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S = contains supplemental resources
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)

Terminology

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

Supplementary Resources (optional)