Curve sketching — systematic approach — Hinglish notes

4.1.30 · Maths › Calculus I — Limits & Derivatives

WHY do we sketch curves systematically?

Ek graphing calculator samajh ko chhupaata hai. Curve sketching ka point yahi hai ki ek formula padho aur uski geometry dekho. Systematic approach ek fixed checklist deta hai taaki koi feature miss na ho (ek hidden asymptote, ek sneaky inflection point). Yeh graphing ka 80/20 hai: ~6 sawaal kisi bhi curve ke ~90% behaviour ko capture karte hain.

WHAT to compute — the 7-step checklist

Definition Systematic checklist

Ek function $y = f (x)$ ke liye:

Domain — $f$ kahan defined hai? (division-by-zero, negative square roots, non-positives ka $lo g$ exclude karo)
Intercepts — $y$ -intercept $f (0)$ ; $x$ -intercepts ke liye $f (x) = 0$ solve karo.
Symmetry — even ( $f (- x) = f (x)$ , $y$ -axis ke baare mein symmetric), odd ( $f (- x) = - f (x)$ , origin ke baare mein symmetric), ya periodic.
Asymptotes — vertical (denominator $\to 0$ ), horizontal ( $lim_{x \to \pm \infty} f$ ), oblique (slant).
First derivative $f^{'}$ — sign se increasing/decreasing pata chalta hai; zeros/undefined se critical points milte hain → local max/min.
Second derivative $f^{''}$ — sign se concavity (up/down) pata chalta hai; sign-change se inflection points milte hain.
Assemble — intercepts, asymptotes, critical & inflection points plot karo, phir saari sign information ka respect karte hue connect karo.

HOW each tool works — derived from first principles

Increasing / decreasing from $f^{'}$

Ek critical point woh hota hai jahan $f^{'} (x) = 0$ ho ya $f^{'}$ undefined ho. First Derivative Test ke anusaar, agar $f^{'}$ $+ \to -$ change kare toh local max hai; $- \to +$ toh local min; koi change nahi → kuch nahi.

Concavity from $f^{''}$

Asymptotes — derived as limiting behaviour

Figure — Curve sketching — systematic approach

Worked Example 1 — a rational function

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

1. Domain. Denominator $x^{2} - 1 = 0$ $x = \pm 1$ par, isliye domain $R ∖ {- 1, 1}$ hai. Yeh step kyun? Woh points exclude karta hai jahan $f$ blow up karta hai.

2. Intercepts. $f (0) = 0$ → origin se guzarta hai. $f (x) = 0 \Rightarrow x^{2} = 0 \Rightarrow x = 0$ sirf. Kyun? Origin picture ke middle ko anchor karta hai.

3. Symmetry. $f (- x) = \frac{x ^{2}}{x ^{2} - 1} = f (x)$ → even, $y$ -axis ke upar mirror. Kyun? Kaam aadha ho jaata hai — $x \geq 0$ sketch karo, reflect karo.

4. Asymptotes. Vertical $x = \pm 1$ par. Horizontal: $lim_{x \to \infty} \frac{x ^{2}}{x ^{2} - 1} = 1$ , isliye $y = 1$ . Kyun? Yeh woh invisible walls/floors hain jinse curve approach karta hai.

5. First derivative. Quotient rule: $f^{'} (x) = \frac{2 x ( x ^{2} - 1 ) - x ^{2} ( 2 x )}{( x ^{2} - 1 ) ^{2}} = \frac{- 2 x}{( x ^{2} - 1 ) ^{2}} .$ Numerator $- 2 x = 0$ $x = 0$ par; denominator hamesha $> 0$ . Isliye $x < 0$ ke liye $f^{'} > 0$ , $x > 0$ ke liye $f^{'} < 0$ . → local max at $(0, 0)$ . Kyun? $f^{'}$ ki sign chadhna/girna batati hai; $x = 0$ ek peak hai.

6. Second derivative. $f^{'} = \frac{- 2 x}{( x ^{2} - 1 ) ^{2}}$ ko differentiate karne par milta hai: $f^{''} (x) = \frac{6 x ^{2} + 2}{( x ^{2} - 1 ) ^{3}} .$ Numerator $6 x^{2} + 2 > 0$ hamesha; denominator $> 0$ jab $∣ x ∣ > 1$ , $< 0$ jab $∣ x ∣ < 1$ . Isliye $∣ x ∣ > 1$ ke liye concave up, $∣ x ∣ < 1$ ke liye concave down. Koi inflection nahi (sign change sirf undefined $x = \pm 1$ par hota hai). Kyun? Har region mein bending batata hai.

