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: Extreme Values

Overview

Graphs of functions have several important points such as the "bottom of a valley" or a "sharp corner". These points can be found using information obtained from the tangent line (derivative) to the graph. Points such as those occurring at the "top of a hill" are particularly useful for optimizing a function, such as finding the largest "extreme" point through which it can pass. We begin our study of important "critical" points and the analysis of curves in this lesson.

Objectives

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

• 3.1.1 Find critical points on a curve by observation
• 3.1.2 Find horizontal tangent line, vertical tangent line, sharp corner, or cusp on a curve by observation
• 3.1.2 Find relative maximum point or relative minimum point on a curve by observation
• 3.1.3 Find critical points algebraically by solving for f'(x)=0 or f'(x)="undefined"
• 3.1.4 Know the difference between a critical point and a critical value
• 3.1.5 Determine if a critical point is a relative extreme point
• 3.1.6 Know the relative extrema theorem and that it only works in one direction (i.e. the contrapositive does not hold)
• 3.1.7 Know the difference between relative extrema and absolute extrema
• 3.1.8 Find absoute extreme values on a curve by observation and be able to confirm algebraically

Terminology

Terms you should be able to define: horizontal tangent line, vertical tangent line, sharp corner, cusp, critical point, critical value, relative extreme point, relative minimum point, relative maximum point, relative extrema, relative extrema theorem, absolute extrema

Mini-Lectures and Examples

Supplementary Resources (optional)

rev. 2020-10-31