# Day 14: Rest, Review, and Week 3 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 | Flashcard Review | | Evening | 7:00 PM - 8:00 PM | 1 hour | Week 3 Preview | **Total Study Time: 3.5 hours (REST DAY)** --- ## 🧘 Rest Day Importance After an intensive week learning differentiation, your brain needs time to: - Consolidate new neural pathways - Process the relationships between concepts - Recover from cognitive load **Honor this rest day—it's part of the learning process.** --- ## 📝 Week 2 Summary Sheet ### The Derivative Definition $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ ### Basic Differentiation Rules | Rule | Formula | |------|---------| | Constant | d/dx[c] = 0 | | Power | d/dx[xⁿ] = nxⁿ⁻¹ | | Constant Multiple | d/dx[cf] = cf' | | Sum | d/dx[f ± g] = f' ± g' | | Product | d/dx[fg] = f'g + fg' | | Quotient | d/dx[f/g] = (f'g - fg')/g² | | Chain | d/dx[f(g(x))] = f'(g(x))·g'(x) | ### Special Derivatives | Function | Derivative | |----------|------------| | eˣ | eˣ | | aˣ | aˣ ln(a) | | ln(x) | 1/x | | sin(x) | cos(x) | | cos(x) | -sin(x) | | tan(x) | sec²(x) | ### Implicit Differentiation Process 1. Differentiate both sides with respect to x 2. Apply chain rule to y terms (multiply by dy/dx) 3. Collect dy/dx terms 4. Solve for dy/dx --- ## 🔄 Self-Assessment Quiz Answer without notes, then check: **Q1:** What is the derivative of f(x) = x³·eˣ?

Answer

f'(x) = 3x²·eˣ + x³·eˣ = eˣ(x³ + 3x²) = x²eˣ(x + 3)

**Q2:** Find dy/dx for x² + y² = 9.

Answer

dy/dx = -x/y

**Q3:** What is d/dx[sin(x²)]?

Answer

d/dx[sin(x²)] = cos(x²)·2x = 2x cos(x²)

**Q4:** Find the equation of the tangent line to y = x² at x = 3.

Answer

f(3) = 9, f'(3) = 6 Tangent: y - 9 = 6(x - 3), or y = 6x - 9

**Q5:** If f(x) = ln(x³ + 1), what is f'(x)?

Answer

f'(x) = 3x²/(x³ + 1)

