__maths__>__Differential Calculus__>__Introduction to Differential Calculus__### Differential Calculus : Understanding Application Scenarios

In this page, the application scenario of derivatives is explained with examples.

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

• mass of an object: `20` gram

• temperature of water: `30^@` Celsius

• the amount of time taken: `3` seconds

• the amount of distance traveled: `20` meter

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

Which one in the following is a measurement?

- predecessor of `7` is `6`
- length of a pen is `10`cm
- length of a pen is `10`cm

The answer is " Length of a pen is `10` centimeters".

A pen can be used to write `30` pages. How many pages one can write with `4` pens?

- `4xx30`
- `120`
- both the above
- both the above

Answer is "both the above".

A pen can be used to write `30` pages.

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

This is an example of *cause and effect pair*.

Can you identify a cause-effect pair in the following?

- 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
- all the above
- all the above

The answer is "all the above".

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

- `14 xx 3/2`
- `14 xx 3/2`
- `14 xx 2/3`

The answer is "`14 xx 3/2`"

• The "area painted" is the effect.

• The "volume of paint" is the cause.

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

`text(area) = text(volume) xx 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?

- `120` coins
- `125` coins
- `125` coins

The answer is "`125` coins".

• The "coins" is the effect.

• The "number of eggs" is the cause.

• The cause-effect relation is defined by a function involving addition of a constant.

`text(coins) =` ` text(number of eggs)` `xx 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?

- speed `=(20m-10m)/(25sec-20sec)`
- speed `=(20m-10m)/(25sec-20sec)`
- speed cannot be computed as only the distance traveled is given

The answer is "speed `=(20m-10m)/(25sec-20sec)`".

• The distance traveled is the effect.

• The speed is cause.

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

`text(speed) = (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?

- `=2m//sec xx 3 sec` `quad + 4m//sec xx 1 sec`
- `=2m//sec xx 3 sec` `quad + 4m//sec xx 1 sec`
- distance cannot be computed as the speed is only given

The answer is "`=2m//sec xx 3 sec` `quad + 4m//sec xx 1 sec`"

• The distance traveled is the effect.

• The speed is cause.

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

`text(distance) = text(speed1) xx text(time1)``quad + text(speed2) xx text(time2)`

From the examples, it is understood that, Definition of function 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 of change**". These two topics are covered in differential and integral calculus respectively. *In the differential calculus, the "rate of change" is explained.*

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