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Slope of a Tangent Line Visually

Author: Sophia

what's covered

In this lesson, you will learn what a tangent line is and how to estimate its slope graphically. Specifically, this lesson will cover:

1. Estimating the Slope of a Tangent Line Graphically
2. Horizontal Tangent Lines

1. Estimating the Slope of a Tangent Line Graphically

A tangent line is a line that touches a graph at one specific point (but does not cross it).

EXAMPLE

The graph of f open parentheses x close parentheses equals 2 square root of x

and its tangent line at open parentheses 4 comma space 4 close parentheses

are shown below. Use this picture to estimate the slope of the tangent line.

In order to estimate the slope of a line, two points are needed. Thus, we need another point on the line besides open parentheses 4 comma space 4 close parentheses

to estimate the slope of this line. Inspecting closely, it looks like the point open parentheses 8 comma space 6 close parentheses

is also contained on the line.

Thus, the slope of the tangent line is approximately $m equals fraction numerator 6 minus 4 over denominator 8 minus 4 end fraction equals 2 over 4 equals 1 half.$ In fact, this is the exact slope of the tangent line.

step by step

If you are given the graph of a function and want to estimate the slope of a tangent line at open parentheses a comma space f open parentheses a close parentheses close parentheses comma

open parentheses a comma space f open parentheses a close parentheses close parentheses comma

do the following:

Sketch the tangent line at the point
Find another point on the line. Visually, it is easiest if the point has whole number coordinates, but sometimes this isn’t possible.
Use the slope formula $m equals fraction numerator y subscript 2 minus y subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction$ to compute the slope of the line.

try it

Consider the graph of f open parentheses x close parentheses

below with its tangent line at the point open parentheses 1 comma space 3 close parentheses.

Identify another point that is on the tangent line.

There are several options, such as open parentheses 2 comma space 2 close parentheses comma

open parentheses 3 comma space 1 close parentheses comma

and

open parentheses 4 comma space 0 close parentheses.

Find the slope of the tangent line.

Using the slope formula with the points open parentheses 1 comma space 3 close parentheses

and

open parentheses 4 comma space 0 close parentheses comma

the slope is $m equals fraction numerator 0 minus 3 over denominator 4 minus 1 end fraction equals short dash 3 over 3 equals short dash 1.$

term to know

Tangent Line

A line that touches (but does not cross) the graph of a function at a specific point.

2. Horizontal Tangent Lines

Tangent lines whose slopes are 0, also known as horizontal tangent lines, are very useful in calculus. It is important (and quite simple) to identify the places on a graph where the tangent line is horizontal.

EXAMPLE

Estimate the slope of the tangent line to the curve f open parentheses x close parentheses equals x cubed minus 3 x plus 4

at the point open parentheses 1 comma space 2 close parentheses.

The graph of f open parentheses x close parentheses

and its tangent line are shown here:

The tangent line appears to be horizontal, which means its slope is zero.

EXAMPLE

Estimate all values of x for which the graph of y equals f open parentheses x close parentheses

below has a horizontal tangent line.

Looking at the graph, the values of x for which the tangent lines are horizontal are about x equals short dash 2.5

and

summary

In this lesson, you learned about tangent lines, which are lines that touch (but do not cross) the graph of a function at a specific point. You learned that given the graph of a function, you can visually estimate the slope of the tangent line graphically. This can be accomplished by estimating another point on the tangent line. You also learned that tangent lines whose slopes are 0 are known as horizontal tangent lines; these are very useful in calculus.

Source: THIS TUTORIAL HAS BEEN ADAPTED FROM CHAPTER 0 OF "CONTEMPORARY CALCULUS" BY DALE HOFFMAN. ACCESS FOR FREE AT WWW.CONTEMPORARYCALCULUS.COM. LICENSE: CREATIVE COMMONS ATTRIBUTION 3.0 UNITED STATES.

Terms to Know

Tangent Line: A line that touches (but does not cross) the graph of a function at a specific point.

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Slope of a Tangent Line Visually

Table of Contents

1. Estimating the Slope of a Tangent Line Graphically

2. Horizontal Tangent Lines