AP Calculus AB/BC

"The limit does not exist." In unit 1 of AP Calc AB/BC, we cover everything you need to know about limits and continuity: limit notation, estimating and calculating limits, L'Hopital's Rule, squeeze theorem, theories of continuity, infinite limits, limits at infinity, and the intermediate value theorem. 

Unit 1 – Limits & Continuity

Limits @ Infinity and Horizontal Asymptotes

Continuity Part II

Continuity Part II - Slides

1.12 Confirming Continuity over an Interval

1.5 Determining Limits Using Algebraic Properties of Limits

1.13 Removing Discontinuities

1.11 Defining Continuity at a Point

1.15 MC Answers and Review

Unit 1 Overview: Limits and Continuity

1.2 Defining Limits and Using Limit Notation

1.4 Estimating Limit Values from Tables

Unit 1 Review - Limits & Continuity

Continuity Part I

1.15 Connecting Limits at Infinity and Horizontal Asymptotes

1.14 Connecting Infinite Limits and Vertical Asymptotes

1.0 Unit 1 Overview

1.16 Working with the Intermediate Value Theorem (IVT Calc)

1.15 Multiple Choice Questions

1.7 Selecting Procedures for Determining Limits

1.3 Estimating Limit Values from Graphs

Graphical Limits

1.1 Introducing Calculus: Can Change Occur at An Instant?

Algebraic Limits

1.6 Determining Limits Using Algebraic Manipulation

What is a Limit?

1.8 Determining Limits Using the Squeeze Theorem

1.10 Exploring Types of Discontinuities

Connecting Multiple Representations of Limits

In unit 2 of AP Calc AB/BC, we introduce the derivative—definitions of the derivative, derivative rules (power rule, quotient rule, product rule, and trig rules), and differentiability of a function. With these resources, you will become a master of differentiation!

Unit 2 – Fundamentals of Differentiation

2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

Helping Students Master Taking Derivatives Part I (for Teachers) 

2.1 Defining Average and Instantaneous Rates of Change at a Point

The Limit Definition of the Derivative

2.11 Multiple Choice Questions

2.3 Estimating Derivatives of a Function at a Point

2.5 Applying the Power Rule

Introduction to Finding Derivatives

2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist

2.7 Derivatives of cos x, sinx, e^x, and ln x

Practicing Derivative Rules

The Product and Quotient Rules

Practicing Derivative Rules - Slides

Unit 2 Review - Differentiation: Definition & Fundamental Properties

2.2 Defining the Derivative of a Function and Using Derivative Notation

2.9 The Quotient Rule

2.8 The Product Rule

Defining and Taking Derivatives

Unit 2 Overview: Differentiation

2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Unit 2 Overview

2.12 MC Answers and Review

In unit 3 of AP Calc AB/BC, we dive deeper into derviatives! We will cover the derivatives of composite, implicit, and inverse functions, including inverse trigonometric functions. We'll delve into higher order derivatives, chain rule, and how to select a procedure to calculate derivatives. 

Unit 3 – Composite, Implicit, & Inverse Functions

3.1 The Chain Rule

Unit 3 Overview: Differentiation: Composite, Implicit, and Inverse Functions

3.5 Selecting Procedures for Calculating Derivatives

Interpreting Derivatives through Tables and in context

Interpreting Derivatives through Tables and in context - Slides

3.6 Calculating Higher-Order Derivatives

Interpreting Derivatives through Graphs and in context

Interpreting Derivatives through Graphs and in context - Slides

Implicit Derivatives

Practicing Derivative Rules II

The Chain Rule

Practicing Derivative Rules II - Slides

3.2 Implicit Differentiation

3.4 Differentiating Inverse Trigonometric Functions

3.7 Multiple Choice Questions

Helping Students Master Taking Derivatives Part II (for Teachers) 

3.3 Differentiating Inverse Functions

3.7 MC Answers and Review

In unit 4 of AP Calc AB/BC, we'll cover derivatives as they relate to position/velocity/acceleration, rates of change, and related rates. You'll learn the contextual meaning of derivatives, approximating values using linearization, and finding limits with L'Hopital's Rule. You'll understand the real-life applications of differentiation!

Unit 4 – Contextual Applications of Differentiation

4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration

4.7 Using L'Hopitals Rule for Determining Limits in Indeterminate Forms

4.9 MC Answers and Review

L'Hospital's rule

Presentation Slides, L'Hospital's Rule

Related Rates

4.1 Interpreting the Meaning of the Derivative in Context

Unit 4 Overview: Contextual Applications of Differentiation

4.4 Intro to Related Rates

4.3 Rates of Change in Applied Contexts other than Motion

4.8 Multiple Choice Questions

4.6 Approximating Values of a Function Using Local Linearity and Linearization

4.5 Solving Related Rates Problems

In unit 5 of AP Calc AB/BC, we talk about mean value theorem, Rolle's theorem, extreme value theorem, global & local extrema, functions increasing & decreasing, first derivative test, concavity, second derivative test, and optimization problems. You'll be ready to analyze all types of graphs using the fundamentals of differentiation!

