Learning Objectives
- Understand rate of change using derivatives
- Learn about increasing/decreasing functions
- Study tangents, normals, and approximations
- Find maxima and minima of functions
Key Concepts
Rate of Change
If y = f(x), then dy/dx represents the rate of change of y with respect to x. If quantity Q changes with time t, then dQ/dt is the rate of change. Example: If area A = πr2, then dA/dt = 2πr(dr/dt). These are called related rates problems.
Increasing and Decreasing Functions
f is strictly increasing on (a,b) if f'(x) > 0 for all x ∈ (a,b). f is strictly decreasing on (a,b) if f'(x) < 0 for all x ∈ (a,b). f is constant if f'(x) = 0 for all x. Critical points: Where f'(x) = 0 or f'(x) is undefined.
Tangent and Normal
At point (x₁, y₁) on curve y = f(x): Slope of tangent = f'(x₁). Equation of tangent: y - y₁ = f'(x₁)(x - x₁). Slope of normal = -1/f'(x₁). Equation of normal: y - y₁ = [-1/f'(x₁)](x - x₁). If f'(x₁) = 0, tangent is horizontal (y = y₁). If f'(x₁) → ∞, tangent is vertical (x = x₁).
Approximations
Linear approximation: f(x + Δx) ≈ f(x) + f'(x)Δx. Differential: dy = f'(x)dx. Useful for computing approximate values like √(25.1), (3.02)4, etc.
Maxima and Minima
First Derivative Test: At critical point c where f'(c) = 0: If f' changes from + to - → local maximum. If f' changes from - to + → local minimum. If f' does not change sign → neither (point of inflection).
Second Derivative Test: At critical point c where f'(c) = 0: If f''(c) < 0 → local maximum. If f''(c) > 0 → local minimum. If f''(c) = 0 → test fails, use first derivative test.
Absolute maxima/minima on [a,b]: Compare f(a), f(b), and f(c) for all critical points c in (a,b). Largest value = absolute maximum; smallest = absolute minimum.
Point of inflection: Where concavity changes; f''(x) = 0 and f'' changes sign.
Summary
Derivatives measure rate of change and determine whether functions are increasing or decreasing. Tangent and normal lines are found using the derivative at a point. Maxima and minima are identified using first or second derivative tests. On a closed interval, absolute extrema occur at critical points or endpoints.
Important Terms
- Critical point: Where f'(x) = 0 or f'(x) doesn't exist
- Local maximum: f'(c) = 0 and f'' changes from + to - (or f''(c) < 0)
- Local minimum: f'(c) = 0 and f'' changes from - to + (or f''(c) > 0)
- Point of inflection: Where concavity changes; f''(x) = 0
- Related rates: Rates connected by a common variable (usually time)
Quick Revision
- f'(x) > 0 → increasing; f'(x) < 0 → decreasing
- Tangent slope = f'(x₁); Normal slope = -1/f'(x₁)
- Approximation: f(x+Δx) ≈ f(x) + f'(x)Δx
- First derivative test: sign change of f' at critical point
- Second derivative test: f''(c) < 0 → max; f''(c) > 0 → min
- Absolute extrema on [a,b]: check f(a), f(b), f(critical points)
- For related rates: differentiate the relationship equation with respect to t