Derivatives & MVT — Real Analysis I

Chapter 5

Differentiation & MVT

Differentiability at a Point

Definition

Let \( f: A \to \mathbb{R} \). We say \( f \) is differentiable at \( c \in A \) if the limit of the secant slopes exists: \[ f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c} = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h} \]

Hinglish: Derivative ka matlab hai curve pe do points (secant) lo, aur unhe paas laate jao. Agar left side se aane wali secant aur right side se aane wali secant ek hi tangent line banayein, toh function differentiable hai. Agar graph mein sharp "corner" hai (jaise \( |x| \)), toh tangents clash karenge aur derivative exist nahi karega!

Theorem — Differentiability \( \Rightarrow \) Continuity

If \( f \) is differentiable at \( c \), then \( f \) is continuous at \( c \). The converse is FALSE!

Hinglish: Differentiable → Continuous (hamesha sach). Lekin ulta sach nahi! Example: \( f(x) = |x| \) at \( x = 0 \): continuous hai ✅ (graph mein koi break nahi) lekin differentiable nahi ❌ (sharp corner hai). Toh "smooth" (differentiable) hona "connected" (continuous) hone se zyada strong condition hai.

Rolle's Theorem (MVT ka Special Case)

Theorem — Rolle's Theorem

If \( f \) is continuous on \( [a,b] \), differentiable on \( (a,b) \), and \( f(a) = f(b) \), then there exists at least one point \( c \in (a,b) \) where \( f'(c) = 0 \).

Hinglish: Agar function start aur end pe same height pe hai (f(a) = f(b)), toh beech mein kahin na kahin ek flat point (slope = 0) zaroor hoga! Example: Ek ball ko oopar phenko — wo same height pe wapas aayegi. Toh beech mein kisi point pe ball ki speed exactly 0 hogi (jab wo apni maximum height pe hogi). Mathematical Example: \( f(x) = x^2 - 4x + 3 \) on \( [1, 3] \). \( f(1) = 0, f(3) = 0 \). \( f'(x) = 2x - 4 = 0 \implies x = 2 \). At \( c = 2 \), slope is 0. ✅

The Mean Value Theorem (MVT)

Theorem

If \( f \) is continuous on \( [a,b] \) and differentiable on \( (a,b) \), then there exists at least one point \( c \in (a,b) \) such that: \[ f'(c) = \frac{f(b) - f(a)}{b - a} \]

Hinglish: MVT geometric terms mein kehta hai: interval ke start aur end points ko jodne wali line (secant) ke parallel, graph pe kahin na kahin ek tangent zaroor milegi. Physical terms mein: Agar tumhari average speed 60 km/h thi, toh safar mein kisi ek point pe tumhari exact speedometer speed exactly 60 km/h rahi hogi.