📚 Key Concepts
Rate of Change- Rate = dy/dx
- Speed = absolute value of velocity
- Growth/Decay rates in real-world problems
Increasing/Decreasing Functions- Increasing: f'(x) > 0
- Decreasing: f'(x) < 0
- Strictly increasing/decreasing based on inequalities
- Points of local change determined by f'(x) = 0
Maxima and Minima- First Derivative Test:
- Maximum if f'(x) changes from + to -
- Minimum if f'(x) changes from - to +
- Second Derivative Test:
- Maximum if f''(x) < 0
- Minimum if f''(x) > 0
🎯 Key Formulas
Tangent equation: y - y₁ = m(x - x₁)where m = f'(x₁)Normal equation: y - y₁ = -1/m(x - x₁)Points of inflection: f''(x) = 0Absolute extrema on [a,b]:- Find f'(x) = 0 points
- Check endpoints a and b
- Compare all values
⚠️ Common Mistakes to Avoid
Not checking endpoints for absolute extremaConfusing local and absolute extremaWrong application of derivative testsForgetting critical pointsIncorrect tangent/normal equationsNot verifying nature of extrema
📖 Knowledge Prerequisites
- Differentiation rules
- Function behavior
- Graphing skills
- Basic calculus concepts
💡 Tips for Students
Always make a sign chart for f'(x)Check both first and second derivative testsList all critical points systematicallyDraw rough graphs to visualizeRemember to check endpointsVerify answers using multiple methods
👉 Practice Recommedations
Find rates of change in real situationsDetermine increasing/decreasing intervalsLocate maxima and minimaSolve optimization problemsFind equations of tangents/normalsPractice application-based questionsSolve word problems using maxima/minima
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