IBDP Math AI HL: Advanced Modelling & Calculus

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Differentiating Sine and Cosine Functions

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**Derivatives:** - \( \frac{d}{dx}(\sin x) = \cos x \) - \( \frac{d}{dx}(\cos x) = -\sin x \) **Significance:** Unlike polynomial functions, trigonometric derivatives are periodic. This is fundamental for modelling Simple Harmonic Motion (SHM) where acceleration is proportional to displacement but directed in the opposite direction (e.g., \( a = -\omega^2 x \)).

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Chain Rule for Composite Functions

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**Formula:** \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \) **Application:** Used when differentiating a function of a function, e.g., \( y = \sin(3x^2) \). Let \( u = 3x^2 \), then \( \frac{dy}{dx} = \cos(3x^2) \cdot 6x \). Essential for determining rates of change in nested physical models.

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Product Rule

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**Formula:** If \( y = u \cdot v \), then \( \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \). **Context:** Necessary when two variable quantities multiply together, such as calculating the instantaneous rate of change of Area (\( A = x \cdot y \)) where both length \( x \) and width \( y \) are changing over time.

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Quotient Rule

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**Formula:** If \( y = \frac{u}{v} \), then \( \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \). **Note:** Often used in calculus-based economics to find marginal costs or velocities defined as ratios (e.g., average velocity \( \bar{v} = \frac{d(t)}{t} \)).

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Optimization: Finding Extrema

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**Procedure:** 1. Differentiate the function \( f(x) \) to find \( f'(x) \). 2. Solve \( f'(x) = 0 \) for stationary points. 3. Use the second derivative \( f''(x) \) to classify: - \( f''(x) > 0 \) \( \rightarrow \) Local Minimum - \( f''(x) < 0 \) \( \rightarrow \) Local Maximum **Real-world:** Maximizing profit or minimizing material surface area.