Name: Concavity
Uploaded: 2019-11-24T23:00:00.000Z
Description: A look at how to determine characteristics about the graphs of a function and its first and section derivatives from what we know, focusing on determining concavity.

AP Calculus AB/BC

Unit 1 – Limits and Continuity

1.12 Confirming Continuity over an Interval

Unit 2 – Fundamentals of Differentiation

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

Unit 3 – Composite, Implicit, and Inverse Functions

3.1 The Chain Rule

Unit 4 – Contextual Applications of Differentiation

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

Unit 5 – Analytical Applications of Differentiation

Unit 5 Overview: Analytical Applications of Differentiation

Unit 6 – Integration and Accumulation of Change

Unit 6 Overview: Integration and Accumulation of Change

Unit 7 – Differential Equations

7.9 Logistic Models with Differential Equations

Unit 8 – Applications of Integration

Unit 8 Overview: Applications of Integration

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

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

Unit 10 – Infinite Sequences and Series (BC Only)

10.10 Alternating Series Error Bound

Frequently Asked Questions

What is U-Sub and how do I use it?

AP Cram Sessions 2021

AP Calc BC Cram Unit 6: BC Concepts (Integration by parts, Using Linear Partial Fractions, and Improper Integrals)

Live Cram Sessions 2020

Tabular Calculus FRQs 

Previous Exam Prep

Helping Students Master Taking Derivatives Part I (for Teachers) 

Study Tools

2024 AP Calculus AB Exam Guide

Exam Skills

AP Calculus Multiple Choice Questions

A look at how to determine characteristics about the graphs of a function and its first and section derivatives from what we know, focusing on determining concavity.

Introduction to Calculus

1.1 Introducing Calculus: Can Change Occur at an Instant?

1.2 Defining Limits and Using Limit Notation

1.3 Estimating from Graphs

1.4 Estimating from Tables

1.5 Using Algebraic Properties of Limits

1.6 Using Algebraic Manipulations

1.8 Squeeze Theorem

1.10 Exploring Types of Discontinuities

1.11 Defining Continuity at a Point

1.12 Confirming Continuity Over an Interval

1.13 Removing Discontinuities

1.14 & 1.15 Asymptotes

1.16 Working with the Intermediate Value Theorem (IVT)

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

How to Determine Continuity at a Point

Examples

Defining Continuity at a Point

Confirming Continuity Over an Open Interval

Confirming Continuity Over a Closed Interval 

Confirming Continuity over an Interval

How to Remove Discontinuities

Removing Discontinuities

Connecting Infinite Limits and Vertical Asymptotes

Limits at Infinity

Objectives:

Essential Knowledge:

Looking Further

Connecting Limits at Infinity and Horizontal Asymptotes

Intermediate Value Theorem (IVT)

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 

Examples:

Determining Limits Using the Squeeze Theorem

Connecting Multiple Representations of Limits

Unit 1 – Limits & Continuity

Unit 2 Overview: Differentiation

What is a Derivative? 🤔

Defining Average and Instantaneous Rates of Change at a Point

Derivatives of Special Functions, Continued!


Example Problems:


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

Understanding the Definition of an Estimated Derivative:


Exploring the Estimated Derivative through Different Methods:


Example Problems:

Estimating Derivatives of a Function at a Point

Differentiability and Continuity


Defining Differentiability and Continuity:


Connecting Differentiability and Continuity:


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

The Power Rule

Applying the Power Rule

Derivative Rules

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

Derivatives of Special Functions 💭

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

The Product Rule

The Quotient Rule

Developing Understanding of Differentiation

Building Mathematical Practices

Preparing for the AP Exam

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

What is the Chain Rule? 🔗

When to Use the Chain Rule 📆

How to Apply the Chain Rule 🤔

Chain Rule Practice 📝

Answers 📝

The Chain Rule

What is Implicit Differentiation?

How to Differentiate Implicitly 🕵

Implicit Differentiation Practice 📝

Implicit Differentiation

What are Inverse Functions? 🙃

How to Differentiate Inverse Functions 🕵

Differentiating Inverse Functions Practice 📝

Differentiating Inverse Functions

What is an Inverse Trigonometric Function? 🤠

How to Differentiate Inverse Trigonometric Functions 🕵

Differentiating Inverse Trigonometric Functions

Calculating Derivatives Practice 📝

Selecting Procedures for Calculating Derivatives

What is a Higher-Order Derivative? 😵

How to Calculate Higher-Order Derivatives 🕵

Calculating Higher-Order Derivatives Practice 📝


Answers 📝

Calculating Higher-Order Derivatives

Unit 3 – Composite, Implicit, & Inverse Functions

🔍 Prerequisite Information 

Unit 4 Overview: Contextual Applications of Differentiation

Review: Implicit Differentiation ✔️

Interpreting the Meaning of the Derivative in Context

Position and Velocity

Acceleration

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

🔍 Examples

📓 Practice

Rates of Change in Applied Contexts other than Motion

What are Related Rates?

The Process 📠

Example Problem:

Draw a Picture

Equation to Relate

Plugging In And Solving

Intro to Related Rates

Special Example: Volume of a Cone 🍦

Solving Related Rates Problems

Linearization

Under and Overestimates

Linearization and Point-Slope

Approximating Values of a Function Using Local Linearity and Linearization

Indeterminate Forms

L'Hopital's Rule

When You Can Use L'Hopitals

Using L'Hopitals Rule for Determining Limits in Indeterminate Forms

Graphical Analysis 

5.1 Using the Mean Value Theorem

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

5.3 Determining Intervals on Which a Function is Increasing or Decreasing

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

5.5 Using the Candidates Test to Determine Absolute (Global) Extrema

5.6 Determining Concavity of Functions Over Their Domains

5.7 Using the Second Derivative Test to Determine Extrema

5.8 Sketching Graphs of Functions and Their Derivatives

5.9 Connecting a Function, Its First Derivative, and Its Second Derivative

5.10 & 5.11 Introduction to & Solving Optimization Problems

Okay, let’s break this down: 🔍

Rolle's Theorem

Using the Mean Value Theorem

add H2 here

Introduction to Optimization Problems

Example Problem 

Practice FRQ

Solving Optimization Problems

Overview

Tips and reminders

Exploring Behaviors of Implicit Relations

Extreme Value Theorem (E.V.T)


Examples:

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



Determining Intervals on Which a Function is Increasing or Decreasing


The First Derivative Test

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

Concavity

Concavity

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