Unit O: AP Calculus Prerequisites
Summer Packet
Unit 1: Limits & Continuity (AB & BC)
Topic 1.1 Can Change Occur at an Instant?
Video 1: Topic 1.1 Can Change Occur at an Instant?
Topic 1.2 Defining Limits and Using Limit Notation
Topic 1.4 Estimating Limit Values from Tables
Video 2: Topic 1.2 & 1.4 Defining Limits + Estimating from Tables
Topic 1.3 Estimating Limit Values from Graphs
Video 3: Topic 1.3 Estimating Limit Values from Graphs
Topic 1.5 Determining Limits Using Algebraic Properties
Video 4: Topic 1.5(A) Mastering Limits with Algebra & Composite Functions
Video 5: Topic 1.5(B) Non-Traditional Composite/Operation Limits
Topic 1.6 Determining Limits Using Algebraic Manipulation
Topic 1.7 Selecting Procedures for Determining Limits
Video 6: Topic 1.6 & 1.7 Mastering Limit Techniques: Algebraic Manipulation & Strategic Selection
Topic 1.8 Determining Limits Using the Squeeze Theorem
Video 7: Topic 1.8(A) Proof of lim (sin x)/x = 1 — Geometry, Squeeze Theorem, and Beyond!
Video 8: Topic 1.8(B) Proof of the Limit of (1 - cos x)/x = 0
Video 9: Topic 1.8(C) Proof of lim (tan x)/x = 1
Video 10: Topic 1.8(D) Crushing Limits with the Squeeze Theorem
Topic 1.9 Connecting Multiple Representations of Limits
Video 11: Topic 1.9 Visualize It All: Mastering Limits with Multiple Representations
Topic 1.10 Exploring Types of Discontinuities
Topic 1.11 Defining Continuity at a Point
Topic 1.12 Confirming Continuity Over an Interval
Topic 1.13 Removing Discontinuities
Video 12: Topic 1.10-1.13 A Deep Dive into Discontinuities & Continuity of Functions
Topic 1.14 Infinite Limits and Vertical Asymptotes
Video 13: Topic 1.14 Crash Course: Infinite Limits & Vertical Asymptotes
Topic 1.15 Limits at Infinity and Horizontal Asymptotes
Video 14: Topic 1.15 Endless Insights: Limits at Infinity & Horizontal Asymptotes
Topic 1.16 Intermediate Value Theorem (IVT)
Video 15: Topic 1.16 Intermediate Value Theorem Demystified
Video 16: Unit 1 Crash Course Everything You Need to Know About the Limits, Continuity, and Intermediate Value Theorem (IVT) for the AP Exam
If you're looking to deepen your understanding of advanced concepts, give this a read.
Unit 2: Differentiation: Definition & Fundamental Properties (AB & BC)
Topic 2.1 Defining Average and Instantaneous Rate of Change at a Point
Video 17: Topic 2.1 From Average to Instant: Mastering Instantaneous Rate of Change
Topic 2.2 Defining the Derivative of a Function and Using Derivative Notation
Topic 2.3 Estimating Derivatives of a Function at a Point
Topic 2.4 Connecting Differentiability and Continuity
Video 18: Topic 2.2-2.4 Sharp Corners, Gaps & Tangents: When Derivatives Don’t Exist!
Topic 2.5 Applying the Power Rule
Topic 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
Video 19: Topic 2.5-2.6 Derivatives in Seconds: The Power Rule and Beyond
Topic 2.7 Derivatives of cos(x), sin(x), e^x, and ln(x)
Video 20: Topic 2.7(A) Proof of d/dx[sin(x)] = cos(x) and d/dx[cos(x)] = -sin(x)
Video 21: Topic 2.7(B) Derivatives of sin, cos, eˣ, and ln(x) – No Memorizing, Just Mastery
Topic 2.8 The Product Rule
Video 22: Topic 2.8(A) Proof of the Product Rule (fg)' = f·g' + f'·g
Video 23: Topic 2.8(B) Don’t FOIL It! Use the Product Rule Like a Calculus Pro
Topic 2.9 The Quotient Rule
Video 24: Topic 2.9(A) Proof of the Quotient Rule Why (f/g)’ Isn’t Just f’/g’
Video 25: Topic 2.9(B) Derivatives of Rational Functions? Use the Quotient Rule!
