Linear approximation and differentials: Visual walkthrough

4.1.32 · D2 · Maths › Calculus I — Limits & Derivatives › Linear approximation and differentials

Pehli line se pehle, teen seedhe words jo hum baar baar use karenge:

Step 1 — Curve draw karo aur do heights mark karo

Figure — Linear approximation and differentials

Step 2 — " $a$ par slope" ka matlab kya hota hai

Step 3 — "close to $f^{'} (a)$ " ko exact equation mein badlo

Step 4 — "height = line + error" mein rearrange karo

Step 5 — Error run se zyada tez kyun khatam hoti hai

Recall Exactly kitna tez? (ek extra assumption chahiye)

Clean rule "error $\approx C (x - a)^{2}$ " free nahi hai — yeh tabhi hold karta hai jab $f$ ka second derivative $f^{''}$ exist kare (yaani $f$ $C^{2}$ ho). Tab Taylor series agla term $\frac{1}{2} f^{''} (a) (x - a)^{2}$ deta hai, toh error $O ((x - a)^{2})$ hai. Ek aisi function ke liye jo sirf differentiable hai (sirf $f^{'}$ exist karta hai), hum sirf upar wali formula se weaker $o (x - a)$ claim kar sakte hain — run ke comparison mein choti, lekin zaruri nahi square-law ho. Concavity and second derivative exactly woh $f^{''}$ hai jo quadratic case govern karta hai.

Step 6 — Differential: wahi rise, ek working point par

Step 7 — Har case: $x$ kahan hai, aur curve kis taraf bend karti hai?

Ek-picture summary

Yeh akela figure poori argument stack karta hai: curve $f$ , tangent line $L$ jo $(a, f (a))$ par touch karti hai, run $(x - a)$ , tangent rise $f^{'} (a) (x - a)$ red mein, aur chhota bacha hua error $ε (x) (x - a)$ — chota isliye kyunki yeh small $\times$ small hai.

Recall Feynman retelling — plain words mein poora walkthrough

Aap spot $a$ par ek curvy pahaadi par khade hain, aur aap wahan do cheezein jaante hain: aap kitne unche hain ( $f (a)$ ) aur zameen kitni steep hai ( $f^{'} (a)$ ). Aap $x$ par, kuch steps door, apni height guess karna chahte hain. Toh aap aankhein band karte hain aur ek bilkul seedhi line mein us direction mein chalte hain jisme zameen tilt thi. $(x - a)$ steps ke run ke baad, seedha chalne se aap slope $\times$ run $= f^{'} (a) (x - a)$ upar uth gaye. Woh shuru wali jagah mein add karo: $f (a) + f^{'} (a) (x - a)$ — yahi aapka guess hai, $L (x)$ . Asli pahaadi thodi si curved thi jab aap seedha chale, toh aap thoda off hain. Lekin yahan magic hai: woh gap "slope kitna drift hua" ( $ε$ ) times "aap kitna chale" ( $x - a)$ hai — do chote numbers multiply hue, toh woh bahut chhota hai, aur woh run se kahin zyada tez shrink karta hai jab aap kam steps lete ho. Isliye thoda seedha chalna almost-perfect shortcut hai.

Recall Memory se derivation rebuild karo

$a$ par slope ki definition se shuru karo ::: $f^{'} (a) = lim_{x \to a} \frac{f ( x ) - f ( a )}{x - a}$ . Secant aur tangent slope ke beech gap ka naam do ::: $ε (x) = \frac{f ( x ) - f ( a )}{x - a} - f^{'} (a)$ , jo $\to 0$ jaata hai. Run $(x - a)$ se multiply karo aur $f (a)$ add karo ::: $f (x) = f (a) + f^{'} (a) (x - a) + ε (x) (x - a)$ . Error $a$ ke paas negligible kyun hai ::: woh $ε (x) (x - a)$ = small $\times$ small hai, toh error $/ (x - a) = ε (x) \to 0$ — error $o (x - a)$ hai. Differential ke liye base point aur run rename karo ::: base $= x$ , step $= d x$ , jisse milta hai $d y = f^{'} (x) d x$ jahan $Δ y \approx d y$ .

Connections

Derivative as a limit — Step 2 literally isi ki definition hai; proof uss limit ko unpack karna hai.
Tangent line — woh line $L (x)$ jo hum Step 4 mein banate hain.
Taylor series — $(x - a)^{2}$ ka agla term (jab $f^{''}$ exist kare) jo Step-5 error ko exactly pin down karta hai.
Newton's method — ek root dhundhne ke liye Step 4 repeat karta hai.
Error propagation — Step 6 ka $d y = f^{'} (x) d x$ lab mein.
Concavity and second derivative — Step 7 mein over- vs under-estimate decide karta hai.