Calculus – Linearization and Differentials

Book Calculus by Thomas/Finney, 9th Edition, Page 248

Linearization: Approximating functions

https://math.stackexchange.com/questions/1385936/approximating-sqrt1-frac1n-by-1-frac12n#:~:text=The%20function%20f(%20x)=%20%E2%88%9A%201+%20x,%E2%88%9A%201+%20x%20for%20x%20%E2%89%A5%200.

Differentials

Differential Estimate of Change:
In calculus, estimating change with differentials is a method from calculus used to approximate how much a function’s output changes when the input changes by a small amount. Instead of calculating the exact change (which can be difficult and messy), we use derivative to find the “easy” approximate change.

The core concept: Imagine we have curve. If we zoom in very close to a specific point, the curve looks like a straight line (which is a tangent line). Thus,

• Actual Change (Δy): The true difference in height on the curve between two points.
• Differential (dy): The difference in height on the tangent line.

To estimate the change in $y$, you use the derivative $f′(x)$ as a multiplier for the change in $x$.

$dy=f′(x)⋅dx$

• $dy$ = The estimated change in y (the differential).
• $f′(x)$ = The derivative (slope of the tangent line) at the starting point.
• $dx$ = The change in x (often written as Δx).

Absolute, relative and percentage change in $f$

Error in Estimation:

Proof of Chaine Rule.

Proof is at Page – 256.