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Equations of Tangent Lines

Author: Sophia
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what's covered
In this lesson, you will use derivative rules to write the equation of a tangent line to a function f open parentheses x close parentheses. Specifically, this lesson will cover:

Table of Contents

1. Writing the Equation of a Tangent Line at a Specific Point

Shown here is the graph of some function y equals f open parentheses x close parentheses and its tangent line at open parentheses a comma space f open parentheses a close parentheses close parentheses.

A graph with an x-axis ranging from −12 to 12 and a y-axis ranging from −4 to 20. A parabolic portion begins at the origin (0, 0) and passes through a marked point at (4, 4), which is labeled ‘(a, f(a))’ in the first quadrant. A dashed line starts from the third quadrant, intersecting the parabolic portion at the marked point (4, 4) by crossing the x-axis between −6 and −4 and the y-axis at (0, 2).

Recall from Unit 1 that writing the equation of a line requires two things:

  • The slope of the line
  • A point on the line
Given a function y equals f open parentheses x close parentheses, this information is known at x equals a colon

  • The slope of the line is f apostrophe open parentheses a close parentheses.
  • A point on the line is open parentheses a comma space f open parentheses a close parentheses close parentheses.
For now, let’s assume that f apostrophe open parentheses a close parentheses is defined, meaning that the tangent line is nonvertical.

Now, use the point-slope form to write the equation of the tangent line:

y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses Use the point-slope form.
y minus f open parentheses a close parentheses equals f apostrophe open parentheses a close parentheses open parentheses x minus a close parentheses open parentheses x subscript 1 comma space y subscript 1 close parentheses equals open parentheses a comma space f open parentheses a close parentheses close parentheses comma space m equals f apostrophe open parentheses a close parentheses
y equals f open parentheses a close parentheses plus f apostrophe open parentheses a close parentheses open parentheses x minus a close parentheses Add f open parentheses a close parentheses to both sides to solve for y.

formula to know
Equation of a Tangent Line to bold italic y bold equals bold italic f open parentheses bold x close parentheses at bold italic x bold equals bold italic a
y equals f open parentheses a close parentheses plus f apostrophe open parentheses a close parentheses open parentheses x minus a close parentheses

2. Different Types of Functions

Now, let’s focus on the mechanics required to write tangent lines for different types of functions.

2a. Power Functions (y = xⁿ)

EXAMPLE

Write the equation of the line tangent to f open parentheses x close parentheses equals x cubed when x equals 2.

First, the line is tangent to the graph at the point open parentheses 2 comma space f open parentheses 2 close parentheses close parentheses, or (2, 8). The derivative is f apostrophe open parentheses x close parentheses equals 3 x squared. Then, the slope of the tangent line is f apostrophe open parentheses 2 close parentheses equals 3 open parentheses 2 close parentheses squared equals 12.

Now, use the tangent line formula:

y equals f open parentheses a close parentheses plus f apostrophe open parentheses a close parentheses open parentheses x minus a close parentheses Use the equation of a tangent line.
y equals f open parentheses 2 close parentheses plus f apostrophe open parentheses 2 close parentheses open parentheses x minus 2 close parentheses a equals 2
y equals 8 plus 12 open parentheses x minus 2 close parentheses f open parentheses 2 close parentheses equals 8 and f apostrophe open parentheses 2 close parentheses equals 12
y equals 8 plus 12 x minus 24 Distribute.
y equals 12 x minus 16 Combine like terms.

In conclusion, the equation of the tangent line is y equals 12 x minus 16.

EXAMPLE

Write the equation of the line tangent to f open parentheses x close parentheses equals 1 over x squared when x equals 1. The line is tangent to the graph at the point open parentheses 1 comma space f open parentheses 1 close parentheses close parentheses, or (1, 1).

First, rewrite f open parentheses x close parentheses equals 1 over x squared with a single exponent: f open parentheses x close parentheses equals x to the power of short dash 2 end exponent. By the power rule, f apostrophe open parentheses x close parentheses equals short dash 2 x to the power of short dash 3 end exponent equals fraction numerator short dash 2 over denominator x cubed end fraction. Then, the slope of the tangent line is f apostrophe open parentheses 1 close parentheses equals fraction numerator short dash 2 over denominator open parentheses 1 close parentheses cubed end fraction equals short dash 2.

