Applications — increasing - decreasing, local extrema (first derivative test): Foundations

4.1.28 · D1 · Maths › Calculus I — Limits & Derivatives › Applications — increasing - decreasing, local extrema (first

Yeh page har woh symbol build karta hai jo parent note use karta hai, "function ka graph kya hota hai" se shuru hokar " $f^{'}$ ka matlab kya hai" tak. Agar koi word ya squiggle wahan upar aaya tha, toh usse pehle yahan earn kiya jayega.

1. Graph — ek function se khiinchi hui sadak

Horizontal axis ko woh zameen samjho jis par tum chalte ho (input $x$ ) aur vertical axis ko height samjho (output $f (x)$ ). Jab tum seedha kadam rakhte ho, dot ek curve ke saath chalti hai — wahi curve sadak hai.

Figure — Applications — increasing - decreasing, local extrema (first derivative test)

Topic ko yeh kyun chahiye: poora topic is baare mein hai ki yeh sadak chalte waqt kaise chadhti aur utarti hai. Baad ke har symbol is ek picture ki kisi feature ko describe karta hai.

$x$ ::: input; zameen par tumhari left-right position.
$f (x)$ ::: output; position $x$ par zameen se tumhari height.

2. Intervals aur symbols $(a, b)$ , $[a, b]$ , $\infty$

Ek fence ki imagine karo. $[a, b]$ mein posts endpoints par hain (woh stretch mein belong karte hain). $(a, b)$ mein posts hata di gayi hain (sirf strictly-andar ki zameen count hoti hai).

$a < b$ ka matlab? ::: position $a$ , position $b$ ke left mein hai.
$[a, b]$ vs $(a, b)$ ? ::: closed (ends include) vs open (ends exclude).

3. Slope — tilt measure karne wala number

Kisi bhi calculus se pehle, slope ek school wala idea hai: har ek kadam daayein chalne par tum kitna upar uthte ho.

Yeh kaisa dikhta hai: do points ko jodne wali seedhi line khincho. Uski steepness yahi fraction hai. Positive value uphill jhukti hai; negative downhill; zero flat hai.

Uphill line ke liye slope ka sign? ::: positive.
$Δ$ ka matlab kya hai? ::: "mein change."

4. Average slope se instantaneous slope tak: derivative $f^{'} (x)$

§3 wala slope poore stretch $[a, b]$ par ek average hai. Lekin sadak modti hai — uski steepness har point par alag hoti hai. Hum ek exact jagah par tilt chahte hain.

Yeh kaisa dikhta hai: do nearby points se hokar guzarti ek slanted line se shuru karo (ek secant). Doosre point ko pehle ki taraf slide karo. Line pivot karti hai jab tak woh curve ko ek point par sirf kiss nahi karti — tangent line. Uski slope $f^{'} (x)$ hai.

Topic ko yeh kyun chahiye: yahi engine hai. Parent note ka har conclusion hai " $f^{'}$ ka sign padhو." Yeh tabhi samajh aata hai jab $f^{'}$ ka matlab "ek point par slope" ho, jo humne abhi build kiya.

$f^{'} (x)$ plain words mein? ::: $x$ single point par graph ki slope.
Uski definition mein limit kyun? ::: akela point $0/0$ deta hai; limit woh slope dhundhta hai jo do nearby points approach karte hain.
$f^{'} (x) = 0$ matlab tangent hai...? ::: flat (horizontal).

5. "Undefined" derivative — corners aur cusps

$f^{'} (x) = 0$ ek tarika hai jis se slope test trigger hota hai. Doosra yeh hai ki derivative simply exist hi nahi karti.

$f (x) = ∣ x ∣$ (absolute value — zero se door, hamesha $\geq 0$ ) ka graph classic V hai. Uska lowest point ek asli valley hai, phir bhi $f^{'} (0)$ kabhi zero nahi hota; woh exist karna fail kar deta hai. Isliye parent critical point ko " $f^{'} (c) = 0$ ya $f^{'} (c)$ undefined" define karta hai — dono cases ko dhundhna padta hai.

$f (x) = ∣ x ∣$ ke liye $f^{'} (0)$ undefined kyun hai? ::: left slope $- 1$ hai, right slope $+ 1$ hai; koi single tangent exist nahi karti.
$∣ x ∣$ ka matlab kya hai? ::: $x$ ki zero se doori (kabhi negative nahi).

6. Symbols glossary — quick reference

Yeh foundations topic ko kaise feed karte hain

Poori chain parent topic Mean Value Theorem mein khatam hoti hai — middle link (average slope → instantaneous behaviour) ko power karti hai, aur Fermat's Theorem on Stationary Points explain karta hai ki extrema sirf critical points par kyun chhipte hain.

Equipment checklist

Right side cover karo aur khud test karo — agar koi line stumble karaaye, toh uska section upar se dobara padho.

Graph ko ek sadak ki tarah padho: horizontal = input $x$ , vertical = height $f (x)$ . ::: §1 — graph.
Do points ke beech slope $\frac{f ( b ) - f ( a )}{b - a}$ se calculate karo. ::: §3 — rise over run.
Explain karo hum $b - a$ se divide kyun karte hain. ::: §3 — har kadam par steepness fairly measure karne ke liye.
Open $(a, b)$ aur closed $[a, b]$ intervals mein fark batao. ::: §2 — brackets ends exclude vs include karte hain.
$lim_{h \to 0}$ ka matlab words mein batao. ::: §4 — woh value jo ek quantity approach karti hai jab $h$ $0$ ki taraf shrink hota hai.
$f^{'} (x)$ kya hai ek sentence mein. ::: §4 — single point $x$ par graph ki slope.
$f^{'}$ ke har sign ko rising/falling/flat se match karo. ::: §4 — $+$ upar, $-$ neeche, $0$ flat.
Ek aisa point do jahan $f^{'}$ undefined ho aur kyun. ::: §5 — $∣ x ∣$ ka corner; left aur right slopes alag hain.
$\pm$ , $\Rightarrow$ , $Δ$ , $\infty$ padho. ::: §6 — glossary.