--- ## 📊 Concept Connections Map ``` LIMITS │ ▼ ┌───────────────┐ │ DERIVATIVE │ │ Definition │ └───────────────┘ │ ┌───────────┼───────────┐ ▼ ▼ ▼ ┌─────────┐ ┌─────────┐ ┌─────────┐ │ Power │ │ Product │ │ Chain │ │ Rule │ │ Rule │ │ Rule │ └─────────┘ └─────────┘ └─────────┘ │ │ │ └───────────┼───────────┘ ▼ ┌───────────────┐ │ IMPLICIT │ │ Differentiation│ └───────────────┘ │ ▼ ┌───────────────┐ │ APPLICATIONS │ │ (Week 3) │ └───────────────┘ ``` --- ## 📚 Flashcard Review Review these key cards: **Card 1:** - Front: Power Rule - Back: d/dx[xⁿ] = nxⁿ⁻¹ **Card 2:** - Front: Product Rule - Back: (fg)' = f'g + fg' **Card 3:** - Front: Quotient Rule - Back: (f/g)' = (f'g - fg')/g² **Card 4:** - Front: Chain Rule - Back: d/dx[f(g(x))] = f'(g(x))·g'(x) **Card 5:** - Front: d/dx[eˣ] - Back: eˣ (only function equal to its derivative!) **Card 6:** - Front: d/dx[ln(x)] - Back: 1/x **Card 7:** - Front: Implicit differentiation key step - Back: When differentiating y, multiply by dy/dx --- ## 🔜 Week 3 Preview: Applications of Derivatives ### Topics Coming Up **Day 15:** Related Rates - How quantities change together - Setting up and solving word problems **Day 16:** Linear Approximation and Differentials - Using tangent lines to approximate - The differential dy = f'(x)dx **Day 17:** Maximum and Minimum Values - Critical points - First and second derivative tests **Day 18:** Optimization Problems - Setting up objective functions - Finding optimal solutions **Day 19:** Week 3 Problem Set **Day 20:** L'Hôpital's Rule and Newton's Method Lab **Day 21:** Rest and Review ### Preview Reading (Optional) Skim Stewart Sections 3.9-3.10: - Just read the examples - Don't worry about details yet ### Key Concepts to Prepare For 1. **Related Rates:** Multiple quantities changing with time - dx/dt, dy/dt connected through an equation - Chain rule is essential! 2. **Extrema:** Finding peaks and valleys - Where f'(x) = 0 or undefined - Local vs. global extrema 3. **Optimization:** Real-world max/min problems - "Find the dimensions that maximize volume" - "Minimize the cost given constraints" --- ## 📓 Reflection Questions Take time to write thoughtful answers: 1. **What was the most challenging concept this week?** 2. **Which differentiation rule do you need more practice with?** 3. **How does the derivative connect geometry (tangent lines) to rates of change?** 4. **Give a real-world example where you'd want to find a derivative.** 5. **What questions do you still have about differentiation?** --- ## 🎯 Week 2 Competency Checklist Before moving to Week 3, ensure you can: - [ ] Compute derivatives from the limit definition - [ ] Apply the power rule fluently - [ ] Use product and quotient rules correctly - [ ] Apply chain rule to nested functions - [ ] Perform implicit differentiation - [ ] Find tangent line equations - [ ] Compute higher derivatives - [ ] Use SymPy for symbolic differentiation - [ ] Implement numerical differentiation in Python **If any box is unchecked, spend extra time reviewing that topic.** --- ## 📈 Progress Check Rate your confidence (1-5): | Topic | Confidence | |-------|------------| | Derivative definition | /5 | | Power rule | /5 | | Product rule | /5 | | Quotient rule | /5 | | Chain rule | /5 | | Implicit differentiation | /5 | | Tangent lines | /5 | | Numerical methods | /5 | **Average score ≥ 4:** Ready for Week 3 **Average score 3-4:** Quick review recommended **Average score < 3:** Extended review needed --- ## 💤 Rest Well Tonight: - Get 7-8 hours of sleep - Avoid heavy studying - Let your brain consolidate Tomorrow begins **Applications of Derivatives**—the payoff for all your hard work! --- ## 📌 Week 2 Quick Reference Card ``` ╔═══════════════════════════════════════════════════════════════╗ ║ DIFFERENTIATION RULES ║ ╠═══════════════════════════════════════════════════════════════╣ ║ Power: (xⁿ)' = nxⁿ⁻¹ ║ ║ Product: (fg)' = f'g + fg' ║ ║ Quotient: (f/g)' = (f'g - fg')/g² ║ ║ Chain: (f∘g)' = f'(g)·g' ║ ╠═══════════════════════════════════════════════════════════════╣ ║ SPECIAL DERIVATIVES ║ ║ (eˣ)' = eˣ (aˣ)' = aˣ ln a (ln x)' = 1/x ║ ║ (sin x)' = cos x (cos x)' = -sin x ║ ╠═══════════════════════════════════════════════════════════════╣ ║ IMPLICIT: Differentiate both sides, dy/dx on y terms ║ ╚═══════════════════════════════════════════════════════════════╝ ``` --- **Week 2 Complete! 🎉** You've mastered the fundamental techniques of differentiation. Week 3 shows you how to apply these tools to solve real problems. *"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful."* — Henri Poincaré

Day 14: Rest, Review, and Week 3 Preparation

Today's Schedule (7 hours)

Day 14: Rest, Review, and Week 3 Preparation

Today's Schedule (7 hours)