Topic 2.10 Derivatives of tan(x), cot(x), sec(x), and csc(x)
Video 26: Topic 2.10(A) Proving the Big Four: tan, cot, sec, and csc Derivatives Made Easy
Video 27: Topic 2.10(B) Mastering Trig Derivatives: tan(x), cot(x), sec(x), and csc(x) with examples
Video 28: Unit 2 Crash Course Everything You Need to Know About the Differentiation for the AP Exam
Unit 3: Differentiation: Composite, Implicit, & Inverse Functions (AB & BC)
Topic 3.1 The Chain Rule
Video 29: Topic 3.1 Chain Rule Hacks to Boost Your AP Calculus Scores
Topic 3.2 Implicit Differentiation
Video 30: Topic 3.2 Implicit Differentiation Demystified – Must-Know for the AP Exam
Topic 3.3 Differentiating Inverse Functions
Video 31: Topic 3.3 Differentiating Inverse Functions with Confidence!
Topic 3.4 Differentiating Inverse Trigonometric Functions
Video 32: Topic 3.4 Arc-attack! Mastering Derivatives of Inverse Trig Functions
Topic 3.5 Selecting Procedures for Calculating Derivatives
Video 33: Topic 3.5 Selecting Procedures for Calculating Derivatives
Topic 3.6 Calculating Higher-Order Derivatives
Video 34: Topic 3.6 Beyond the First Derivative: Mastering Higher-Order Derivatives!
Video 35: Unit 3 Crash Course Everything You Need to Know About the Differentiation of Composite, Implicit, & Inverse Function for the AP Exam
Unit 4: Contextual Applications of Differentiation (AB & BC)
Topic 4.1 Interpreting the Meaning of the Derivative in Context
Video 36: Topic 4.1 Interpreting Derivatives in Context with AP Style Examples!
Topic 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
Video 37: Topic 4.2 Position, Velocity & Acceleration—The Ultimate Motion Breakdown with AP-Style Examples!
Topic 4.3 Rates of Change in Applied Contexts Other Than Motion
Video 38: Topic 4.3 Rates of Change in Applied Contexts Other Than Motion
Topic 4.4 Introduction to Related Rates
Topic 4.5 Solving Related Rates Problems
Video 39: Topic 4.4-4.5(A) The #1 Related Rates Video You Need for the AP Calc Exam! (Part 1)
Video 40: Topic 4.4-4.5(B) Master Related Rates with These Real AP Calc Problems! (Part 2)
Topic 4.6 Approximating Values of a Function Using Local Linearity and Linearization
Video 41: Topic 4.6 Linear Approximations | Master Linearization & Local Linearity
Topic 4.7 Using L'Hopital's Rule for Determining Limits of Indeterminate Forms
Video 42: Topic 4.7 Using L'Hopital's Rule for Determining Limits of Indeterminate Forms
Video 43: Unit 4 Crash Course Everything You Need to Know About the Contextual Applications of Differentiation for the AP Exam
Unit 5: Analytical Applications of Differentiation (AB & BC)
Topic 5.1 Using the Mean Value Theorem (IVT)
Video 44: Topic 5.1 Mastering the Mean Value Theorem (IVT) & the Rolle's Theorem
Topic 5.2 Extreme Value Theorem (EVT), Global Versus Local Extrema, and Critical Points
Video 45: Topic 5.2 Extreme Value Theorem (EVT), Global vs. Local Extrema, and Critical Points
Topic 5.3 Determining Intervals on Which a Function is Increasing or Decreasing
Video 46: Topic 5.3 Determine Intervals on Which a Function is Increasing or Decreasing
Topic 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
Video 47: Topic 5.4 First Derivative Test | Relative Maximum | Relative Minimum
Topic 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
Video 48: Topic 5.5 Master the Candidates Test | Global Maximum & Minimum
Topic 5.6 Determining Concavity of Functions over Their Domains
Video 49: Topic 5.6 Determining Concavity of Functions over Their Domains | Inflection Points | Second Derivative
Topic 5.7 Using the Second Derivative Test (SDT) to Determine Extrema
Video 50: Topic 5.7 Using the Second Derivative Test (SDT) to Determine Extrema
Topic 5.8 Sketching Graphs of Functions and Their Derivatives
Topic 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
Video 51: Topic 5.8-5.9(A) Sketching f′(x) from f(x) — A Visual Guide to Derivatives
Video 52: Topic 5.8-5.9(B) Graphing f(x) from f′(x) — The Reverse Engineering of Calculus!
