Visual walkthrough — Curve sketching — systematic approach

4.1.30 · D2 · Maths › Calculus I — Limits & Derivatives › Curve sketching — systematic approach

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Step 1 — Machine ko chalane ki permission kahan hai? (Domain)

KYA. Humari machine hai $f(x)=\frac{x^2}{x^2-1}$. Bar ka matlab hai "divide karo". Zero se division ek aisi cheez hai jo arithmetic mein bilkul mana hai — iska koi jawab nahi hota. Toh pehle hum koi bhi aisa $x$ dhoondenge jo bottom (denominator) ko zero banata ho.

$$x^2-1=0⟹\underset{\text{a square}}{\underbrace{x^2}}=\underset{\text{the number }1}{\underbrace{1}}⟹x=\pm 1.$$

• $x^2$ matlab $x$ khud se multiply.
• Do dangerous inputs hain $x=+1$ aur $x=-1$, kyunki $(\pm 1{)}^2=1$ aur $1-1=0$.

YEH STEP PEHLE KYUN. Agar hum domain bhool gaye toh hum shayad curve ko $x=1$ se guzarte hue khushi-khushi draw kar dein — lekin wahan literally koi point hai hi nahi. Poori picture jhooth hogi. (Dekho parent ka mistake box: domain hamesha pehle aata hai.)

PICTURE. Do dashed vertical "forbidden lines" number line ko teen rooms mein tod deti hain — curve ko inhi ke andar rehna hoga.

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Step 2 — Curve axes ko kahan touch karta hai? (Intercepts)

KYA. Do aasaan anchor dots.

• $y$-intercept: $x=0$ feed karo. $f(0)=\frac{0^2}{0^2-1}=\frac{0}{-1}=0.$ Toh curve $(0,0)$ se guzarti hai.
• $x$-intercepts: output zero kab hai? Ek fraction tab zero hota hai jab uska top zero ho: $x^2=0\Rightarrow x=0$. Wohi point phir se.

$$f(x)=0⟺\underset{\text{top}}{\underbrace{x^2}}=0⟺x=0.$$

KYUN. Intercepts poore graph ke sabse saste sachche dots hote hain — sirf arithmetic, koi calculus nahi. Yeh picture ke beech ko pin karte hain.

PICTURE. Origin par ek akela pinned dot, donon forbidden walls ke beech mein.

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Step 3 — Kya picture mirror image hai? (Symmetry)

KYA. Hum test karte hain ki kya hoga agar hum $x$ ko $-x$ mein flip karein:

$$f(-x)=\frac{(-x{)}^2}{(-x{)}^2-1}=\frac{x^2}{x^2-1}=f(x).$$

• $(-x{)}^2=x^2$ kyunki negative times negative positive hota hai. Toh kuch bhi nahi badla.

Kyunki $f(-x)=f(x)$ hai, machine even hai: $x$ par jo kuch bhi karti hai, wahi $-x$ par bhi karta hai.

KYUN. Yeh literally humara kaam aadha kar deta hai. Ek baar jab hum $x\ge 0$ ki shape jaante hain, hum $y$-axis ke paar flip karke $x\le 0$ free mein paate hain.

PICTURE. Ek titli: seedha aadha aur uska mirror-reflected ulta aadha.

Recall Even vs odd — ek-line reminder

Even test ::: $f(-x)=f(x)$, $y$-axis ke paar mirror. Odd test ::: $f(-x)=-f(x)$, origin ke baare mein $180^{\circ }$ ghoomao.

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Step 4 — Andeekhi deewarein aur zameen (Asymptotes)

KYA — vertical walls. $x=1$ ke paas bottom $x^2-1$ zero ki taraf simat ta jaata hai jabki top $x^2$ $1$ ke paas rehta hai. Ek fixed number jab chhoote aur chhoote number se divide hota hai toh size mein blast ho jaata hai. Toh output $\pm \infty$ ki taraf bhaagta hai: line $x=1$ (aur uska mirror $x=-1$) ek vertical asymptote hai — ek deewar jise curve chumta hai par kabhi chhoota nahi.

