what exactly is

Question

what exactly differentiation and integration is????how its helpful in our life.

Answer 1

Hey Kumar!

Calculus is the mathematics of change. Slope in time is about your future heading. Integration in time is about where you've been -- past history. Integration is simply a "fancy way of adding." Think about your car's dashboard: the speedometer measures velocity. How do you "integrate" the speed? Look at the odometer (an "integrator") -- it adds, or accumulates, the distance traveled. Automatic feedback controls for all kinds of devices and processes can use slope detectors or integrators. Say your car's cruise control is set for 60 mph. You head up a steep hill; car speed drops quickly. That sudden drop in speed can activate a differentiator or slope detector to "hit the gas" as well as downshift the "tranny" to bring the speed back to 60 mph. This correction "burst" process can "overshoot" the 60 mph target somewhat. Then, an integrator -- which adds up how "off" (+ and -) the speed HAS BEEN recently -- triggers some small, final adjustments to smooth-out the overshoot. So, the slope detector ANTICIPATES error and is quick to fix things; however, "noise" in the measurement line having extra "zig-zaggy" slopes can over-activate the differential controller. Integrators correct PAST errors, smoothing-out noise, but are SLOW.  Integrator slowness or lag can actually cause  instability ... it could "step on the gas" at the very moment the car is headed DOWNHILL, creating a bigger  error to fix!  Wishing you the best ... :)

Answer 2

Differentiation if the process of deriving a formula for the rate of change.

The most common occurrence of derivatives is perhaps in driving a car.

Looking at odometer measures distance travel, so if you check the odometer reading at various time points, you get a function for distance travelled as a function of time. Calling that x(t).

The speedometer measures your speed, which is v(t) = dx/dt.

An accelerometer doesn't come with the car by default. However, if it did, it would measure acceleration, a(t) = dv/dt = d²x/dt².

Integration would be the reverse. The one note is that now you get a family of functions as you can add any constant.

∫a(t)dt = v₀ + v(t).

∫v(t)dt = x₀ + x(t).

∫∫ a(t) dt dt = ∫ (v₀ + v(t)) dt = x₀ + v₀t + x(t).