# Day 21: Rest, Review, and Week 4 Preparation ## 📅 Schedule Overview | Block | Time | Duration | Activity | |-------|------|----------|----------| | Morning | 10:00 AM - 11:30 AM | 1.5 hours | Concept Review | | Afternoon | 2:00 PM - 3:00 PM | 1 hour | Self-Assessment | | Evening | 7:00 PM - 8:00 PM | 1 hour | Week 4 Preview | **Total Study Time: 3.5 hours (REST DAY)** --- ## 🧘 Rest Day Philosophy You've completed three intensive weeks covering: - Limits and continuity - Differentiation techniques - Applications of derivatives Your brain needs consolidation time. Honor this rest day. --- ## 📝 Week 3 Summary Sheet ### Related Rates Strategy 1. Draw diagram, label variables 2. Write equation relating variables 3. Differentiate with respect to time t 4. Substitute known values AFTER differentiating 5. Solve for unknown rate ### Linear Approximation $$f(x) \approx f(a) + f'(a)(x - a)$$ ### Differentials $$dy = f'(x) \cdot dx$$ ### Finding Extrema **Critical points:** Where f'(x) = 0 or f'(x) DNE **Closed Interval Method:** 1. Find critical points in (a, b) 2. Evaluate f at critical points and endpoints 3. Compare values **First Derivative Test:** - f' changes + to − → local max - f' changes − to + → local min **Second Derivative Test:** - f''(c) > 0 → local min - f''(c) < 0 → local max ### Optimization Strategy 1. Identify objective function 2. Write constraints 3. Reduce to single variable 4. Find critical points 5. Verify max/min ### L'Hôpital's Rule For 0/0 or ∞/∞ forms: $$\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$$ ### Newton's Method $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ --- ## 🔄 Self-Assessment Quiz **Q1:** Water fills a conical tank at 3 m³/min. If height = 2 × radius, find dh/dt when h = 4m.

Answer

V = (1/3)πr²h = (1/12)πh³ dV/dt = (1/4)πh²(dh/dt) 3 = (1/4)π(16)(dh/dt) dh/dt = 3/(4π) m/min

**Q2:** Find critical points of f(x) = x⁴ - 4x³.

Answer

f'(x) = 4x³ - 12x² = 4x²(x - 3) Critical points: x = 0 and x = 3

**Q3:** Evaluate: $\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$

Answer

Using L'Hôpital twice: = lim (eˣ - 1)/(2x) = lim eˣ/2 = 1/2

--- ## 📊 Concept Map: Applications of Derivatives ``` DERIVATIVES │ ┌───────────────┼───────────────┐ │ │ │ ▼ ▼ ▼ ┌─────────┐ ┌──────────┐ ┌──────────┐ │ Related │ │ Linear │ │ Extrema │ │ Rates │ │ Approx │ │ Finding │ └─────────┘ └──────────┘ └──────────┘ │ │ │ │ │ ▼ │ │ ┌──────────┐ │ │ │Optimization│ │ │ └──────────┘ │ │ │ └───────────────┼───────────────┘ │ ▼ ┌──────────────┐ │ L'Hôpital's │ │ Rule │ └──────────────┘ │ ▼ ┌──────────────┐ │ Newton's │ │ Method │ └──────────────┘ ``` --- ## 🔜 Week 4 Preview: Integration ### The Big Picture We've learned to find rates of change (derivatives). Now we reverse the process: **Given:** The rate of change f'(x) **Find:** The original function f(x) This is **antidifferentiation** or **integration**. ### Topics Coming Up **Day 22:** Antiderivatives - Reversing differentiation - Basic antiderivative rules **Day 23:** Area and the Definite Integral - Riemann sums - Area under curves **Day 24:** The Fundamental Theorem of Calculus - The profound connection between derivatives and integrals **Day 25:** Integration Techniques I - Substitution method **Day 26:** Week 4 Problem Set **Day 27:** Computational Lab **Day 28:** Rest and Review ### Key Formulas Preview | Derivative | Antiderivative | |------------|----------------| | xⁿ | x^(n+1)/(n+1) + C | | cos(x) | sin(x) + C | | sin(x) | -cos(x) + C | | eˣ | eˣ + C | | 1/x | ln|x| + C | ### Quantum Mechanics Connection Integration is essential in QM: - Normalization: ∫|ψ|² dx = 1 - Expectation values: ⟨x⟩ = ∫ψ* x ψ dx - Probability: P(a < x < b) = ∫ₐᵇ |ψ|² dx --- ## 📈 Progress Tracker: Month 1 | Week | Topic | Status | |------|-------|--------| | 1 | Limits & Continuity | ✅ Complete | | 2 | Differentiation | ✅ Complete | | 3 | Applications | ✅ Complete | | 4 | Integration | 🔄 Starting | **You are 75% through Month 1!** --- ## 💪 Motivation You've now mastered the differential calculus. This is a major milestone! The integral calculus (starting tomorrow) completes the picture. Together, they form the foundation of all advanced mathematics and physics. Every quantum mechanics equation you'll encounter relies on what you've learned and what's coming next. --- ## 📓 Reflection Questions 1. Which application of derivatives was most challenging? 2. How do related rates and optimization differ in their setup? 3. Why is the chain rule so important for related rates? 4. What real-world problems could you now solve that you couldn't before? --- ## ✅ Checklist Before Week 4 - [ ] Can set up and solve related rates problems - [ ] Can find linearizations and use differentials - [ ] Can find extrema on closed intervals - [ ] Can solve optimization problems - [ ] Can apply L'Hôpital's Rule - [ ] Can implement Newton's Method - [ ] Feel ready for integration! --- **Week 3 Complete! 🎉** Tomorrow begins the integral calculus—where we learn to "undo" differentiation and discover the beautiful connection between the two operations. *"The integral calculus is the most useful and powerful mathematical tool ever invented."*

Day 21: Rest, Review, and Week 4 Preparation

Today's Schedule (7 hours)

Day 21: Rest, Review, and Week 4 Preparation

Today's Schedule (7 hours)