Now, use the tangent line formula:

y equals f open parentheses a close parentheses plus f apostrophe open parentheses a close parentheses open parentheses x minus a close parentheses Use the equation of a tangent line.
y equals f open parentheses 1 close parentheses plus f apostrophe open parentheses 1 close parentheses open parentheses x minus 1 close parentheses a equals 1
y equals 1 minus 2 open parentheses x minus 1 close parentheses f open parentheses 1 close parentheses equals 1 and f apostrophe open parentheses 1 close parentheses equals short dash 2
y equals 1 minus 2 x plus 2 Distribute.
y equals short dash 2 x plus 3 Combine like terms.

In conclusion, the equation of the tangent line is y equals short dash 2 x plus 3.

try it
Consider the function f open parentheses x close parentheses equals x to the power of 3 divided by 2 end exponent
Write the equation of the line tangent to the graph of this function at x = 4.
The equation of the tangent line is y equals 3 x minus 4. Here is why:

First, find f apostrophe open parentheses x close parentheses equals 3 over 2 x to the power of 1 divided by 2 end exponent.

Then, f apostrophe open parentheses 4 close parentheses equals 3 over 2 open parentheses 4 close parentheses to the power of 1 divided by 2 end exponent equals 3 over 2 open parentheses 2 close parentheses equals 3.

At this point, we have the slope of the line.

Since f open parentheses 4 close parentheses equals 4 to the power of 3 divided by 2 end exponent equals 8 comma space the line contains the point open parentheses 4 comma space 8 close parentheses.

Now, use point-slope form to write the equation of the line, then solve for y:

y minus 8 equals 3 open parentheses x minus 4 close parentheses</dd></dl> y minus 8 equals 3 x minus 12 y equals 3 x minus 4

2b. y = sinx and y = cosx

Let’s look at an example involving a trigonometric function.

EXAMPLE

Write the equation of the line tangent to the graph of f open parentheses x close parentheses equals cos x at the point open parentheses straight pi over 2 comma space 0 close parentheses.

First, recall that f apostrophe open parentheses x close parentheses equals short dash sin x. Then, the slope of the tangent line is f apostrophe open parentheses straight pi over 2 close parentheses equals short dash sin straight pi over 2 equals short dash 1.

Now, use the tangent line formula:

y equals f open parentheses a close parentheses plus f apostrophe open parentheses a close parentheses open parentheses x minus a close parentheses Use the equation of a tangent line.
y equals f open parentheses straight pi over 2 close parentheses plus f apostrophe open parentheses straight pi over 2 close parentheses open parentheses x minus straight pi over 2 close parentheses a equals straight pi over 2
y equals 0 plus open parentheses short dash 1 close parentheses open parentheses x minus straight pi over 2 close parentheses f open parentheses straight pi over 2 close parentheses equals 0 and f apostrophe open parentheses straight pi over 2 close parentheses equals short dash 1
y equals short dash x plus straight pi over 2 Distribute and simplify.

Thus, the equation of the tangent line is y equals short dash x plus straight pi over 2.

summary
In this lesson, you learned how to write the equation of the tangent line at a specific point, noting that this equation can be found for a function f open parentheses x close parentheses at x equals a as long as f apostrophe open parentheses a close parentheses is defined. You also learned how to write tangent lines for different types of functions, such as power functions (bold italic y bold equals bold italic x to the power of bold n) and trigonometric functions (bold italic y bold equals bold sin bold italic x and bold italic y bold equals bold cos bold italic x). This is a gateway for a wider variety of applications that will be discussed later in this chapter once we learn how to find derivatives of more functions.

Source: THIS TUTORIAL HAS BEEN ADAPTED FROM CHAPTER 2 OF "CONTEMPORARY CALCULUS" BY DALE HOFFMAN. ACCESS FOR FREE AT WWW.CONTEMPORARYCALCULUS.COM. LICENSE: CREATIVE COMMONS ATTRIBUTION 3.0 UNITED STATES.

Formulas to Know
Equation of a Tangent Line to y = f(x) at x = a

y equals f open parentheses a close parentheses plus f apostrophe open parentheses a close parentheses open parentheses x minus a close parentheses

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