Dear Suyash

Differential calculus can be used for finding the local slope and curvature of a function and important points on a curve: maximum, minimum, inflection points, etc. These points and slopes can be the solution to a wide variety of problems related to physics, mechanical and electrical design, perhaps chemistry and other fields where a problem can be represented by an equation with variables. One example, if you can derive total fuel consumption as a function of acceleration there may be a minimum point of the function that allows you to reach your target speed with the least fuel - this is especially important with airplanes and rockets.

Integral calculus can be used to find the area under curves which can represent the amount of work done, distance gone, energy exerted, stress, angle and deflection of beams and a host of other scientific and mechanical properties that can be characterized by equations.

One overly simple example: if I push on a 1 Kg rock in space with a constant force of "A" newtons in the same direction where A represents a constant.

The acceleration (a) as a function of time: a = A meters per second squared. The graph will be a horizontal line with respect to time (t).

The speed (v for velocity) is the integral of the acceleration: v = A * t + B where t is time (how many seconds) and B is the speed it was going when we started. At+B is the integral of A with respect to time. The graph will be a line sloping upward with slope of A units with respect to t.

The distance traveled is the integral of the speed. One half A * t squared + B * t + C where C is how far away it was when we started. ( .5At^2 + Bt + C) The graph will be a parabola with increasing slope with respect to t.

Now, if a rocket starts with a mass of 60,000 Kg of which 50,000 is fuel and uses 500 Kg of fuel per second and the force generated by the fuel burn increases with altitude up to 15,000 m and decreases thereafter and the air resistance goes up with speed and down with altitude, you will need some calculus to determine how fast it will be going and how far it has gone when it runs out of fuel.

All the best.                                                           

AKASH GOYAL

AskiitiansExpert-IIT Delhi

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