Linear approximation and differentials — Hinglish notes

4.1.32 · Maths › Calculus I — Limits & Derivatives

WHAT is it?

WHY does it work? (derive from scratch)

Derivative defined hai ek limit se: $f^{'} (a) = lim_{x \to a} \frac{f ( x ) - f ( a )}{x - a} .$

"Limit equals $f^{'} (a)$ " ka matlab hai: $a$ ke karib $x$ ke liye, difference quotient $f^{'} (a)$ ke karib hota hai. Us closeness ko ek error term $ε (x)$ se likho jo $x \to a$ par $\to 0$ hoti hai: $\frac{f ( x ) - f ( a )}{x - a} = f^{'} (a) + ε (x) .$

Dono sides ko $(x - a)$ se multiply karo: $f (x) - f (a) = f^{'} (a) (x - a) + ε (x) (x - a) .$

Toh $f (x) = L (x) f (a) + f^{'} (a) (x - a) + error ε (x) (x - a) .$

Yeh kyun great hai: error hai $ε (x) \cdot (x - a)$ — ek chhoti number times ek chhoti number. Yeh $(x - a)$ se bhi zyada tezi se shrink karta hai. Isi liye $a$ ke paas line $L (x)$ $f$ ka ek excellent substitute hai.

HOW to use it — a recipe

Ek base point $a$ chuno apne target ke paas jahan $f (a)$ aur $f^{'} (a)$ easy/exact hon.
$f (a)$ aur $f^{'} (a)$ compute karo.
$L (x) = f (a) + f^{'} (a) (x - a)$ mein plug in karo.
Apne target $x$ par evaluate karo. (Differentials ke liye: $d x = x - a$ , phir $f (x) \approx f (a) + d y$ .)

Worked examples

Common mistakes (steel-manned)

Recall Feynman: 12-saal ke bachche ko explain karo

Ek tedhi-medhi sadak socho. Jahan tum khade ho, wahan sadak jis direction mein jaati hai (uski slope) usi direction mein chalo. Kuch kadam tak tum almost sadak par hi rehte ho, chahe woh dheere-dheere curve karte jaaye. Toh mushkil curvy math karne ki bajaye, tum "thoda seedha chalte ho." $f (a)$ hai jahan tum shuru karte ho, $f^{'} (a)$ hai woh direction jisme tum point kar rahe ho, aur $(x - a)$ hai kitne kadam tum lete ho. Jitna zyada chalo, utna zyada asli sadak tumhare seedhe raaste se door curve ho jaati hai — wahi error hai.

Active recall

What is the linearization of

f

a

L (x) = f (a) + f^{'} (a) (x - a)

Why does linear approximation work (one line)?

Limit definition se force hota hai ki

f (x) = L (x) + ε (x) (x - a)

jahan error

(x - a)

se bhi tezi se shrink karta hai.

Define the differential

d y

d y = f^{'} (x) d x

, tangent line par rise jab input

d x

badhlta hai.

Difference between

Δ y

and

d y

Δ y

= curve ki asli change;

d y

= tangent line par change;

Δ y \approx d y

Approximate

(1 + x)^{n}

for small

x

1 + n x

(

a = 0

par linearize karke).

How do you pick the base point

a

Target ke sabse paas wala value jahan

f (a)

aur

f^{'} (a)

exact/easy hon.

Estimate

4.1

using

a = 4

2 + \frac{1}{4} (0.1) = 2.025

Relative error in volume of a sphere from radius error?

\frac{d V}{V} = 3 \frac{d r}{r}

sin x

e^{x}

ln (1 + x)

near

0

sin x \approx x

e^{x} \approx 1 + x

ln (1 + x) \approx x

Connections

Derivative as a limit — woh definition jo yahan proof hai.
Tangent line — geometric object jo $L (x)$ hai.
Taylor series — linear approximation degree-1 Taylor polynomial hai; agle terms error ko kam karte hain.
Newton's method — $f (x) = 0$ solve karne ke liye baar-baar linearize karna.
Error propagation — physics/lab mein $d y = f^{'} (x) d x$ ka use.
Concavity and second derivative — control karta hai ki $L$ over-estimate karta hai ya under-estimate.