Linear approximation and differentials: Foundations

4.1.32 · D1 · Maths › Calculus I — Limits & Derivatives › Linear approximation and differentials

Yeh page assume karta hai ki tumne Linear approximation and differentials ki koi bhi notation pehle nahi dekhi. Hum har symbol ko upar se banate hain, usi order mein jis order mein topic ko actually zaroorat hai.

1. Ek function $f$ aur uska graph

Ise ek grid par ek curve ke roop mein picture karo. Horizontal axis input $x$ hai; vertical axis output $f (x)$ hai. Har input $x$ height $f (x)$ par ek dot mark karta hai; saare dots ko jodo aur tumhe curve milta hai.

Figure — Linear approximation and differentials

2. Base point $a$ aur value $f (a)$

3. Slope — curve kitna steep hai

Ek curve ka slope samajhne se pehle, ek straight line ka slope samjho.

$2$ ka slope matlab hai "har $1$ step right ke liye, $2$ upar jao." $0$ ka slope flat hai; negative slope downhill point karta hai.

4. Difference quotient — curve ka slope jise tum approach karte ho

Ek curve bend karta hai, isliye uska koi ek slope nahi hota. Lekin tum uske do points ko join karne wali straight line ka slope measure kar sakte ho.

5. Limit $lim$ — $x$ ko $a$ mein slide karna

Yeh tool precisely ek secant (do points) ko controlled squeeze se ek tangent (ek point) mein turn karne ke liye exist karta hai. Dekho Derivative as a limit.

6. Derivative $f^{'}$ — har input ke liye ek slope

7. Output ka naam: $y$ , phir do tarah ke changes $Δ y$ aur $d y$

Ab input-symbols ko line up karte hain taaki koi kho na jaaye. Ab tak $a$ humara fixed base tha aur $x$ ek nearby input. Is section mein hum ek starting input se thoda moved input tak ke step ki parwah karte hain, isliye hum cheezein rename karte hain us step ko spotlight karne ke liye:

8. "Approximately equal" symbol $\approx$

9. Letters ko saath rakho: $L (x)$

Ab $L (x) = f (a) + f^{'} (a) (x - a)$ ka har piece defined hai:

Yeh foundations kaise stack up karte hain

Neeche di gayi figure ek study checklist visually hai: har box upar ka ek section hai, aur har arrow ka matlab hai "arrow ke peechhe wala box arrow ke aage wale box se pehle chahiye." Ise top to bottom trace karo yeh confirm karne ke liye ki tumne Linear approximation and differentials tackle karne se pehle har prerequisite meet kar li hai — agar koi box shaky lagta hai, woh section dobara padho.

Equipment checklist

Neeche har line ek mini flashcard hai: ::: se pehle wala part question hai, baad wala part answer hai. Daayein side cover karo, zor se jawab do, phir check karne ke liye reveal karo.

Ek sentence mein

f (x)

ka kya matlab hai?

Woh output jo ek function-machine deta hai jab use input

x

diya jaaye; curve ki height

x

ke upar.

a

kya hai, aur

f (a)

kya hai?

a

ek fixed chosen input hai;

f (a)

uske upar curve ki height hai — humara exact starting point.

"Slope" koi formula ke bina batao, phir ek ke saath.

Rise over run — up-distance divided by across-distance;

slope = \frac{rise}{run}

Slope (rise over run) kab number dene mein fail karta hai?

Vertical tangent par — run

0

hota hai, toh slope infinite/undefined hai; linear approximation wahan use nahi ho sakta.

Difference quotient likho aur batao yeh kya measure karta hai.

\frac{f ( x ) - f ( a )}{x - a}

—

(a, f (a))

aur

(x, f (x))

ke beech secant line ka slope.

Difference quotient mein

x = a

simply plug in kyun nahi kar sakte?

Yeh

\frac{0}{0}

deta hai; limit zaroori hai

x \to a

slide karne ke liye bina do points ke collide hue.

Two-sided limit

lim_{x \to a}

exist karne ke liye kya true hona chahiye?

Left-hand aur right-hand limits exist aur agree karein;

a

ke dono taraf curve hona chahiye.

f^{'}

define karo (sirf

f^{'} (a)

nahi).

Slope machine: ek function jo jo bhi input doh us par

f

ka tangent slope return karta hai, jab woh slope finite ho.

y = f (x)

kya kehta hai?

y

output height ka nickname hai; vertical axis par value.

Kya

d x

Δ x

ke same size ka hai?

Haan — same chosen input step; do naam sirf output mein "tangent-line change" vs "true-curve change" flag karte hain.

Δ y

aur

d y

mein farq karo.

Δ y

= curve ke saath

y

ka true change;

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

= straight tangent ke saath

y

ka change; yeh sirf approximately equal hain.

\approx

silently kya promise karta hai?

Approximate equality jo step jitna chhota hota hai utni sach hoti jaati hai.

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

ko teen plain words mein padhho.

Start + Slope × Step.

Connections

Derivative as a limit — §5 aur §6 iska poora content hai; yeh page on-ramp hai.
Tangent line — geometric object jise $L (x)$ trace karta hai.
Linear approximation and differentials — parent topic jisme yahan har symbol feed hota hai.