Now, use the follow link to explore the graphs of derivatives.
Topic 5.10 Introduction to Optimization Problems
Topic 5.11 Solving Optimization Problems
Video 53: Topic 5.10-5.11 Master Optimization Problems in Minutes!
Topic 5.12 Exploring Behaviors of Implicit Relations
Video 54: Topic 5.12 Exploring Behaviors of Implicit Relations
Video 55: Unit 5 Crash Course Everything You Need to Know About the Analytical Applications of Differentiation for the AP Exam
If you're looking to deepen your understanding of advanced concepts, give this a read.
Unit 6: Integration & Accumulation of Change (AB & BC)
Topic 6.1 Exploring Accumulation of Change
Video 56: Topic 6.1 From Rate to Total: Mastering Accumulations of Change
Topic 6.2 Approximating Areas with Riemann Sums
Video 57: Topic 6.2 Approximating Areas with Left, Right, and Midpoint Riemann Sums as well as Trapezoidal Rules
Topic 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
Video 58: Topic 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
Topic 6.4 The Fundamental Theorem of Calculus (FTC) and Accumulation Functions
Topic 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
Video 59: Topic 6.4-6.5A Unlocking the Fundamental Theorem of Calculus (FTC) & Net Change
Video 60: Topic 6.4-6.5B Mastering Rate In–Rate Out Problems
Topic 6.6 Applying Properties of Definite Integrals
Video 61: Topic 6.6 Applying Properties of Definite Integrals
Topic 6.7 The Fundamental Theorem of Calculus (FTC) and Definite Integrals
Topic 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
Video 62: Topic 6.7-6.8 The Fundamental Theorem of Calculus (FTC), Definite Integrals, and Indefinite Integrals
Topic 6.9 Integrating Using Substitution
Video 63: Topic 6.9 The Secret to Solving Tricky Integrals: u-Substitution Explained
Topic 6.10 Integrating Functions Using Long Division and Completing the Square
Must watch Videos 64 - 67 since they are prerequisites for Topic 6.10.
Video 64: Long Division & Synthetic Division with 8 Specific Examples - The easy way!
Video 65: Dividing Polynomials using 3 methods: Factoring - Long Division - Synthetic Division.
Video 66: Factoring Special Cases of polynomials.
Video 67: Solving Quadratic Equations by Taking Square Root and Completing Squares.
Video 68: Topic 6.10 Tame the Toughest Integrals: Long Division & Completing the Square Explained!
Topic 6.11 Integrating Using Integration by Parts (BC only)
Video 69: Topic 6.11 Master Integration by Parts Step-by-Step
Topic 6.12 Integrating Using Linear Partial Fractions (BC only)
Video 70: Topic 6.12 Integrating Using Partial Fraction Decomposition
Topic 6.13 Evaluating Improper Integrals (BC only)
Video 71: Topic 6.13 Infinite Limits? No Problem! Evaluating Improper Integrals
Topic 6.14 Selecting Techniques for Antidifferentiation
Video 72: Topic 6.14 Selecting the Best Techniques for Antidifferentiation
Video 73: Unit 6 Crash Course Everything You Need to Know About Integration for the AP exam
If you're looking to deepen your understanding of advanced concepts, give this a read.