Infinity ka kaun sa sign? Har taraf bottom ka sign check karo, kyunki top $x^2>0$ hamesha positive hai:

approach karte waqt	$x^2-1$ ka sign	$f=\frac{+}{\text{that}}$
$x\to 1^{+}$ (1 se thoda upar)	$+$ (jaise $1.1^2-1>0$)	$+\infty$
$x\to 1^{-}$ (1 se thoda neeche)	$-$ (jaise $0.9^2-1<0$)	$-\infty$

KYA — horizontal floor/ceiling. $x$ ko bahut bada push karo. Top aur bottom dono ko $x^2$ se divide karo:

$$f(x)=\frac{x^2}{x^2-1}=\frac{1}{1-\frac{1}{x^2}}\underset{x\to \pm \infty }{\to }\frac{1}{1-0}=1.$$

• Term $\frac{1}{x^2}$ bade number ka denominator hai, isliye yeh $0$ mein fade ho jaata hai.
• Output $1$ ki taraf settle hota hai: line $y=1$ ek horizontal asymptote hai.

KYUN. Deewarein aur zameen "cage" hai — yeh hamare teen rooms mein se har ek ko batati hain ki curve ko kis taraf jaana hai. Yeh saare limit statements hain; dekho Limits and continuity aur Rational functions and asymptotes.

PICTURE. Cage: $x=\pm 1$ par do vertical dashed walls, $y=1$ par ek horizontal dashed line, aur chote arrows jo dikhate hain ki curve har wall ke paas kis taraf shoot karti hai.

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Step 5 — Kahan chadta hai, kahan girta hai? ($f^{\prime }$)

KYA. Derivative $f^{\prime }(x)$ tangent line ka slope hai — har point par steepness. Positive slope = chadna, negative slope = girna. $\frac{x^2}{x^2-1}$ par quotient rule use karte hue:

$$f^{\prime }(x)=\frac{\stackrel{\text{deriv of top}}{\overbrace{2x}}(x^2-1)-\stackrel{\text{top}}{\overbrace{x^2}}\stackrel{\text{deriv of bottom}}{\overbrace{(2x)}}}{(x^2-1{)}^2}=\frac{-2x}{(x^2-1{)}^2}.$$

Ab signs padho, ek factor at a time:

• Denominator $(x^2-1{)}^2$ kuch squared hai, isliye yeh $\ge 0$ hai — hamesha positive (sivaay walls par jahan undefined hai). Yeh sign kabhi nahi palatega.
• Numerator $-2x$: positive jab $x<0$, negative jab $x>0$.

$$f^{\prime }(x)>0\text{ for }x<0(\text{climbing}),f^{\prime }(x)<0\text{ for }x>0(\text{falling}).$$

$x=0$ par slope exactly $0$ hai aur yeh $+\to -$ switch karta hai: First Derivative Test se yeh ek local maximum hai, $(0,0)$ par ek peak. (Yeh First derivative and monotonicity ka action hai; underlying guarantee hai Mean Value Theorem.)

KYUN. Slope extra points plot kiye bina travel ki direction batata hai. Ek sign line poori upar/neeche ki kahani pakad leti hai.

PICTURE. Ek slope sign-line: $0$ ke left mein green "$+$ climbing" tangents, right mein coral "$-$ falling" tangents, peak par flat tangent.

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Step 6 — Yeh kaise moda hua hai? ($f^{\prime \prime }$)

KYA. Second derivative $f^{\prime \prime }$ slope ke change ki rate hai — yeh bending measure karta hai. $f^{\prime }=\frac{-2x}{(x^2-1{)}^2}$ ko ek baar aur differentiate karo:

$$f^{\prime \prime }(x)=\frac{6x^2+2}{(x^2-1{)}^3}.$$

Phir se sign detective work:

• Numerator $6x^2+2$: ek square $6$ se multiply, plus $2$ — hamesha $\ge 2>0$. Kabhi negative nahi.
• Denominator $(x^2-1{)}^3$: ek odd power, toh yeh $x^2-1$ ka sign rakhta hai.
  • $|x|>1$ (outer rooms): $x^2-1>0\Rightarrow$ cube $>0\Rightarrow f^{\prime \prime }>0$ → concave up $⌣$.
  • $|x|<1$ (middle room): $x^2-1<0\Rightarrow$ cube $<0\Rightarrow f^{\prime \prime }<0$ → concave down $⌢$.

