Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the tangent line to \(y=\tan x\) at \(x=0\).

Step-by-Step Solution

Answer

The equation of the tangent line is \( y = x \).

Step 1: Understand the Problem

To find the equation of the tangent line to the function \( y = \tan x \) at \( x = 0 \), we need to find the slope of the tangent line at this point and the point of tangency.

Step 2: Find the Point on the Curve

Evaluate the function \( y = \tan x \) at \( x = 0 \). Since \( \tan 0 = 0 \), the point on the curve is \( (0, 0) \).

Step 3: Calculate the Derivative

To find the slope of the tangent line, we need to compute the derivative of \( y = \tan x \). The derivative of \( \tan x \) is \( \sec^2 x \).

Step 4: Evaluate the Derivative at Point

Evaluate the derivative \( \sec^2 x \) at \( x = 0 \). Since \( \sec 0 = 1 \), it follows that \( \sec^2 0 = 1^2 = 1 \). So, the slope of the tangent line is 1.

Step 5: Write the Equation of the Tangent Line

The equation of a line can be written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We know the slope \( m = 1 \) and the point on the line \( (0, 0) \). Substituting \( y = mx + b \) with \( x = 0 \), \( y = 0 \), and \( m = 1 \), we get \( 0 = 1 \cdot 0 + b \). Thus, \( b = 0 \). The equation is \( y = x \).

Key Concepts

Understanding Derivatives

Derivatives are a fundamental concept in calculus. They measure how a function changes as its input changes.
More specifically, they represent the rate of change or the slope of the function at any given point. Imagine it as the steepness of the graph at a particular location.

• The mathematical symbol for a derivative is often shown as \( f'(x) \) or \( \frac{dy}{dx} \).
• The process of finding a derivative is called differentiation.

The derivative of a function can be thought of as the slope of the tangent line to the function at any given point. In our example, we were asked to find the derivative of \( y = \tan x \). This derivative turns out to be \( \sec^2 x \), which shows how steep the function is at various values of \( x \).
Calculating a derivative requires practice, as it involves specific rules and methods. But knowing it provides deep insights into the behavior of different functions.

Exploring Tangent Lines

A tangent line is a straight line that "just touches" a curve at a certain point.
It represents the best linear approximation of the function at that point.

• The tangent line's slope is identical to the derivative of the function at the intersection point.
• It's important because it shows how the curve behaves at a specific point.

In our exercise, we found the tangent line at the point where \( x = 0 \) for the function \( y = \tan x \). The slope of this tangent line was determined to be 1, which was the value of the derivative \( \sec^2 x \) at \( x = 0 \).
With this slope, we could easily write the equation of the tangent line as \( y = x \). All tangent lines always lie flat against their tangent point, showcasing the closest straight-line behavior to the curve at that point.

Understanding Trigonometric Functions

Trigonometric functions are essential in calculus and many other areas of mathematics. They relate the angles of a triangle to the lengths of its sides and are periodic.
They often appear in problems dealing with waves or cycles.

• Common trigonometric functions include sine, cosine, and tangent.
• Tangent, denoted as \( \tan x \), is the ratio of sine to cosine \( \tan x = \frac{\sin x}{\cos x} \).

In the original problem, we worked with \( \tan x \), a key trigonometric function that can have some interesting properties. At \( x = 0 \), \( \tan x = 0 \), providing a convenient starting point for solving problems. Keep in mind that trigonometric functions are continuous and differentiable, making them relevant for calculus problems that involve growth rates and acceleration.
Mastering these functions provides a solid foundation for exploring more complex mathematical concepts and real-world applications.