Unit 7: Differential Equations (AB & BC)
Topic 7.1 Modeling Situations with Differential Equations
Video 76: Topic 7.1 How to Model the Real World with Differential Equations
Topic 7.2 Verifying Solutions for Differential Equations
Video 77: Topic 7.2 Verifying Solutions for Differential Equations
Topic 7.3 Sketching Slope Fields
Topic 7.4 Reasoning Using Slope Fields
Video 78: Topic 7.3-7.4 Unveiling the Invisible: Sketching & Reasoning with Slope Fields
Topic 7.5 Euler's Method (BC only)
Video 79: Topic 7.5 Use Euler’s Method to Approximate the Particular Solution to a Differential Equation.
Topic 7.6 General Solutions Using Separation of Variables
Topic 7.7 Particular Solutions using Initial Conditions and Separation of Variables
Video 80: Topic 7.6-7.7 Solving Differential Equations: Separation of Variables & Initial Conditions
Topic 7.8 Exponential Models with Differential Equations
Video 81: Topic 7.8 Exponential Models with Differential Equations
Topic 7.9 Logistic Models with Differential Equations (BC only)
Video 82: Topic 7.9 Logistic Models with Differential Equations
Video 83: Unit 7 Crash Course Everything You Need to Know About Differential Equations for the AP exam
If you're looking to deepen your understanding of advanced concepts, give this a watch.
Video 84: Introduction to Differential Equations - Order - ODE - PDE - Solutions.
Video 85: Find n-Parameter Family of Solutions, General Solution, Particular Solution.
Unit 8: Applications of Integration (AB & BC)
Topic 8.1 Average Value of a Function on an Interval
Video 86: Topic 8.1 Average Value of a Function on an Interval
Topic 8.2 Position, Velocity, and Acceleration Using Integrals
Video 87: Topic 8.2 Motion Mastered: Position, Velocity & Acceleration with Integrals
Topic 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
Video 88: Topic 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
Topic 8.4 Area Between Curves (with respect to x)
Video 89: Topic 8.4 Area Between Curves (w.r.t x) — Visualize, Set Up, Solve!
Topic 8.5 Area Between Curves (with respect to y)
Video 90: Topic 8.5 Area Between Curves (w.r.t y) — Visualize, Set Up, Solve!
Topic 8.6 Area Between Curves - More than Two Intersections
Must watch Videos 91 - 92 since they are prerequisites for Topic 8.6.
Video 91: Determine if the Function is EVEN or ODD?
Video 92: Graphing Sine & Cosine Functions.
Video 93: Topic 8.6A Area Between Curves with More than Two Intersections
Video 94: Topic 8.6B Proof of Integral Symmetry: Even & Odd Functions
Video 95: Topic 8.6C Area Between Curves with Even & Odd Functions
Topic 8.7 Cross Sections: Squares and Rectangles
Topic 8.8 Cross Sections: Triangles and Semicircles
Video 96: Topic 8.7-8.8
Topic 8.9 Disc Method: Revolving Around the x- or y- Axis
Video 97: Topic 8.9
Topic 8.10 Disc Method: Revolving Around Other Axes
Video 98: Topic 8.10
Topic 8.11 Washer Method: Revolving Around the x- or y-Axis
Video 99: Topic 8.11
Topic 8.12 Washer Method: Revolving Around Other Axes
Video 100: Topic 8.12
Topic 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC only)
Video 101: Topic 8.13
Video 102: Unit 8 Crash Course
Unit 9: Parametric Equations, Polar Coordinates, & Vector-Valued Functions (BC only)
Topic 9.1 Defining and Differentiating Parametric Equations
Video 100: Topic 9.1
Topic 9.2 Second Derivatives of Parametric Equations
Video 101: Topic 9.2
Topic 9.3 Arc Lengths of Curves (Parametric Equations)
Video 102: Topic 9.3
Topic 9.4 Defining and Differentiating Vector-Valued Functions
Video 103: Topic 9.4
Topic 9.5 Integrating Vector-Valued Functions
Video 104: Topic 9.5
Topic 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
Video 105: Topic 9.6
Topic 9.7 Defining Polar Coordinates and Differentiating in Polar Form
Must watch Videos 106 - 114 since they are prerequisites for Topic 9.7.
