Unit 2A

Derivatives Part 1

Lesson 1: Limits, Secants, and Tangents

Skill Builder

Slope

A linear relation's rate of change corresponds to the slope of the line formed when that relation is graphed. The slope of a line takes into consideration the line's steepness and direction.

Slope the ratio of the vertical change to the horizontal change of a line or line segment

The slope of a line measures how steep a line is by comparing the vertical change of the line to the horizontal change of the line. The variable «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»m«/mi»«/mstyle»«/math» is often used to represent slope. The following two formulas are equivalent.

Lines with a positive slope increase from left to right.

Lines with a negative slope decrease from left to right.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#8710;«/mo»«/mstyle»«/math» is the Greek letter delta and is used to represent a change. «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#8710;«/mo»«mi»y«/mi»«/mstyle»«/math» represents the amount of vertical change. «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#8710;«/mo»«mi»x«/mi»«/mstyle»«/math» represents the amount of horizontal change.

The rise, or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#8710;«/mo»«mi»y«/mi»«/mstyle»«/math», can be determined by subtracting the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values of any two points on a line or line segment. This is often written as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mrow»«/mstyle»«/math».

The run, or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#8710;«/mo»«mi»x«/mi»«/mstyle»«/math», can be determined by subtracting the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-values of the same two points on the line or line segment. This is often written as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mstyle»«/math».

When a line is horizontal, the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-values for all points on the line are equal, so the numerator of the slope formula will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math», which means the slope will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math».


When a line is vertical, the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-values for all points on the line are equal, so the denominator of the slope formula will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math». Division by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»0«/mn»«/mstyle»«/math» is undefined, so the slope of a vertical line is said to be undefined.