Foundations — Net change theorem

4.2.5 · D1 · Maths › Calculus II — Integration › Net change theorem

Parent note ek hi saanch mein bohot saari notation fire karta hai: $F$, $F^{\prime }$, ${\int }_a^b$, $dx$, $\Delta x$, $\sum$, $\lim$, $v(t)$, $|v|$, subscripts $x_i$ aur $x_i^{\ast }$. Agar inme se ek bhi fog mein hai, toh poori derivation fog mein hai. Yeh page inhe ek-ek karke bilkul scratch se build karta hai, uss order mein jisme woh ek doosre par rely karte hain.

1. Ek quantity aur uska naam: $F(x)$

Picture ek graph hai: horizontal axis input $x$ hai (aksar time), vertical axis output $F(x)$ hai (amount). Curve par ek dot kehta hai "is time par, itni cheez".

Topic ko $F$ ki zaroorat kyun hai? Kyunki "net change" $F$ mein change hai — kisi cheez ki change ki baat tab tak nahi ho sakti jab tak tum cheez ko naam nahi de sakte.

2. Ek choti change: $\Delta$ (delta)

Isse pehle ki hum change ki rate ki baat karein, humein change ke liye hi ek seedha word chahiye.

Picture: $F$ ke graph par do kareeb points chuno. $\Delta x$ se seedha chalo; curve $\Delta F$ se upar (ya neeche) jaati hai. Do chote arrows — ek horizontal, ek vertical — exactly inhe measure karte hain.

Topic ko $\Delta$ ki zaroorat kyun hai: aage aane wali har cheez — rate, slice width, tiny change — "end minus start" se bani hai.

3. Settle karna: limit $\lim$

Agle step mein jo rate chahiye woh actually ek $\frac{\Delta F}{\Delta x}$ ratio hai jab step kuch nahi ho jaata. "Kuch nahi ho jaana" ek limit hai, isliye hum woh word ab define karte hain.

Picture: "kuch" mein $h=0.1$, phir $0.01$, phir $0.001$ punch karo. Agar answers ek value par home in karte hain, woh value limit hai. Yeh honest tarika hai "instant par" kehne ka bina kabhi zero se divide kiye.

Topic ko iski zaroorat kyun hai: derivative (§4) aur integral (§8) dono limits hain — approximations jo shrinking se exact ho jaati hain.

4. Change ki rate: $F^{\prime }(x)$

Picture: ek chota seedha ruler curve ke saath flat rakhna jaise woh ek point par bas graze kare — tangent line. Uski slope hi $F^{\prime }(x)$ hai. Steep uphill = bada positive $F^{\prime }$; downhill = negative $F^{\prime }$; flat = zero.

Topic ko $F^{\prime }$ ki zaroorat kyun hai: net change theorem ek rate se shuru hoti hai aur amount recover karti hai. $F^{\prime }$ rate hai; $F$ amount hai. Prime symbol unke beech ka bridge hai.

5. Slice edges ko label karna: subscripts $x_0,x_1,\dots ,x_n$

Kai choti changes add karne ke liye pehle hum $[a,b]$ ko $n$ equal slices mein kaatke har cut ko naam dete hain.

Star version $x_i^{\ast }$ (padho "$x_i$ star") ek sample point hai — slice number $i$ ke andar koi chuna hua spot (no star = ek edge; star = andar ka ek point). §7 mein iska exact kaam milega.

Topic ko subscripts ki zaroorat kyun hai: kai slices add karne ke liye tumhe "slice $i$" ke baare mein baat karni hogi bina har baar naya letter invent kiye.

6. Kai cheezein add karna: sum symbol $\sum$

Likh kar, $\sum _{i=1}^3{a}_i=a_1+a_2+a_3$. Kuch mysterious nahi — yeh ek compact "inhe add karo" hai.

Topic ko $\sum$ ki zaroorat kyun hai: net change kai choti changes added up hai; $\sum$ maths mein "added up" word hai.

7. Ek slice ki change ko rate × width mein badalna (Mean Value Theorem)

§6 ki telescoping sum sach muchi choti changes $F(x_i)-F(x_{i-1})$ add karti hai. Lekin hum chaahte hain woh changes rate $F^{\prime }$ ke terms mein likhein, taki ek sum of rates aaye. Yeh guarantee dene wala tool Mean Value Theorem hai.

Yahi §5 mein promised star point ka exact matlab hai: woh spot hai jahan small change = rate wahan × width, har sach muchi change ko ek rate contribution mein badal deta hai.

Topic ko iski zaroorat kyun hai: yeh woh hinge hai jo $F(x_i)-F(x_{i-1})$ ko $F^{\prime }(x_i^{\ast })\Delta x$ se swap karta hai, jisse raw changes ki sum ki jagah rates ki sum aa jaaye.

8. Definite integral: ${\int }_a^b{F}^{\prime }(x)dx$

Ab saare pieces assemble karo. Boxed Mean-Value line ko §6 ki telescoping sum mein daalo: $F(b)-F(a)={\sum }_{i=1}^n[F(x_i)-F(x_{i-1})]={\sum }_{i=1}^n{F}^{\prime }(x_i^{\ast })\Delta x.$ Far-right side ek Riemann sum hai: (rate × width) strips ka ek dher. §3 ki limit use karke slices ko infinitely thin jaane do, aur yeh dher definite integral ban jaata hai: $\boxed{{\int }_a^b{F}^{\prime }(x)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^n{F}^{\prime }(x_i^{\ast })\Delta x=F(b)-F(a)}$

Woh ek line hai jahan is page ka har symbol milta hai. Pieces padho:

Toh poora symbol literally matlab hai: rate curve aur axis ke beech signed area, $a$ se $b$ tak. Kyunki har strip (rate)×(width) = ek tiny change hai, strips add karne se total change milti hai. Yahi Riemann-sum idea hai, aur poore parent topic ka punchline.

9. Signed area, aur bars: $|v(t)|$

Parent displacement ($\int v$, signs rakhta hai — axis ke neeche wala area negative count hota hai) aur distance ($\int |v|$, below-axis area upar flip karta hai taki sab kuch positive count ho) mein split karta hai. Dekho Displacement vs Distance. Velocity $v(t)$ sirf ek $F^{\prime }$ hai jiska $F$ position hai — same machinery, physics costume.

Topic ko iski zaroorat kyun hai: signed vs unsigned sabse zyaada tested distinction hai, aur yeh poori tarah is baat mein rehta hai ki bars integral ke andar jaate hain ya nahi.

Prerequisite map

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle jawab do.

Connections

• Net change theorem — woh parent jiske liye yeh page tumhe prepare karta hai.
• Definite Integral as a Riemann Sum — jahan $\int$, $\sum$, aur $\lim$ sab milte hain.
• Fundamental Theorem of Calculus — $F^{\prime }$ ko $F$ se link karta hai.
• Telescoping Sums — §6 ka cancellation.
• Mean Value Theorem — sample point $x_i^{\ast }$ supply karta hai (§7).
• Displacement vs Distance — signed vs unsigned, §9 ke bars.
• Marginal Cost and Revenue — $F^{\prime }$ ka economics wala face.