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: First Derivative Test & Increasing/Decreasing Functions

Overview

The first derivative gives information about the nature of a function including critical points and increasing/decreasing regions of a curve. Critical points (CPs) are points where the first derivative is zero or undefined thereby providing information about the tangent line at a CP which could be horizontal, vertical, or impossible. A region of a curve that is going up up up as you look at it from left to right is increasing and, similarly, one that goes down down down is decreasing. The first derivative test enables us to determine if a critical point is a relative minimum or relative maximum using information about the increasing/decreasing nature of the curve where the CP lies.

Objectives

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

• 3.3.1 Find increasing/decreasing regions on a curve by observation
• 3.4.2 Determine if a critical point is also a relative extrema by observation of a curve
• 3.4.3 Use the first derivative to find critical points
• 3.4.4 Use the first derivative to classify a critical point as having a horizontal tangent, vertical tangent, cusp, or sharp corner
• 3.4 5 Use the first derivative to find regions where the graph of a function is increasing and where it is decreasing
• 3.4.6 Apply the first derivative test to determine if a critical point is a relative minimum, relative maximum, or neither

Terminology

Terms you should be able to define: increasing, decreasing, strictly monotonic, critical point (CP), horizontal tangent line (HTL), vertical tangent line (VTL), cusp, sharp corner, first derivative test

Mini-Lectures and Examples

Supplementary Resources (optional)

rev. 2020-10-31