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A Knowledge Hub by Rahul Anand

Mathematical Marvels: Insights from the Field

Illustrating the concept of real numbers decimal expansion, this image displays wooden numbers from 1 to 9 arranged horizontally on a red surface, with small, colorful blocks placed beneath each number.

Do Real Numbers Always Have a Decimal Expansion?

Discuss whether real numbers are any number with a decimal expansion. Learn the precise relationship between real numbers, terminating decimals, repeating decimals, and irrational non-repeating decimals.

Calculus Castle

Find the limit: \( \lim_{x \to 3} (2x + 5) \)

Find \( \lim_{x \to 3} (2x + 5) \) Solution:To solve this limit, we substitute the value of \( x \) directly because the function is continuous at \( x = 3 \).\( \lim_{x \to 3} (2x + 5) = 2(3) + 5 = 6 + 5 = 11 \)

Morning Refresher – 5 Basic Problems in Limits to Boost Your Mind

5 Basic Problems on Limits just to refresh your mind. Problem 1 Find the limit: \( \lim_{x \to 2} (3x – 4) \) Solution: To solve this limit, we substitute the value of \(x\) directly because the function is continuous at \(x = 2\). \[ \lim_{x \to 2} (3x – 4) = 3(2) -…

THEOREM# \( \lim_{\theta\to0} \dfrac{sinθ}{θ} \) = 1

We have  \( \lim_{\theta\to0} { \sin\theta \over \theta } \) = 1 Consider the below diagram. We have r = radius of the circle.A = centre of the circle.The sector ⌔ formed by the arc BD subtends an angle θ at the centre. Case 1 : θ > 0 i.e. θ is +ve Let 0 ≤ θ ≤ \(…

Theorem# Limit of tanθ as θ → 0

Proof : We have, lim\(_{θ\to 0} { \dfrac {\mathrm tan \mathrm θ}{ \mathrm θ} }   \) = lim\(_{θ\to 0} { \dfrac {\mathrm \sin \mathrm θ} {\mathrm θ \mathrm \cos\mathrm θ} }   \)      \( \{∵ \tan\theta =  \dfrac…

Theorem# Limit of cosθ as θ → 0

As θ → 0, we have cosθ → 1 Proof : When θ = 0, We have, lim\(_{θ\to 0} \cos \)θ = cos0 = 1   { ∵ cos0 = 1 } Hence, lim\(_{θ\to 0} \cos \)θ = 1

Derivative of \(\mathsf { x^{n} }\) using the First Principle

Let y = \(\mathsf {x^{n} }\) ∴ y + δy = \(\mathsf { {(x + δx)^{n}} }\) ∴ δy = y + δy – y = \(\mathsf { (x + δx)^{n} }\) – \(\mathsf { x^{n} }\) or δy = \(\mathsf { [\text{ }^{n}C_0 x^{n}{(δx)}^{0} }\) + \(\mathsf {\text{ }^{n}C_1 x^{n-1}{(δx)}^{1}}\) + \(\mathsf…

Derivative of \({e}^x\) using First Principle 

Derivative of \({e}^x\) using the First Principle Let \(y\) = \({e}^x\)∴ \(y + δy\) = \({e}^{x + δx}\)∴ \(δy\) = \({e}^{x + δx}\) – \({e}^x\)or \(δy\) = \({e}^{x}\) . \( [ {e}^{δx} – 1 ]\)Dividing each side by δx </h3>or \(\dfrac {δy}{δx}\) = \( \dfrac { {e}^{x}…

Insight into the Laws of Physics

phonon-based mechanical memory

Can Mechanical Vibrations Replace Magnetic Memory in Future Devices?

Mechanical resonators could transform how future devices store and process information, offering coherent phonon states, hybrid quantum interfaces, and potentially lower-power operation. Their real challenge is not carrying information, but preserving it reliably against heat, damping, fabrication defects, and costly readout.

NUMERICAL SOLUTIONS IN PHYSICS

RELATIVE AND ABSOLUTE MOTION NUMERICAL

Swimmer in a River: RELATIVE AND ABSOLUTE MOTION NUMERICAL PROBLEMS

A swimmer is crossing a river that flows at 2 m/s. The swimmer’s speed relative to the water is 4 m/s.

Determine the swimmer’s speed relative to the riverbank (absolute motion) when swimming directly downstream.
Determine the swimmer’s speed relative to the riverbank when swimming directly upstream.

RELATIVE AND ABSOLUTE MOTION NUMERICAL

A Boat is moving downstream on a river. Determine the speed of the boat relative to the riverbank (RELATIVE AND ABSOLUTE MOTION NUMERICAL PROBLEMS)

A boat is moving downstream on a river. The river flows at a speed of 3 m/s relative to the riverbank. The boat’s engine propels it at a speed of 7 m/s relative to the water.

Determine the speed of the boat relative to the riverbank (absolute motion).
If the boat turns around and starts moving upstream, what will be its speed relative to the riverbank?

Displacement, Time & Velocity

Drive, Walk, Refuel: Find Displacement, Time & Velocity

You drive a car on a straight road at a constant speed of 70 km/h for a distance of 8.4 km, after which the car runs out of fuel. You then walk a further distance of 2 km for 30 minutes to reach the gas station. a) What is the total displacement from the beginning of your drive to your arrival at the gas station? b) What is the total time interval from the beginning of your drive to your arrival at the gas station? c) What is the average velocity from the beginning of your drive to your arrival at the gas station?

Atom to Atom: Insights from the Chemical Universe

Life’s Mysteries Revealed: Exploring Biology

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