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Introduction to Differential Calculus

Introduction to Differential Calculus

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Application Scenario of Derivative


 »  cause-effect relation in quantities.

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

    →  eg: speed is "rate-of-change" of displacement

Differential Calculus : Understanding Application Scenarios

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


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Starting on learning "Application Scenario of Derivative". In this page, the application scenario of derivatives is explained with examples.

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

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

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

    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 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

    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 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`
    • 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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    comprehensive information for quick review

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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 degree 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;7;6;seven;six
    predecessor of 7 is 6
    length;pen;centi;meter;centimeter;10;ten
    length of a pen is 10 centimeter
    The answer is length of a pen is 10 centimeter
    A pen can be used to write 30 pages. How many pages one can write with 4 pens?
    4;30;times
    4 times 30
    120;20
    one hundred and twenty
    both;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;paint;area
    Volume of Paint and painted area
    sold;tickets;money;collected
    Number of tickets sold and the money collected in the sale
    speed;car;distance;covered
    speed of a car and distance covered in an hour
    all;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?
    3 by 2; three by two;by 3; 3 by; by two; three by; 3 buy 2; three buy two;buy 2; 3 buy; buy two; three buy
    14 times 3 by 2
    2 by 3; two by three;by 3; 2 by; by three; two by; 2 buy 3; two buy three;buy 3; 2 buy; buy three; two buy
    14 times 2 by 3
    The answer is 14 times 3 by 2 ;; The area painted is the effect. ;; The volume of paint is the cause. ;; The cause-effect is relation defined by a function involving multiplication by a constant. area equals volume in to 3 by 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
    120 coins
    125
    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. ; coins ; equals; number of eggs ; in to ; price per egg ; plus; 5
    A car is moving in a straight line at constant speed. It is at a distance 10 meter at 20 second and at a distance 20 meter at 25 second. The "effect" distance is given and the "cause" speed is to be computed. What is the speed.
    =;equals;minus;meter;second;2nd;-
    speed equals 20 meter minus 10 meter by 25 second minus 20 second
    cannot;computed;only;distance;given
    speed cannot be computed as only the distance traveled is given
    The answer is "speed equals 20 meter minus 10 meter by 25 second minus 20 second". ;; The distance traveled is the effect. ;; The speed is cause. ;; The cause-effect relation is defined by a function involving rate of change speed ; equals ; speed two minus speed one ; by ; time two minus time one.
    A car is moving in a straight line at constant speed. It has a velocity of 2 meter per second for first 3 seconds and 4 meter per second for the next 1 second. What is the distance traveled in the 4 seconds.
    equals; meter;second; into; plus;per
    distance equals 2 meter per second into 3 second plus 4 meter per second into 1 second
    cannot;computed;speed;only;given
    distance cannot be computed as the speed is only given
    The answer is "distance equals 2 meter per second into 3 second plus 4 meter per second into 1 second" ;; The distance traveled is the effect. ;; The speed is cause. ;; The cause-effect is relation defined by a function involving aggregate of change. ; distance ; equals ; speed one into time1 ; + ; speed 2 into 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 this topic "rate of change" is explained.

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