Edge case — kya koi inflection point hai? Ek inflection ke liye $f^{\prime \prime }$ ko curve ke kisi real point par sign change karna hoga. Yahan $f^{\prime \prime }$ sirf $x=\pm 1$ ke paas sign flip karta hai — lekin woh holes hain (walls), curve par points nahi. Toh koi inflection point nahi hai. Yahi wajah hai ki Step 4 matter karta tha: forbidden $x$ par sign changes count nahi hote.

KYUN. Concavity decide karti hai ki chadna/girna smile par hota hai ya frown par, jo humare anchor points ke beech exact curvature fix karta hai. Dekho Second derivative test.

PICTURE. Teen rooms concavity se shaded — cup, cap, cup — har ek mein $⌣/⌢/⌣$ label.

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Step 7 — Har clue ko ek curve mein assemble karo

KYA. Room ke facts ko saath mein rakh do. Ek aur sign check batata hai ki har room floor $y=1$ ke kis taraf hai, bottom ka sign use karke (top $x^2\ge 0$ hamesha):

• Outer rooms $|x|>1$: bottom $>0$, aur $x^2>x^2-1$, toh $f>1$ → curve floor ke upar baitha hai, concave up, walls ki taraf $+\infty$ mein ghuosta hua.
• Middle room $|x|<1$: bottom $<0$, toh $f<0$ (sivaay peak $(0,0)$ ko chhoone ke) → curve floor ke neeche baitha hai, concave down, ek single hump jo $(0,0)$ par peak karta hai aur donon walls par $-\infty$ mein gota lagata hai.

KYUN. Pehle ka har jawab — domain, intercept, symmetry, walls, slope, bend — exactly ek consistent picture mein snap ho jaata hai. Koi freedom bacha nahi; curve forced hai.

PICTURE. Finished graph, teeno pieces draw kiye, cage lines dashed.

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Ek-picture summary

Sab kuch ek saath: forbidden walls (Step 1), origin dot + peak (Steps 2, 5), mirror symmetry (Step 3), floor $y=1$ (Step 4), climb/fall arrows (Step 5), $⌣/⌢/⌣$ shading (Step 6).

Recall Feynman retelling — poora walkthrough simple shabdon mein

Mere paas ek dividing machine hai, $\frac{x^2}{x^2-1}$. Pehle main poochhta hoon: kya yeh kabhi toot sakti hai? Haan — $x=\pm 1$ par bottom zero hai, toh main wahan do andeekhi walls paint karta hoon aur duniya ko teen rooms mein tod deta hoon. Phir saste dots: yeh origin se guzarti hai. Main $x$ ko $-x$ mein flip karta hoon aur kuch nahi badlta, toh picture ek mirror butterfly hai — mujhe bas seedha aadha chahiye. Door se, ek bade number se divide karte waqt, machine $1$ par shant ho jaati hai, toh $y=1$ par ek andeeki zameen hai. Phir main slope feel karta hoon: $x<0$ ke waqt chadta hai, origin par peak par tip karta hai, aur $x>0$ ke waqt girta hai. Phir main bend feel karta hoon: walls ke bahar yeh muskurata hai (cup), andar yeh sar jhukaa leta hai (cap), aur kyunki smile-to-frown switch sirf forbidden walls par hota hai, koi real inflection nahi hai. Aakhir mein main heights check karta hoon: do outer pieces floor ke upar float karte hain aur walls par rocket ho jaate hain; beech waala piece zero se neeche ka ek akela hump hai jo donon andar walls par neeche ghus jaata hai. Chhe saare jawaab saath mein snap karo aur sirf ek curve hai jo fit hoti hai — kaam khatam, calculator ki zaroorat nahi.

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Active Recall