4.3.17 · D1Maclaurin series of eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ — deri
Ek smooth curve kisi ek point ke paas apni height, slope, bend, aur us point par har deeper rate-of-change se poori tarah describe hoti hai . Ek Maclaurin series simple powers 1 , x , x 2 , x 3 , … x = 0
Yeh page kuch bhi assume nahi karta . Parent note (parent topic) mein jo bhi symbols aate hain, unhe yahan unpack kiya gaya hai, ek aisi order mein jahan har idea sirf pehle waale ideas par tikaa hai.
Function ek machine hai: tum ek number x f ( x ) f x f x
Picture. Inputs ke liye ek horizontal number line banao aur outputs ke liye ek vertical. Har input x f ( x ) curve banta hai (uska graph).
Intuition Topic ko yeh kyun chahiyeJo kuch bhi hum banate hain — e x sin x curve ko copy karne ke baare mein hai ek simpler machine (polynomial) se. "Function" ka notion nahi ⇒ copy karne ke liye kuch nahi.
Definition Power notationx n x n x 2 = x ⋅ x x 3 = x ⋅ x ⋅ x n exponent hai. Special cases: x 1 = x x 0 = 1 1
Picture. Jaise n x = 0.1 chhoti input ke liye x n 0.1 , 0.01 , 0.001 , … x = 0
Intuition Isko kyun chahiyePolynomial — jis cheez se hum approximate karte hain — sirf powers ka ek weighted sum hai: a 0 + a 1 x + a 2 x 2 + ⋯ x n
Derivative f ′ ( x ) f x x slope hai: wahan height kitni tezi se chadh ya utar rahi hai. Ise d x d f ( x )
Picture. Curve mein itna zoom karo ki woh ek straight line jaisi lagey — us line ki steepness (rise over run) f ′ ( x )
Intuition Yeh tool kyun aur koi nahi?Hum ek point par curve ko pin down karna chahte hain. Height f ( 0 ) f ′ ( 0 ) Taylor series (general centre a) dekho.
Definition Repeated derivativesSlope ki slope lo: f ′′ ( x ) second derivative ) bend / curvature measure karta hai. Phir se karo f ′′′ ( x ) f ( n ) ( x ) f n ( n ) counter hai, power nahi.
f ( n ) f n f ( 3 ) ( x ) nahi ( f ( x ) ) 3 3
n ! n 1 n 3 ! = 1 ⋅ 2 ⋅ 3 = 6 , 4 ! = 1 ⋅ 2 ⋅ 3 ⋅ 4 = 24 , 0 ! = 1. 0 ! = 1 1 x 0 = 1
Picture. Factorials explode karte hain: 1 , 1 , 2 , 6 , 24 , 120 , 720 , …
Intuition Topic ko factorials kyun chahiyeJab tum x n n n ⋅ ( n − 1 ) ⋯ 1 = n ! n ! d x d x 3 = 3 x 2 , d x d 3 x 2 = 6 x , d x d 6 x = 6 = 3 ! .
Symbol n = 0 ∑ ∞ n = 0 n = 1 n = 2 n = 0 start hai; upar ∞
Isko slowly padho:
∑ n = 0 ∞ n ! x n = n = 0 0 ! x 0 + n = 1 1 ! x 1 + n = 2 2 ! x 2 + ⋯ = 1 + x + 2 x 2 + ⋯
Intuition Isko kyun chahiye"+ ⋯ ∑ exact rule ek saath batata hai. Yeh shorthand hai, nayi maths nahi.
Infinitely many cheezein add karna dangerous lagta hai — surely total infinite hoga? Hamesha nahi. Agar pieces itni tezi se shrink hoon, toh running total ek finite number par aa jaata hai .
Picture. Running total (jise partial sum kehte hain) 1 , 2 , 3 , … converges karti hai. Agar woh infinity ki taraf bhaagein ya jump karein, toh diverges karti hai.
Definition Interval of convergenceInput values x interval of convergence hai. Iske bahar, polynomial nonsense hai chahe original function theek ho. Yeh apne aap mein ek poora topic hai: Radius & interval of convergence .
ln ( 1 + x ) x = − 1 x = − 1 − 1 − 2 1 − 3 1 − ⋯ − ∞ ln ( 0 ) = − ∞
Definition Geometric seriesEk geometric series ek fixed ratio r 1 + r + r 2 + r 3 + ⋯ ∣ r ∣ < 1 1 1 + r + r 2 + ⋯ = 1 − r 1 ( ∣ r ∣ < 1 ) .
Picture. Har term apne pehle wale ka ek fixed fraction hai, toh terms ek shrinking staircase banate hain jiska total area finite hai. Poori derivation Geometric series mein hai.
Intuition Topic iske liye kyun jaata haiHum ln ( 1 + x ) free mein milega 1 + x 1 1 + x 1 ek geometric series hai ratio r = − x 1 + x 1 = 1 − ( − x ) 1 = 1 + ( − x ) + ( − x ) 2 + ⋯ = 1 − x + x 2 − x 3 + ⋯ r = − x
Integrating differentiation ko ulta chalana hai: ek slope function diya, height function recover karo. Symbol: ∫ ∫ x n d x = n + 1 x n + 1 + C . + C
Intuition Topic ko yeh kyun chahiye1 + x 1 ln ( 1 + x ) C ln ( 1 + 0 ) = ln 1 = 0
Definition Binomial coefficient( k n ) n k ( k n ) = k ! n ( n − 1 ) ( n − 2 ) ⋯ ( n − k + 1 ) ( exactly k factors upar ) . n ≥ 0 fractional n 2 1
Intuition Topic ko yeh kyun chahiye( 1 + x ) n ∑ ( k n ) x k Binomial theorem (integer n) hai jo kisi bhi real power tak stretch kiya gaya hai — parent ke 1 + x n = 2 1
Ek function even hai agar uska graph vertical axis par mirror image ho (f ( − x ) = f ( x ) cos x odd hai agar iske paas origin ke baare mein 180° rotational symmetry ho (f ( − x ) = − f ( x ) sin x
Picture. Even = reflection (jaise x 2 x 3 x n n n
Intuition Yeh kaam kyun bachata haisin x sirf odd powers ho sakti hain — har even-power coefficient zero hona chahiye. Isi tarah cos x kisi bhi calculation se pehle .
Definition Imaginary uniti i 2 = − 1 e i x = cos x + i sin x Euler's formula ) mention hota hai; panch core series derive karne ke liye tumhe i zaroorat nahi , lekin jab symbol aaye toh iska yahi matlab hai.
Slope = derivative f prime
Coefficient rule an = f n at 0 over n!
Where the series is valid
Which terms vanish in sin cos
Maclaurin series topic 4.3.17
Khud ko test karo — right side cover karo aur zyabaan se jawab do.
f ′ ( x ) Wahan curve ki slope (steepness).
f ( 4 ) ( x ) f
4 ! 1 ⋅ 2 ⋅ 3 ⋅ 4 = 24
0 ! 0 ! = 1 1
d x n d n x n n ! n ! 1
n = 0 ∑ 2 n ! x n 1 + x + 2 x 2
1 + r + r 2 + ⋯ Jab ∣ r ∣ < 1 1 − r 1
1 + x 1 1 − x + x 2 − x 3 + ⋯ r = − x
∫ x n d x n + 1 x n + 1 + C
( 2 1/2 ) 2 ! 2 1 ⋅ ( − 2 1 ) = − 8 1
Kya sin x Odd; uski series mein sirf odd powers bachti hain.
"Interval of convergence" tumhe kya batata hai? Exactly woh set of x
