Insurance and Annuity

Answer

Step 1: Find dy/dx = 2x + 2 Step 2: At (1, 3), slope m = 2(1) + 2 = 4 Step 3: Using point-slope form: y - 3 = 4(x - 1) Step 4: Simplifying: y = 4x - 1 Therefore, equation of tangent is y = 4x - 1…

Answer

If m is the slope of tangent at a point, then: • Slope of normal = -1/m (provided m ≠ 0) • This is because tangent and normal are perpendicular to each other • For perpendicular lines: m₁ × m₂ = -1 • …

Answer

Step 1: Given dr/dt = 3 cm/sec, find dA/dt when r = 5 cm Step 2: A = πr², so dA/dt = 2πr(dr/dt) Step 3: When r = 5 cm: dA/dt = 2π(5)(3) = 30π cm²/sec Therefore, area increases at 30π cm²/sec when radi…

Answer

Formula: f(a + h) ≈ f(a) + h·f'(a) Where: • a is a convenient value near the required point • h is a small increment • f'(a) is the derivative at x = a Use when: • h is small (typically < 0.1) • f'(a…

Answer

Step 1: Let f(x) = √x, so f'(x) = 1/(2√x) Step 2: Choose a = 64, h = 0.1 (since 64.1 = 64 + 0.1) Step 3: f(64) = √64 = 8, f'(64) = 1/(2√64) = 1/16 Step 4: Using formula: f(64.1) ≈ 8 + (0.1)(1/16) = 8.

Answer

If function f(x) satisfies: 1. f is continuous on closed interval [a, b] 2. f is differentiable on open interval (a, b) 3. f(a) = f(b) Then there exists at least one c ∈ (a, b) such that f'(c) = 0 G…

Answer

Step 1: Check continuity - polynomial, so continuous on [1, 3] ✓ Step 2: Check differentiability - polynomial, so differentiable on (1, 3) ✓ Step 3: Check f(1) = f(3): f(1) = 1 - 4 + 3 = 0 f(3) …

Answer

If function f(x) satisfies: 1. f is continuous on closed interval [a, b] 2. f is differentiable on open interval (a, b) Then there exists at least one c ∈ (a, b) such that: f'(c) = [f(b) - f(a)]/(b -…