Applications of Derivatives

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The derivative f'(a) represents the slope of the tangent line to the curve y = f(x) at point P(a, b). It is obtained as the limit: f'(a) = lim(h→0) [f(a+h) - f(a)]/h. The tangent line has equation (y …

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Step 1: Find dy/dx = 2x + 4 Step 2: At x = -1, slope = 2(-1) + 4 = 2 Step 3: Using point-slope form: (y - (-2)) = 2(x - (-1)) Step 4: Simplify: y + 2 = 2x + 2 Step 5: Final answer: 2x - y = 0…

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The normal is perpendicular to the tangent at point P(a, b). Slope of normal = -1/f'(a) (negative reciprocal of tangent slope) Equation of normal: (y - b) = -1/f'(a)(x - a) Note: Normal exists only wh…

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A function f(x) is increasing in interval (a, b) if: 1. f(x₂) > f(x₁) whenever x₂ > x₁ for all x₁, x₂ ∈ (a, b) 2. Mathematically: f'(x) > 0 for all x ∈ (a, b) 3. Geometrically: The curve rises as we m…

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Step 1: Find f'(x) = 3x² - 6x + 3 Step 2: Factor: f'(x) = 3(x² - 2x + 1) = 3(x - 1)² Step 3: Since (x - 1)² ≥ 0 for all x ∈ R Step 4: f'(x) = 3(x - 1)² ≥ 0 for all x ∈ R Step 5: f'(x) > 0 for all x ≠ …

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Step 1: Find f'(x) = 6x² - 18x + 12 Step 2: Factor: f'(x) = 6(x² - 3x + 2) = 6(x - 1)(x - 2) Step 3: For decreasing function: f'(x) < 0 Step 4: 6(x - 1)(x - 2) < 0 Step 5: (x - 1)(x - 2) < 0 Step 6: T…

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For function f(x) at point x = c: Local Maximum: - First derivative test: f'(c) = 0 and f'(x) changes from + to - - Second derivative test: f'(c) = 0 and f''(c) < 0 Local Minimum: - First derivative…

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Step 1: f'(x) = 9x² - 18x - 27 = 9(x² - 2x - 3) = 9(x + 1)(x - 3) Step 2: f'(x) = 0 gives x = -1, x = 3 Step 3: f''(x) = 18x - 18 Step 4: At x = -1: f''(-1) = -36 < 0 → Maximum Step 5: f(-1) = 3(-1) -…