Unit 5 – Analytical Applications of Differentiation

Unit 5 Overview: Analytical Applications of Differentiation

5.1 Using the Mean Value Theorem

5.4 Using the First Derivative Test to Determine Relative (Local) Extrema

f, f', f''

Existence Theorems

Presentation Slides, Existence Theorms

Concavity

Concavity - Slides

5.6 Determining Concavity

5.13 MC Answers and Review

5.3 Determining Intervals on Which a Function is Increasing or Decreasing

Increasing and Decreasing Functions

5.12 Multiple Choice Questions

5.7 Using the Second Derivative Test to Determine Extrema

5.10 Introduction to Optimization Problems

Optimization Problems

5.11 Solving Optimization Problems

5.2 Extreme Value Theorem, Global vs Local Extrema, and Critical Points

Sketching Graphs of Functions and Their Derivatives

In unit 6 of AP Calc AB/BC, we wil cover accumulation change with definite integrals, Riemann sums, the Fundamental Theorem of Calculus, u-substitution, antiderivatives, indefinite integrals, and properties of integrals, integration by parts, and partial fraction decomposition. You'll understand the fundamentals of integration and its applications!

Unit 6 – Integration & Accumulation of Change

Integration Techniques Part II

Using FTC and Integral Defined Functions

AP Calculus: Riemann Sums

6.14 MC Answers and Review

Unit 6 Overview: Integration and Accumulation of Change

6.1 Unit 6: Integration and Accumulation of Change

6.11 Integrating Using Integration by Parts

6.12 Using Linear Partial Fractions

Presentation Slides, FTC and Integral Defined Functions

U-Substitution

6.13 Multiple Choice Questions

In unit 7 of AP Calc AB/BC, we delve into differential equations, slope fields, Euler's method, finding particular solutions using initial conditions & separation of variables, and logistic models for exponential growth and decay. With these resources, you'll develop a deeper understanding of differential equations as mathematical models.

Unit 7 – Differential Equations

Separable Differential Equations

Separable Differential Equations Slides

7.10 Multiple Choice Questions

7.9 Logistic Models with Differential Equations

7.5 Approximating Solutions Using Euler’s Method

7.3 Sketching Slope Fields

7.4 Reasoning Using Slope Fields

7.6 Finding General Solutions Using Separation of Variables

7.2 Verifying Solutions for Differential Equations

7.11 MC Answers and Review

7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables

Unit 7 Overview: Differential Equations

In unit 8 of AP Calc, we dive into integration as it relates to the average value of a function, net changes, particle motion (position/velocity/acceleration), areas between curves, and volumes (with cross sections, disc method, & washer method). You'll develop a solid understanding of integration and its applications in real life!

Unit 8 – Applications of Integration

Unit 8 Overview: Applications of Integration

8.11 Volume with Washer Method: Revolving Around the x- or y-Axis

8.1 Finding the Average Value of a Function on an Interval

8.12 Volume with Washer Method: Revolving Around Other Axes

8.8 Volumes with Cross Sections: Triangles and Semicircles

8.5 Finding the Area Between Curves Expressed as Functions of y

8.6 Finding the Area Between Curves That Intersect at More Than Two Points

Interpreting the Meaning of a Derivative/Integral

Presentation Slides, Derivative Integral

8.9 Volume with Disc Method: Revolving Around the x- or y-Axis

8.4 Finding the Area Between Curves Expressed as Functions of x

8.7 Volumes with Cross Sections: Squares and Rectangles

8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals

8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts

8.10 Volume with Disc Method: Revolving Around Other Axes

Position, Velocity & Acceleration

8.14 MC Answers and Review

8.13 Multiple Choice Questions

In unit 9 of AP Calc BC, we review parametric equations, arc lengths, polar coordinates, vector-valued functions, and areas under polar curves. With these resources, you'll ace the AP Calc BC exam!

Unit 9 – Parametric Equations, Polar Coordinates, & Vector-Valued Functions (BC Only)

9.9 Finding the Area of the Region Bounded by Two Polar Curves

9.2 Second Derivatives of Parametric Equations

9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions

Unit 9 Overview: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

9.5 Integrating Vector-Valued Functions

9.3 Finding Arc Lengths of Curves Given by Parametric Equations

9.7 Defining Polar Coordinates and Differentiating in Polar Form

9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

Polar Coordinates and Calculus (for BC teachers)

9.1 Defining and Differentiating Parametric Equations

9.4 Defining and Differentiating Vector-Valued Functions

9.10 Multiple Choice Questions

9.11 MC Answers and Review

Let's learn about series in unit 10 of AP Calc BC. We cover infinite sequences and series, including all the divergence and convergence tests: nth term test, integral test, comparison tests, alternating series tests, and Taylor polynomials. 

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