Taylor's remainder theorem — error estimation

4.3.18 · Maths › Calculus III — Sequences & Series

1. Problem setup karna

KYA chahiye humein: $R_{n} (x)$ ka ek usable formula. KYU hum ise directly compute nahi kar sakte: agar hume $f (x)$ exactly pata hoti toh approximation ki zaroorat hi nahi hoti. Isliye humein ek bound chahiye.

2. Lagrange remainder ko scratch se derive karna

Hum ise Mean Value Theorem se derive karte hain, generalized form mein. Yeh sabse clean tarika hai.

Derivation (Cauchy MVT trick). $x$ ko fix karo. $a$ ko variable $t$ maano aur define karo: $g (t) = f (x) - \sum_{k = 0}^{n} \frac{f ^{(k)} ( t )}{k !} (x - t)^{k} .$

Yeh step kyun? $g (t)$ bas woh remainder hai jab hum $t$ par center karte hain. Note karo $g (x) = 0$ ( $x$ par centered Taylor polynomial $f (x)$ ke barabar hai) aur $g (a) = R_{n} (x)$ .

$g (t)$ ko differentiate karo. Lagbhag sab kuch telescope ho jaata hai: $g^{'} (t) = - \frac{f ^{(n + 1)} ( t )}{n !} (x - t)^{n} .$

Yeh step kyun? Consecutive $k$ ke product-rule terms cancel ho jaate hain — yahi Taylor structure ka kamaal hai. Sirf top derivative bachti hai.

Ab $h (t) = (x - t)^{n + 1}$ introduce karo, toh $h (x) = 0$ , $h (a) = (x - a)^{n + 1}$ , aur $h^{'} (t) = - (n + 1) (x - t)^{n}$ .

Cauchy Mean Value Theorem ko $g$ aur $h$ par $[a, x]$ par apply karo: koi $c$ exists karta hai $a$ aur $x$ ke beech jisme: $\frac{g ( a ) - g ( x )}{h ( a ) - h ( x )} = \frac{g ^{'} ( c )}{h ^{'} ( c )} .$

Yeh step kyun? Cauchy MVT do functions ki rates compare karta hai; isse unknown $c$ appear hota hai.

Sab kuch substitute karo: $\frac{R _{n} ( x ) - 0}{( x - a ) ^{n + 1} - 0} = \frac{- \frac{f ^{(n + 1)} ( c )}{n !} ( x - c ) ^{n}}{- ( n + 1 ) ( x - c ) ^{n}} = \frac{f ^{(n + 1)} ( c )}{( n + 1 )!} .$

$(x - c)^{n}$ cancel ho jaata hai, aur exactly Lagrange form milta hai. $■$

3. Error BOUND (jo part tum actually use karte ho)

Tumhe $c$ nahi pata. Lekin tumhe chahiye bhi nahi — tumhe bas $∣ f^{(n + 1)} ∣$ ka upper bound chahiye.

HOW to use it (recipe):

Center $a$ , degree $n$ , aur target point $x$ choose karo.
$f^{(n + 1)}$ nikalo aur ek bound $M = max ∣ f^{(n + 1)} ∣$ interval $[a, x]$ par dhundho.
Box mein plug karo.

Figure — Taylor's remainder theorem — error estimation

4. Worked examples

5. Common mistakes

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tum memory se kuch pencil strokes mein ek smooth hill draw kar rahe ho (yahi Taylor polynomial hai). Remainder theorem ek rule hai jo kehta hai: "tera sketch itne se zyada galat nahi hai." Woh "itna" nikalne ke liye, tum dekhte ho ki hill kitni wildly bend karti hai (next derivative) jitni stretch tumne draw ki, usse kitna door gaye ussse multiply karo, aur ek bade factorial number se divide karo jo jaldi error ko chhota bana deta hai. Toh kuch strokes se short walk ke liye almost perfect guarantee milti hai.

Flashcards

Taylor remainder

R_{n} (x)

ki definition kya hai?

R_{n} (x) = f (x) - P_{n} (x)

, degree-

n

approximation ki exact error.

Remainder ka Lagrange form batao.

R_{n} (x) = \frac{f ^{(n + 1)} ( c )}{( n + 1 )!} (x - a)^{n + 1}

kisi

c

ke liye jo

a

aur

x

ke beech hai.

Standard error bound kya hai?

∣ R_{n} (x) ∣ \leq \frac{M}{( n + 1 )!} ∣ x - a ∣^{n + 1}

jahan

M \geq ∣ f^{(n + 1)} ∣

interval par.

Bound mein kaun sa derivative aur factorial aata hai?

(n + 1)

-th derivative aur

(n + 1)!

— polynomial degree

n

se ek step aage.

c

ki specific value kyun nahi le sakte?

c

unknown hai; tumhe poore interval par

∣ f^{(n + 1)} ∣

ko uske max

M

se bound karna padta hai.

e^{x}

ke liye

a = 0

par, error ko bound karna easy kyun hai on

[0, 1]

Saari derivatives

e^{t} \leq e < 3

hain, isliye

M = 3

har

n

ke liye kaam karta hai.

e

1 0^{- 6}

accuracy tak dene ke liye kitne terms chahiye?

3/ (n + 1)! < 1 0^{- 6} \Rightarrow (n + 1)! > 3 \times 1 0^{6} \Rightarrow n = 9

chahiye.

Derivation mein kaun sa MVT result use hota hai?

Cauchy (generalised) Mean Value Theorem jo

g (t)

aur

h (t) = (x - t)^{n + 1}

par apply hota hai.

Connections

Taylor & Maclaurin Series — polynomial $P_{n}$ jise yeh bound karta hai.
Mean Value Theorem / Cauchy Mean Value Theorem — derivation ka engine.
Radius of Convergence — series tab converge hoti hai jab $R_{n} (x) \to 0$ .
Alternating Series Estimation Theorem — alternating series ke liye alternative error bound.
Big-O and asymptotic error — $R_{n} = O (∣ x - a ∣^{n + 1})$ near $a$ .