# Thought Process to Discover Knowledge : unique proposition of nubtrek

Books and education websites treat knowledge as “matter-of-fact”. In comparison to those, nubtrek provides a thought process to discover the knowledge. In this post, the two are is explained for differential calculus.

### matter-of-fact knowledge

Derivative as slope of curve : Reference Khan Academy

Formal definition of derivatives : Reference Khan Academy

In these the knowledge is presented as “matter-of-fact” for students to learn.

### Thought Process to Discover Knowledge from nubtrek

Nubtrek starts with rate-of-change in numerical problems.

• If a car travels a distance of 20m in 2 seconds, the speed of the car is 10m/sec. The speed is the rate of change of distance traveled.

Students are lead to understand that the quantities can be function of variables.

• if a car is moving at 10m/sec, the distance covered in t sec is 10t meter. The expression 10t is a function of variable t.
• If a car’s position of displacement is given as a function of a variable, How would one go about finding the speed? This is the problem we try to solve in differentiation. Car’s displacement is given as s=3t×t+2.

nubtrek takes the students in a thought-process to discover knowledge.

• Person A does the following: At t=2 sec, the distance traveled is 3×2×2+2 = 14m. So the speed should be 14/2 = 7 m/sec
• Person B does the following: At t=1 sec, the distance traveled is 3×1+2 = 5m. So the speed should be 5/1 = 5 m/sec

Why these two are different? What is being calculated by Person A and B? These students are calculating the average speed as a numerical value.

Students are lead to realise that displacement is a function of variable (s=3t×t+2), the speed is also a function of variable.

nubtrek continues to take students in a thought process to discover knowledge.

• Knowing that speed has to be computed as a function of time…
• Person A calculated rate of change for 1 sec interval as [s(t+1) — s(t)] / 1 = 6t+3
• Person B calculated rate of change for 2 sec interval as [s(t+2) — s(t)] / 2 = 6t+6
• Person C calculated rate of change for 0.5 sec interval as [s(t+.5) — s(t)] / .5 = 6t+1.5

Students are lead to understand what instantaneous speed is.

• A speedometer measures instantaneous speed and we would like to calculate that as a function of given variable.
• Instantaneous speed is when the rate of change is calculated for a time difference of 0

nubtrek challenges the young brains with questions to discover using limits.

• considering the time interval as δ, the speed is [s(t+δ) — s(t)] / δ
• For δ=0, the speed becomes [s(t+0) — s(t)] / 0 = 0/0
• What would be the value of function evaluating to 0/0?
• Students should shout USE LIMITS, if they have learnt the limits in nubtrek
• Thus the differentiation in first principle is discovered as limit δ tending to 0 of [s(t+δ) — s(t)] / δ

The thought process in nubtrek is designed to be