Video 106: How to Find Multiple Representations of Polar Coordinates.
Video 107: How to Convert between Polar and Rectangular Coordinates.
Video 108: Tests for Symmetry.
Video 109: Graphs of Polar Equations of Circles.
Video 110: Graphs of Limacon Curves.
Video 111: Graphs of Rose Curves.
Video 112: Graphs of Lemniscate Curves.
Video 113: Graphs of Archimedean Spirals: TI-Nspire, TI-84, & Desmos Made Easy!
After learning how to graph polar functions by hand from Videos 105–108, use the following link to practice graphing them on Desmos.
Video 114: Polar Function Graph Related Problems.
Here's a link to a Desmos demonstration that beautifully illustrates the relationship between polar and rectangular graphs on the same xy-coordinate plane - definitely worth exploring!
Video 115: Topic 9.7
Topic 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
Video 116: Topic 9.8
Topic 9.9 Finding the Area of the Region Bounded by Two Polar Curves
Video 117: Topic 9.9
Video 118: Unit 9 Crash Course
Unit 10: Infinite Sequences & Series (BC only)
Must watch Videos 119 since it is a prerequisite for Topic 10.1-10.2.
Video 119: Change in Arithmetic and Geometric Sequences.
Topic 10.1 Defining Convergent and Divergent Infinite Series
Video 120: Topic 10.1
Topic 10.2 Working with Geometric Series
Video 121: Topic 10.2
Topic 10.3 The nth Term Test for Divergence
Video 122: Topic 10.3
Topic 10.4 Integral Test for Convergence
Video 123: Topic 10.4
Topic 10.5 Harmonic Series and p-Series
Video 124: Topic 10.5
Topic 10.6 Comparison Tests for Convergence
Video 125: Topic 10.6
Topic 10.7 Alternating Series Test for Convergence
Video 126: Topic 10.7
Topic 10.8 Ratio Test for Convergence
Video 127: Topic 10.8
Topic 10.9 Determining Absolute or Conditional Convergence
Video 128: Topic 10.9
Topic 10.10 Alternating Series Error Bound
Video 129: Topic 10.10
Topic 10.11 Finding Taylor Polynomial Approximations of Functions
Video 130: Topic 10.11
Topic 10.12 Lagrange Error Bound
Video 131: Topic 10.12
Topic 10.13 Radius and Interval of Convergence of Power Series
Video 131: Topic 10.13
Topic 10.14 Finding Taylor Maclaurin Series for a Function
Video 132: Topic 10.14
Topic 10.15 Representing Functions as a Power Series
Video 133: Topic 10.15
Video 134: Unit 10 Crash Course
If you're looking to deepen your understanding of advanced concepts, give this a read.
AP Calculus Exam Prep - Road to a 5!
Past AP Calculus AB/BC Exams
Use the past AP Calculus AB/BC exams below to sharpen your skills and get familiar with the test format. Reviewing real exam questions helps build familiarity with the format, reinforce key concepts, and boost confidence. Practicing with these resources is one of the best ways to identify strengths and target areas for growth - use them to your full advantage! If there's any concept you've missed or want to strengthen your understanding of, be sure to check out the corresponding videos above.
Video 1: Full Solutions to the AP Calculus AB FRQ 2023.
Video 2: Full Solutions to the AP Calculus BC FRQ 2023.
Video 3: Full Solutions to the AP Calculus AB FRQ 2024.
