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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue
mathsDifferential CalculusIntroduction 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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