

•  The "volume of paint" is the cause.

•  The "area painted" is the effect.

•  This cause-effect relation is defined by a function involving multiplication by a constant.
$\text{area}=\text{volume}×\frac{3}{2}$$\textrm{a r e a} = \textrm{v o l u m e} \times \frac{3}{2}$.

Everyday, a hotel sends a worker to buy eggs from market. The eggs are priced at $1$$1$ coin each and the worker charges $5$$5$ coins for the travel to buy eggs. How many coins are to be given to buy $120$$120$ eggs?

The answer is "$125$$125$ coins".

•  The "number of eggs" is the cause.

•  The "coins" is the effect.

•  This cause-effect relation is defined by a function involving addition of a constant.
$\text{coins}=$$\textrm{c o \in s} =$ $\text{number of eggs}$$\textrm{\nu m b e r o f e g g s}$ $×\phantom{\rule{1ex}{0ex}}\text{price per egg}$$\times \textrm{p r i c e p e r e g g}$ $+5$$+ 5$

A car is moving in a straight line at constant speed. It is at a distance $10$$10$m at $20$$20$sec and at a distance $20$$20$m at $25$$25$sec. The "effect" distance is given and the "cause" speed is to be computed. What is the speed?

The answer is "speed $=\frac{20m-10m}{25\mathrm{sec}-20\mathrm{sec}}$$= \frac{20 m - 10 m}{25 \sec - 20 \sec}$".

•  The speed is cause.

•  The distance traveled is the effect.

•  This cause-effect relation is defined by a function involving rate of change.

$\text{speed}=\frac{\text{speed2}-\text{speed1}}{\text{time2}-\text{time1}}$$\textrm{s p e e d} = \frac{\textrm{s p e e d 2} - \textrm{s p e e d 1}}{\textrm{t i m e 2} - \textrm{t i m e 1}}$

A car is moving in a straight line at constant speed. It has a velocity of $2$$2$ m/sec for first $3$$3$ seconds and $4$$4$ m/sec for the next $1$$1$ sec. What is the distance traveled in the $4$$4$ seconds?

The answer is "$=2m/\mathrm{sec}×3\mathrm{sec}$$= 2 m / \sec \times 3 \sec$ $\quad + 4 m / \sec \times 1 \sec$"

•  The speed is cause.

•  The distance traveled is the effect.

•  The cause-effect relation is defined by a function involving aggregate .

$\text{distance}=\text{speed1}×\text{time1}$$\textrm{\mathrm{di} s \tan c e} = \textrm{s p e e d 1} \times \textrm{t i m e 1}$$\quad + \textrm{s p e e d 2} \times \textrm{t i m e 2}$

summary

From the examples, it is understood that, Definition of a relation as an expression involves

•  multiplication and division

•  exponents and roots

Apart from these arithmetic operations, quantities may be related by "rate of change" and "aggregate ". These two topics are covered in differential and integral calculus respectively.

In the differential calculus, the "rate of change" is explained.

Outline

The outline of material to learn "Differential Calculus" is as follows.

•   Detailed outline of Differential Calculus

→   Application Scenario

→   Differentiation in First Principles

→   Graphical Meaning of Differentiation

→   Differntiability

→   Algebra of Derivatives

→   Standard Results