Foundations — Fundamental Theorem of Calculus — Part 1 and Part 2 — full proofs

4.2.4 · D1 · Maths › Calculus II — Integration › Fundamental Theorem of Calculus — Part 1 and Part 2 — full p

Yeh page kuch bhi assume nahi karta. parent topic mein proofs padhne se pehle, har squiggle, letter, aur idea jis par woh tika hai — yahan neeche se upar tak, ek order mein banaya gaya hai jahan har piece apne pehle wale par tika ho.

1. Function $f$ aur uski curve

Picture karo. Koi $x$ chuno, curve tak jao, height padho — woh height hi $f(x)$ hai.

Topic ko yeh kyun chahiye. FTC ek continuous curve $f$ ke baare mein statement hai: yeh us curve ke neeche ke area ko us curve ki height se relate karta hai. Dono ideas ek hi picture par rehte hain.

2. Curve ke neeche area, aur symbol $\int$

Hum curvy area kaise measure karte hain? Hum ise patli rectangles se approximate karte hain, phir rectangles ko infinitely patla hone dete hain. Woh summing-up process ek special stretched-S symbol se likhi jaati hai.

Picture karo. Near-zero width ka ek tall column, area $f(x)dx$, $a$ se $b$ tak slide karta hai; unhe sab add karo aur total shaded area milta hai.

Topic ko yeh kyun chahiye. FTC Part 2 ka left side yahi symbol hai. Iske baare mein kuch bhi prove karne ke liye hume pata hona chahiye ki iska matlab "limit of rectangle sums" hai — exactly yahi Riemann Sums and the Definite Integral mein poori tarah se banta hai.

3. Dummy variable $t$ vs. real variable $x$

Picture karo. $t$ ek finger hai jo base par $a$ se right ki taraf slide karta hai; $x$ woh jagah hai jahan tum finger rok dete ho. Jahaan roko woh badlo, aur shaded area badal jaata hai — isliye area $x$ ka function hai, aur isiliye hum ise $F(x)$ kehte hain.

Topic ko yeh kyun chahiye. $F(x)$ — "ab-tak-ka-area" function — Part 1 ka star hai. Dono letters ko seedha samajhna valid proof aur nonsense ke beech ka fark hai.

4. Continuity — woh ek hypothesis jo dono proofs ko chahiye

Picture karo. Ek unbroken pen-stroke vs. ek curve jo achanak ek nayi height par jump karti hai (jump) — sirf unbroken wali continuous hai.

Topic ko yeh kyun chahiye. Dono proofs mein yeh quietly zaroor hai:

• Min/max rectangles (agla section) exist hi isliye karte hain kyunki $f$ closed interval par continuous hai.
• Squeeze at the end isliye kaam karta hai kyunki "inputs paas ⇒ outputs paas" literally continuity hi hai.

Precise meaning Continuity mein master karo.

5. Sliver ko trap karne ke tools: EVT aur Squeeze

Picture karo. Kisi bhi closed stretch ki unbroken curve par ek sachcha peak aur ek sachcha valley hota hai — curve "ek aisi value ki taraf nahi bhaag sakti jo woh kabhi actually touch na kare."

Topic ko yeh kyun chahiye. $x$ se $x+h$ tak ke patле sliver ka area sabse chhoti possible rectangle $m_h\cdot h$ aur sabse badi $M_h\cdot h$ ke beech dabba hua hai. EVT guarantee karta hai ki woh dono heights exist karti hain. Extreme Value Theorem dekho.

Picture karo. Ek sandwich: upar ki slice aur neeche ki slice ko saath dabao aur jo bhi beech mein hai uske paas koi choice nahi — woh usi jagah milega.

Topic ko yeh kyun chahiye. Jab $h\to 0$, dono $m_h$ aur $M_h$ $f(x)$ ki taraf slide karte hain; Squeeze Theorem difference quotient (jo unke beech hai) ko $f(x)$ ke barabar hone par force karta hai — yahi Part 1 hai. Squeeze Theorem mein detail se diya hai.

6. Derivative $F^{\prime }(x)$ aur difference quotient

Picture karo. $\frac{F(x+h)-F(x)}{h}$ $F$ ke graph par do nearby points ko jodne wali line ka slope hai; jab $h\to 0$ woh line tangent ban jaati hai, aur uska slope $F^{\prime }(x)$ hai.

Topic ko yeh kyun chahiye. Part 1 claim karta hai $F^{\prime }(x)=f(x)$. Ise prove karne ke liye hume $F^{\prime }$ ki definition — is limit — se shuru karna hoga, aur kuch nahi. Symbol ${\lim }_{h\to 0}$ ka matlab sirf itna hai "woh value jo $h$ ke $0$ tak shrink hone par approach ho rahi hai."

7. Antiderivatives aur constant $C$

Topic ko yeh kyun chahiye. Part 2 kehta hai: ${\int }_a^{b}f$ paane ke liye, koi bhi antiderivative $G$ dhundo aur $G(b)-G(a)$ compute karo. Reverse rules se $G$ dhundhna Antiderivatives and Indefinite Integrals ka poora content hai.

8. Mean Value Theorem — woh honest hinge

Picture karo. Start point se end point tak straight chord khiincho; kahin beech mein, curve us chord ke parallel chalta hai.

Topic ko yeh kyun chahiye. Part 2 ko chahiye ki "agar $(F-G{)}^{\prime }=0$ har jagah hai toh $F-G$ constant hai." Yeh obvious nahi hai — MVT ise prove karta hai: $G^{\prime }(c)=0$ ke saath, formula deta hai $G(b)-G(a)=0$ kisi bhi endpoints ke liye, toh $G$ kabhi change nahi karta. Mean Value Theorem mein poori treatment hai.

9. Chain Rule — moving limits ke liye

Topic ko yeh kyun chahiye. Jab upper limit plain $x$ nahi balki $x^2$ ya $x^3$ jaisi koi cheez ho, jaise $\frac{d}{dx}{\int }_0^{x^2}f$, Part 1 "$H^{\prime }$" piece handle karta hai aur Chain Rule extra factor $\frac{d}{dx}(x^2)=2x$ deta hai. Miss karo toh galat answer milega.

10. (Optional) Jab region infinite ho

Prerequisite map

Equipment checklist

Khud ko test karo — har ek sirf answer bol dene ke baad reveal karo.