Limits & Derivatives — Applied Maths 11

🔍 1. Limits — Paas Pahunchna

Limit batata hai ki jab x kisi value (a) ke paas pahunchta hai toh f(x) kahan pahunchta hai — chahe wahan f(a) defined ho ya na ho!

lim f(x) = L x→a Matlab: x ko a ke paas lete jaate hain (left se bhi, right se bhi), aur dekh te hain f(x) kis value L ke paas jaata hai. Left limit: lim f(x) = L⁻ (left side se aaye) x→a⁻ Right limit: lim f(x) = L⁺ (right side se aaye) x→a⁺ Limit exists tabhi jab L⁻ = L⁺

Standard Limits

lim (xⁿ − aⁿ)/(x − a) = n × aⁿ⁻¹ [Very important!] x→a lim sin(x)/x = 1 (x radians mein, x→0) x→0 lim (1 + 1/n)ⁿ = e ≈ 2.71828 n→∞ lim (eˣ − 1)/x = 1 x→0 lim (log(1+x))/x = 1 x→0

Example 1: Direct Substitution

lim (3x² + 2x − 1) as x→2

= 3(4) + 2(2) − 1 = 12 + 4 − 1 = 15

Example 2: 0/0 Form — Factor karo

lim (x² − 4)/(x − 2) as x→2

Direct substitution: 0/0 — factor karein!

= lim (x+2)(x−2)/(x−2) = lim (x+2) = 2+2 = 4

Example 3: Standard Limit Use

lim (x⁵ − 32)/(x − 2) as x→2

= lim (x⁵ − 2⁵)/(x − 2) = 5 × 2⁵⁻¹ = 5 × 2⁴ = 5 × 16 = 80

📐 2. Continuity

Function f(x) point a par continuous hai agar teen conditions satisfy hon:

f is continuous at x = a iff: 1. f(a) defined hai (exists) 2. lim f(x) exists (left aur right limits equal hain) x→a 3. lim f(x) = f(a) (limit = actual function value) x→a Agar koi bhi condition fail ho → discontinuity!

🚀 3. Derivatives — Instantaneous Rate of Change

Derivative = function ka slope at a point. Business mein: marginal cost = ΔCost/ΔQuantity ka instantaneous version!

First Principle (Definition): f'(x) = lim [f(x+h) − f(x)] / h h→0 Notations: dy/dx = f'(x) = Df(x) = ẏ — sab same hain!

Differentiation Rules — Yaad Karo!

1. Constant: d/dx (c) = 0 2. Power: d/dx (xⁿ) = nxⁿ⁻¹ ← MOST USED! 3. Constant×fn: d/dx (cf) = c·f' 4. Sum/Diff: d/dx (f±g) = f'±g' 5. Product: d/dx (fg) = f'g + fg' ← ILATE trick nahi, yeh hai! 6. Quotient: d/dx (f/g) = (f'g − fg')/g² 7. Chain Rule: d/dx [f(g(x))] = f'(g(x))·g'(x)

Standard Derivatives Table

f(x)	f'(x)	f(x)	f'(x)
xⁿ	nxⁿ⁻¹	eˣ	eˣ
√x = x^(1/2)	1/(2√x)	aˣ	aˣ·ln a
1/x = x⁻¹	−1/x²	ln x	1/x
c (constant)	0	logₐx	1/(x·ln a)

Example 1: Power Rule

y = 5x³ − 3x² + 7x − 2. Find dy/dx.

dy/dx = 5(3x²) − 3(2x) + 7(1) − 0

= 15x² − 6x + 7

Example 2: Product Rule

y = x²·eˣ. Find dy/dx.

Let f = x², g = eˣ. Then f' = 2x, g' = eˣ.

dy/dx = f'g + fg' = 2x·eˣ + x²·eˣ = eˣ(2x + x²) = xeˣ(x+2)

Example 3: Chain Rule

y = (3x² + 1)⁵. Find dy/dx.

Outer: d/du (u⁵) = 5u⁴, Inner: d/dx (3x²+1) = 6x

dy/dx = 5(3x²+1)⁴ × 6x = 30x(3x²+1)⁴

💼 4. Business Applications of Derivatives

Applied maths mein derivatives ka use real business problems solve karne mein hota hai!

Cost Function: C(x) = total cost, x = quantity produced Marginal Cost (MC) = dC/dx = rate of change of cost per unit Revenue Function: R(x) = p × x (price × quantity) Marginal Revenue (MR) = dR/dx Profit: P(x) = R(x) − C(x) Marginal Profit = dP/dx = MR − MC Maximum Profit when: MR = MC (aur d²P/dx² < 0)

Example: Profit Maximization

R(x) = 40x − x², C(x) = x³/3 − 8x² + 50x + 10. Find profit-maximizing quantity.

1

P(x) = R−C = 40x−x² − (x³/3−8x²+50x+10) = −x³/3 + 7x² − 10x − 10

2

P'(x) = −x² + 14x − 10. Set P'=0: x² − 14x + 10 = 0

3

x = (14 ± √(196−40))/2 = (14 ± √156)/2 ≈ (14 ± 12.49)/2

4

x ≈ 13.24 ya x ≈ 0.75. Check P''(x) = −2x+14. At x≈13.24: P''<0 ✓ Maximum!

🔬 Calculus Lab Open Karo →