7. Assemble: origin par peak, $y = 1$ asymptote ki taraf neeche jaata hai $∣ x ∣ < 1$ side ke liye... actually $∣ x ∣ < 1$ ke liye, $x^{2} - 1 < 0$ isliye $f < 0$ ; $∣ x ∣ > 1$ ke liye, $f > 1$ . Curve walls ke beech $y = 1$ se neeche aur bahar $y = 1$ ke upar baithta hai.

Worked Example 2 — a polynomial

$f (x) = x^{3} - 3 x$ sketch karo.

Domain: poora $R$ . Intercepts: $f (0) = 0$ ; $x^{3} - 3 x = x (x^{2} - 3) = 0 \Rightarrow x = 0, \pm 3$ . Symmetry: $f (- x) = - x^{3} + 3 x = - f (x)$ → odd, origin ke baare mein symmetric. Asymptotes: koi nahi (polynomial; ends $\to \pm \infty$ ). $f^{'} = 3 x^{2} - 3 = 3 (x^{2} - 1) = 0$ $x = \pm 1$ par. Sign: $+$ for $∣ x ∣ > 1$ , $-$ for $∣ x ∣ < 1$ . → local max at $(- 1, 2)$ , local min at $(1, - 2)$ . Yeh step kyun? Donon turning points identify karta hai. $f^{''} = 6 x = 0$ $x = 0$ par, sign $- \to +$ change hota hai → inflection at $(0, 0)$ . Assemble: badhta hai, $(- 1, 2)$ par peak, origin se girta hai (bending switch), $(1, - 2)$ par bottom, phir badhta hai.

Common Mistakes

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho tum apne dost ko phone par ek roller-coaster track describe kar rahe ho, lekin tum sirf kuch key facts bata sakte ho. Tum kehte ho: "Yeh yahan se shuru hota hai (intercepts), poori cheez left-right mirror-image hai (symmetry), do invisible walls hain jinhe yeh kabhi touch nahi karta (asymptotes), yeh UPAR jaata hai phir NEECHE yahaan (first derivative), aur yeh yahaan smile ki tarah curve karta hai aur wahaan frown ki tarah (second derivative)." Sirf un facts se tumhara dost almost exact ride draw kar sakta hai — yahi curve sketching hai!

Active Recall

What is a critical point?

Ek point jahan

f^{'} (x) = 0

ho ya

f^{'}

undefined ho.

What does

f^{'} (x) > 0

tell you?

Function wahan increasing hai (tangent upar ki taraf slope karta hai).

What does

f^{''} (x) > 0

indicate?

Curve concave up hai (cup shaped,

⌣

What is an inflection point?

Ek point jahan concavity change hoti hai, yaani

f^{''}

sign change karta hai.

First Derivative Test for a local max?

f^{'}

us point par

+

-

mein change hota hai.

Condition for a horizontal asymptote

y = L

lim_{x \to \pm \infty} f (x) = L

How to get the slope

m

of an oblique asymptote?

m = lim_{x \to \infty} f (x) / x

, phir

c = lim (f (x) - m x)

Why isn't

f^{'} (x) = 0

enough for an extremum?

f^{'}

ka sign change hona zaroori hai; jaise

x^{3}

mein

f^{'} (0) = 0

hai par koi extremum nahi.

First step in systematic curve sketching?

Domain nikalao (undefined points exclude karo).

Can a curve cross its horizontal asymptote?

Haan — asymptote sirf infinity par behaviour govern karta hai.

Test for even symmetry?

f (- x) = f (x)

; graph

y

-axis ke baare mein symmetric hai.

Test for odd symmetry?

f (- x) = - f (x)

; graph origin ke baare mein symmetric hai.

Connections

Limits and continuity — asymptotes limits ke zariye define hoti hain.
First derivative and monotonicity — increasing/decreasing intervals.
Second derivative test — concavity aur extrema classification.
Mean Value Theorem — justify karta hai kyun $f^{'}$ ki sign trend control karti hai.
Rational functions and asymptotes — vertical/oblique cases.
Optimization — extrema dhundhna usi critical-point machinery ka reuse karta hai.