Differential Calculus : Understanding Application Scenarios

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 $10$cm

• mass of an object: $20$ gram

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

• the amount of time taken: $3$ seconds

• the amount of distance traveled: $20$ meter

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

A pen can be used to write $30$ pages. With $4$ pens, one can write $4\times 30=120$ 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 "$14\times \frac{3}{2}$"

• 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.

$\text{area}=\text{volume}\times \frac{3}{2}$.

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

The answer is "$125$ 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.

$\text{coins}=$
$\text{number of eggs}$
$\times \phantom{\rule{1ex}{0ex}}\text{price per egg}$
$+5$

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

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

• The speed is cause.

• The distance traveled is the effect.

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

$\text{speed}=\frac{\text{speed2}-\text{speed1}}{\text{time2}-\text{time1}}$

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

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

• 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}\times \text{time1}$$+\text{speed2}\times \text{time2}$

summary

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

• addition and subtraction

• 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__