BC Review & Connections

Part 1 of 7 — The Big Picture

Core Themes of Calculus

1. Limits → foundation of everything
2. Derivatives → rates of change (local behavior)
3. Integrals → accumulation (global behavior)
4. Series → infinite sums (representation of functions)

The FTC Connects Them

rac{d}{dx}int_a^x f(t),dt = f(x)

Differentiation and integration are inverse operations.

Big Picture 🎯

Key Takeaways — Part 1

Everything in calculus is built on limits.

BC Review

Part 2 of 7 — Derivative Rules Summary

Must-Know Derivatives

Plus: product rule, quotient rule, chain rule!

Derivatives 🎯

Key Takeaways — Part 2

Know all basic derivative formulas cold. Chain rule applies to all of them.

BC Review

Part 3 of 7 — Integration Techniques Summary

Integration Toolkit

1. Direct: power, trig, exponential
2. u-Substitution: $int f(g(x))g'(x),dx$
3. Integration by Parts (BC): $int u,dv = uv - int v,du$
4. Partial Fractions (BC): for rational functions
5. Improper Integrals (BC): limits at $pminfty$ or discontinuities
6. Series Integration (BC): term-by-term for power series

Integration 🎯

Key Takeaways — Part 3

Master all six integration techniques. Know when to use each.

BC Review

Part 4 of 7 — Series Convergence Summary

Convergence Tests Quick Reference

Convergence 🎯

Key Takeaways — Part 4

Factorials → ratio. Powers → comparison/$p$-test. Alternating → AST.

BC Review

Part 5 of 7 — Applications Summary

BC Applications Checklist

• Area between curves: $int (f - g),dx$
• Volume (disk/washer): $piint [R^2 - r^2],dx$
• Volume (shell): $2piint x cdot f(x),dx$
• Arc length: $int sqrt{1 + (f')^2},dx$ or $int sqrt{(x')^2 + (y')^2},dt$
• Polar area: rac{1}{2}int r^2,d heta
• Accumulation: $F(x) = F(a) + int_a^x f(t),dt$

Applications 🎯

Key Takeaways — Part 5

Know every application formula. Draw pictures!

BC Review

Part 6 of 7 — Practice Workshop

Mixed Review 🎯

Workshop Complete!

BC Review & Connections — Final

Part 7 of 7 — You're Ready!

What You've Mastered

• Limits, continuity, and the definition of derivative
• All differentiation rules and applications
• All integration techniques (including BC-exclusive)
• Differential equations (separable, Euler, logistic)
• Parametric, polar, and vector calculus
• Sequences, series, and convergence tests
• Taylor/Maclaurin series and error bounds
• Applications: area, volume, arc length, accumulation

You are prepared for the AP Calculus BC exam. Good luck! 🎓

Final Question 🎯

AP Calculus BC — Complete! 🎓✅

You have completed all interactive lessons for AP Calculus BC. Go ace that exam!