Parent note ki ek bhi line padhne se pehle, tumhe har woh symbol apna banana hoga jo woh tumhare samne pheinkta hai. Yeh page har ek symbol ko bilkul zero se build karta hai, is order mein jahan har idea pichle idea pe lean karta hai. Yahan koi assumption nahi ki tumne notation pehle dekha hai.
Ek curve socho jo graph paper par bani ho. Horizontal axis input x carry karti hai; vertical axis height y=f(x) carry karti hai. Har input x curve par ek height pick karta hai.
Figure s01 — "Ek function ek input ko ek height mein badle." Ek cyan curve y=f(x) blueprint grid par. Ek amber input value x horizontal axis par dashed line se curve tak trace ki gayi hai, phir us curve se uski height y=f(x) vertical axis par — yeh single input-to-output mapping dikhata hai, aur white line y=0.
Graph par, ek root woh point hai jahan curve horizontal axis ko touch ya cross karti hai — kyunki axis exactly line y=0 hai ("height =0," §1 se).
Figure s02 — "Root woh jagah hai jahan curve line y=0 se milti hai." Ek cyan curve jo white horizontal axis ko do amber dots par cross karti hai; har dot par x∗ label hai jisme f(x∗)=0 likha hai, yeh dikhata hai ki root exactly curve aur zero line ka crossing point hai.
Figure s05 — "Continuity vs ek jump." Left: ek smooth cyan curve jo ek unbroken stroke mein draw ki gayi hai (continuous — pen kabhi nahi uthta). Right: ek amber curve jo axis ke neeche se seedha upar tak jump karti hai — ek open circle neeche aur ek filled circle upar jump ko mark karte hain, ek arrow ke saath "no crossing here" note karta hai, yeh dikhata hai ki discontinuity curve ko y=0 ko touch kiye bagair skip karne deti hai.
Figure s03 — "Opposite signs ek root ko bracket karte hain." Grid par ek cyan straight-through curve; amber dots left endpoint a ko mark karte hain jahan f(a)<0 (white axis ke neeche) aur right endpoint b ko jahan f(b)>0 (uske upar). Unke beech shaded band aur axis par white dot dikhata hai ki root [a,b] ke andar trapped hai jab bhi endpoint heights ke opposite signs hon.
Figure s04 — "Derivative tangent line ka slope hai." Ek cyan curve jisme ek amber tangent line use point x=p par touch karti hai. Tangent ke neeche ek white right-triangle apna "rise" aur "run" label karta hai, aur ek annotation likhta hai f′(p)=rise/run — derivative ko us point par curve ki steepness ke roop mein define karta hai.
Neeche diagram dikhata hai ki yeh foundations topic ko kaise feed karte hain: plain function idea do branches mein split hoti hai (sign-tracking → bisection, slope-tracking → Newton), aur dono parent topic Root-Finding from Scratch ko feed karte hain.
Main f(x) ko aloud padh sakta hoon aur keh sakta hoon iska matlab kya hai
"f of x": input x machine f ko do, curve se output height padho.
Main jaanta hoon y aur line y=0 kya hain
y vertical coordinate hai (height f(x)); y=0 horizontal axis hai, saare points jo height zero par hain.
Main jaanta hoon x∗ kya hai
Exact true root — woh input jahan f(x∗)=0; star bas "real answer" mark karta hai, guesses se alag.
Main xn aur xn+1 mein subscripts decode kar sakta hoon
"Current guess" aur "next guess" ke labels; subscript ek name tag hai, power nahi.
Main [a,b] par continuous function define kar sakta hoon
Woh jo poore closed interval [a,b] mein pen uthaye bagair draw ki ja sake — usmein kahin bhi koi jumps ya holes nahi.
Main jaanta hoon [a,b] aur uske endpoints kya hain
Left endpoint a se right endpoint b tak inputs ka interval; f(a),f(b) wahan heights hain.
Main jaanta hoon f(a)⋅f(b)<0 mujhe kya batata hai — aur uska edge case
Endpoints par opposite signs, toh continuous curve andar zero cross karti hai; lekin agar ek endpoint khud root hai toh f(a)⋅f(b)=0, toh ≤0 use karo ya pehle endpoints check karo.
Main Intermediate Value Theorem words mein state kar sakta hoon
Agar f poore [a,b] par continuous hai aur axis se neeche se upar jaati hai, toh beech mein kahin zero se zaroor guzarni padegi.
Main jaanta hoon ∣en∣ aur ε ka matlab kya hai
∣en∣ guess aur true root ke beech gap ki size hai; ε "close enough" tolerance hai jahan hum rokте hain.
Main slope aur tangent line explain kar sakta hoon
Slope = rise over run; tangent woh straight line hai jo curve ki direction se ek point par match karti hai.
Main jaanta hoon f′(x) kya hai aur kahan se aata hai
Derivative: ek aur machine jo x par curve ki steepness (tangent slope) deti hai; jaise x2−2 ke liye yeh 2x hai.
