One of the questions we have about power series approximations of functions is where the approximation is valid, or in other words, where the power series converges. For a given x, we can find the 

interval of convergence for a power series

For a Taylor series centered at x = a, the only place where we are sure that it converges for now is x = a, but we can expand this to a greater range using our knowledge of the ratio test. Let’s make an example to demonstrate this!

We have to test the endpoints by plugging them into x as sometimes it may converge or diverge at the endpoints.

Find the interval, radius of convergence, and the center of the interval of convergence.

Radius and Interval of Convergence of Power Series

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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

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: Differentiation: Definition & Fundamental Properties

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: Differentiation: Composite, Implicit & Inverse Functions

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 the Differentiation

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

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 and Accumulation of Change

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

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

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)

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. 

Unit 10: Infinite Sequences and Series (BC Only)

Check out these resources for the multiple choice section of the AP Calculus exam. With helpful videos reviewing multiple choice questions, and the best strategies that will help. Covering all sorts of problems and all the topics of AP Calculus, these resources will give you a better understanding of what to expect on the AP Calculus exam.

Multiple Choice Questions (MCQ)

Learn about the structure of the free response questions on the AP Calculus exam. With helpful resources, practice problems and reviews of previous FRQs you will feel comfortable with all of the different types of FRQs. And learn about strategies to ensure that you know how to approach each FRQ no matter what topic it covers. 

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Your guide to the 2021 AP Calculus AB exam

Format of the 2021 AP AP Calculus AB exam

When is the 2021 AP Calculus AB exam and how do I take it?

How should I prepare for the exam?

Pre-work: set up your study environment

AP Calculus AB 2021 Study Plan

👑 UNIT 1: Limits and Continuity

🤓 UNIT 2: Differentiation: Definition and Basic Derivative Rules


🤙🏽 UNIT 3: Differentiation: Composite, Implicit & Inverse Functions


👀 UNIT 4: Contextual Applications of Differentiation


✨ UNIT 5: Analytical Applications of Differentiation


🔥 UNIT 6: Integration and Accumulation of Change


💎 UNIT 7: Differential Equations


🐶 UNIT 8: Applications of Integration

Your guide to the 2021 AP Calculus BC exam

Format of the 2021 AP Calculus BC exam 

When is the 2021 AP Calculus BC exam and how do I take it?

AP Calculus BC 2021 Study Plan


🤓 UNIT 2: Differentiation: Definition and Basic Derivative Rules

👀 UNIT 4: Contextual Applications of Differentiation


🦖 UNIT 9: Parametric Equations, Polar Coordinates & Vector-Valued Functions


♾ UNIT 10: Infinite Sequences and Series

2021 AP Calculus BC Exam Guide

Overview

📖 Content

✋ Justification

💯 Scoring

Earning Points

♾️ Derivatives

♾️ Integrals

♾️ New Representation

♾️ Appropriate Units

Part A - Calculator Section

📖 Worked Example

📊 Past Trends

📱 A Note on Calculators

Part B - Non-Calculator Section

FAQs About FRQs

Calculus Free Response Questions

Table of Contents 📊

🧠 Exam Format and Logistical Details

❓ Types of Problems 

⭕️ How to Approach MC Questions 

Calculus Multiple Choice Questions

Multiple Choice Help (MCQ)

Overview ♾️

Format 📄

Tips and Tricks ✏️

Where to Practice 📍

Closing Thoughts 💭

AP Calculus AB/BC Multiple Choice Help (MCQ)

Exam Weighting ⚖

Developing Understanding 🧠

Mathematical Practices ➗

Main Ideas for this Unit 💬

Unit 10 Overview: Infinite Series and Sequences

What is a Sequence?

The Limit of A Sequence

Some Notation and Terminology on Sequences

What is a Series?

Convergence and Divergence of Series

Problems

Defining Convergent and Divergent Infinite Series

How do we find the error bound?

Alternating Series Error Bound

What are Taylor Series, and how do you find them?

Finding Taylor Polynomial Approximations of Functions

What is the Lagrange Error Bound and how do we find it?

Lagrange Error Bound

Some Important Maclaurin Series to Remember

Finding Taylor or Maclaurin Series for a Function

What is a power series?

How can we manipulate Power Series?


Conclusion and What’s Next?

Sources

Representing Functions as Power Series

What is a Geometric Series?

The Geometric Series Test

Working with Geometric Series

What is the nth Term Test?

The nth Term Test for Divergence

What is the Integral Test?

Integral Test for Convergence

P-Series Test

Harmonic Series and p-Series

Intro to the Comparison Tests

What is the Direct Comparison Test?


What is the Limit Comparison Test?

Comparison Tests for Convergence

What are Alternating Sequences?

What is the Alternating Series Test?

Alternating Series Test for Convergence

What is the Ratio Test?

Review Flowchart

Ratio Test for Convergence

Determining Absolute or Conditional Convergence

What will you learn? 🤔

Unit 1 Overview: Limits and Continuity

Limits

Introducing Calculus: Can Change Occur at An Instant?

Discontinuities

Jump Discontinuity ↔

Point Discontinuity 🕳

Essential Discontinuity ♾️

Removable Discontinuity 🚫

Exploring Types of Discontinuities

Conditions for Continuity

Example Problem: Using Conditions to Determine Continuity

Defining Continuity at a Point

Confirming Continuity Over an Open Interval 🔓

Confirming Continuity Over a Closed Interval 🔐

Confirming Continuity over an Interval

Removing Discontinuities Using Factoring 

Removing Discontinuities Using Rationalization

Removing Discontinuities

Asymptotes ✔

Connecting Infinite Limits and Vertical Asymptotes

Asymptotes, Continued

Connecting Limits at Infinity and Horizontal Asymptotes

The Intermediate Value Theorem (IVT)

IVT FRQ Practice

Working with the Intermediate Value Theorem (IVT Calc)

Defining Limits and Using Limit Notation

Estimating a Limit Value: Step by Step 

Estimating Limit Values from Graphs

Example Problem ❓

Estimating Limit Values from Tables

Example Problem: Using the Sum Rule to Determine a Limit ➕

Example Problem: Using the Constant Multiple Rule to Determine a Limit ➡

Example Problem: Using the Product Rule to Determine a Limit ✖️

Example Problem: Using the Exponent/Power Rule to Determine a Limit 💥

Determining Limits Using Algebraic Properties of Limits

Example Problem: Determining a Limit Using Factoring 

Example Problem: Determining a Limit Using Alternate Forms of Trigonometric Functions 

Determining Limits Using Algebraic Manipulation

Selecting Procedures for Determining Limits

The Squeeze Theorem 🥪

Example Problem 💡

Determining Limits Using the Squeeze Theorem

Unit 2 Overview

Main Topics and Sections:


An Introduction to Differentiation ⤴

Unit 2 Overview: Differentiation

What is a Derivative? 🤔

Defining Average and Instantaneous Rates of Change at a Point

Derivatives of Special Functions, Continued!💬

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

Derivatives Defined

Defining the Derivative of a Function and Using Derivative Notation

Estimating Derivatives of a Function at a Point

Differentiability Rules 📌

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

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