Video 4: Full Solutions to the AP Calculus BC FRQ 2024.
Video 5: Full Solutions to the AP Calculus AB FRQ 2025.
Video 6: Full Solutions to the AP Calculus BC FRQ 2025.
Assorted Advanced Concepts
Dive into a collection of advanced math concepts!
This series of videos covers a variety of challenging topics designed to deepen your understanding and sharpen your problem-solving skills. Whether you’re preparing for competitions, exams, or just love advanced math, these concepts will push your thinking and expand your toolkit. Ready to level up? Let’s get started!
Video 1: Crack the Code: Solve the Integral, Get the Wi-Fi
Video 2: This Trig Integral Broke My Brain…
Video 3: The Most Satisfying Integral You’ll See Today – Putnam Challenge
Video 4: Only True Math Nerds Can Solve This MIT Integral.
Video 5: Crack the Code: Evaluating a Hard Trigonometric Integral!
Video 6: Oxford MAT 1998’s Toughest Integral — Can You Solve It?
Video 7: A Stunning Definite Integral from Vietnam – Can You Solve It?
Video 8: Can You Solve This Tough JEE Advanced Limit?
Video 9: Dare to Solve This Radical Trig Integral?
Video 10: Double Your Skills: 2 Ways to Find Tough Limits
Video 11: This Integral Stumped Math Olympiad Students — Can You Solve It?
Video 12: Can You Crack This Putnam-Level Integral?
Video 13: This Trick Makes Integrating cos⁶xdx Simple!
Video 14: Level Up Your Integration Skills with Square Root of tanxdx
Video 15: This Rational Function Looks Scary… But You’ve Got This! Integrate 1/(x^6 - 1)
Video 16: How to Integrate cosⁿx Like a Math Pro!
Video 17: How to Integrate (3 + cos⁵x)dx — The Smart Way!
Video 18: Solve This Rational Integral FAST or Lose the Challenge! ⏳
Video 19: Mind-Blowing Integral Trick! 🔥 Solving a Tough Cos² Integral Step-by-Step
Video 20: This Rational Integral Stumps Most Calculus Students!
Video 21: Disk & Washer Methods Made Easy — AP Calc Must-Know!
Video 22: JEE Advanced Killer Integral – Only the Best Can Solve It!
Video 23: 2024 JEE Advanced: The Insane Integral Everyone's Talking About!
Video 24: The Integral That Scares Students — Integral of ln(tanx)/sinxcosxdx
Video 25: This Tough Composite Integral Stumped Students!
Video 26: This Radical Integral Looks Scary… But Watch How Easy It Gets! 😲
Video 27: MIT Integration Bee 2025: Can You Solve This Before Me?
Video 28: MIT Bee Final Boss: Insane Trig Integral DECODED!
Video 29: How to Tackle Hard Trig Integrals — Integration Bee Strategy
Video 30: Stop Struggling! Integrate Composite Functions Like a Pro!
Video 31: This Tricky Integral Stumped the MIT Semifinals | MIT Integration Bee 2025
Video 32: Cracking an MIT Final Round Integral with Just One Clever Trick
Video 33: MIT Integration Bee 2022: Tough Integral, Tougher Competition!
Video 34: MIT’s “Impossible” Integral Cracked with One Simple Trick! MIT Integration Bee 2022
Video 35: Mastering 20 Advanced Integrals from Olympiads & MIT Integration Bee
Video 36: JEE Advanced Trig Integral That Stumps Everyone 🤯
Video 37: 20 Hard Integrals from the 2025 MIT Integration Bee (Can You Survive?)
Video 38: MIT Integration Bee 2025: Infinite Series Meets Calculus
Video 39: A Deep Dive into a Brutal Trig Integral from the MIT Integration Bee 2025
Video 40: MIT Integration Bee Meets Math Olympiad: TOP 20 Killer Integrals
...and many more. Explore my playlist of nearly 1000 Math Olympiad problem videos, with new content added regularly. Dive in and challenge yourself!
