Foundations — Taylor's remainder theorem — error estimation

4.3.18 · D1 · Maths › Calculus III — Sequences & Series › Taylor's remainder theorem — error estimation

Is page par assume kiya gaya hai ki aapne pehle kuch nahi dekha. parent topic padhne se pehle, neeche diye har symbol par aapki pakad honi chahiye. Hum inhe order mein build karte hain — baad wala har ek pehle wale par lean karta hai.

1. Ek function $f(x)$ — woh machine

Picture. Neeche blue curve dekho. Apni ungali kisi input $x$ par flat axis pe slide karo; seedha uske upar curve ki height $f(x)$ hai.

Topic ko iske zarorat kyun hai. Poora game hai "ek curve ko polynomial se copy karo". Curve nahi, to game nahi.

2. Centre $a$ — jahaan hum khade hain

Picture. Upar wali figure mein, $a$ horizontal axis par orange dot hai, aur $f(a)$ uski height (orange) hai. Hamara copycat polynomial $a$ par bilkul perfect hoga aur jaise-jaise aap door jaate ho, drift karta jaayega.

Topic ko iske zarorat kyun hai. Taylor approximation local hai: $a$ ke paas excellent, door jaane par worse. $a$ se aap jitna chaalte ho use $x-a$ kehte hain, aur wahi poori error ko control karta hai.

3. Slope, aur derivative $f^{\prime }(x)$

Symbols se pehle, ek picture: kisi bhi smooth curve mein itna zoom karo aur woh ek straight line jaisi dikhne lagti hai. Us line ki steepness hi slope hai.

Picture. Neeche, green line blue curve ko sirf $a$ par kiss karti hai (bina cross kiye touch karti hai). Uski steepness hi $f^{\prime }(a)$ hai.

Topic ko iske zarorat kyun hai. Pehla useful copycat tangent line $f(a)+f^{\prime }(a)(x-a)$ hai: $a$ par $f$ jaisi height aur slope. Taylor bas aur matching qualities add karta jaata hai.

4. Higher derivatives $f^{\prime \prime }$, $f^{\prime \prime \prime }$, ..., $f^{(k)}$

Agar derivative khud ek machine hai (har point par slope), to hum uski derivative le sakte hain — slope kitni tezi se change ho rahi hai. Yeh bendiness (curvature) measure karta hai.

Picture. Neeche dekho: teen curves jinka $a$ par same height aur slope hai lekin bending alag-alag hai. $f^{\prime \prime }$ hi inhe alag karta hai — woh pehli cheez hai jo sirf slope capture nahi kar sakta.

Topic ko iske zarorat kyun hai. Ek degree-$n$ Taylor polynomial $f$ aur polynomial ko $f(a),f^{\prime }(a),\dots ,f^{(n)}(a)$ share karaata hai — yeh sab $n+1$ matched qualities. Error phir agla unmatched wala, $f^{(n+1)}$, govern karta hai.

5. Factorial $k!$ — tezi se badhne wala divisor

Picture. Factorials explode karte hain. Neeche, $k$ ke against plot mein dekh sakte ho ki $k!$ ordinary powers se kitni tezi se upar jaata hai — woh kisi bhi fraction mein upar wali cheez ko crush kar deta hai.

Topic ko iske zarorat kyun hai. Error bound $\frac{M}{(n+1)!}|x-a{|}^{n+1}$ mein, denominator mein factorial itni violently badhta hai ki ek bada top bhi near zero squash ho sakta hai — yahi wajah hai ki kuch Taylor terms add karne se error tiny ho jaata hai.

6. Sigma notation $\sum$ — ek compact "in sab ko add karo"

Topic ko iske zarorat kyun hai. Taylor polynomial $P_n(x)={\sum }_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a{)}^k$ terms ka ek poora pile hai; $\sum$ unhe sab ek saath likhta hai.

7. Absolute value $|\cdot |$ — size, sign ignore karo

Topic ko iske zarorat kyun hai. Error positive ho sakta hai (undershoot) ya negative (overshoot); hum sirf kitna bada hai yeh care karte hain. Isliye theorem $|R_n(x)|$ bound karta hai, gap ki magnitude.

8. Bound $M$ aur "$\le$" — ek ceiling jo hum guarantee kar sakte hain

Picture idea. Interval par $|f^{(n+1)}|$ ka curve imagine karo; $M$ ek flat horizontal ceiling hai uske highest peak ke upar bani hui. Hume exact peak ki zaroorat nahi — koi bhi honest ceiling kaam karti hai.

Topic ko iske zarorat kyun hai. $M$ hi woh cheez hai jo un-computable exact remainder ko ek usable, provable error estimate mein badal deta hai.

9. Symbols ko saath mein rakhna

Ab headline formula ka har piece plain words mein padhta hai:

Parent mein [!mnemonic] padho — "Next derivative, Next factorial, Next power" — ab jab aap jaante ho ki har word kis cheez ki taraf point karta hai.

Prerequisite map

Equipment checklist

Reveal karne se pehle plain-words mein zor se jawaab do.