Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Calculus I
Topic: Analysis of Curves
Subtopic: Second Derivative Test & Concavity

Overview

The second derivative gives information about the nature of a function including inflection points and concave up/down regions of a curve. Concavity is describes the shape of a region of a curve, basically measuring if it is like a right side up cup or an upside down cup. Inflection points (IPs) are points where the concavity of a curve changes. The second derivative test enables us to determine if a critical point is a relative minimum or relative maximum using information about concavity of the curve where the CP lies.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 3.4.1 Find inflection points and regions of concavity on a curve by observation
• 3.4.2 Determine if an inflection point is also a critical point by observation of a curve
• 3.4.3 Use the second derivative to find possible inflection points
• 3.4.4 Determine which of the possible inflection points are actual inflection points
• 3.4 5 Use the second derivative to find regions where the graph of a function is concave up and where it is concave down
• 3.4.6 Apply the second derivative test to determine if a critical point is a relative minimum, relative maximum, or neither

Terminology

Terms you should be able to define: concavity, inflection point (IP), second derivative test

Mini-Lectures and Examples

Supplementary Resources (optional)

rev. 2020-10-31