Question

Find all points on the graph of \(y=\tan ^{2} x\) where the tangent line is horizontal.

Short answer

The points are \((n\pi, 0)\) where \(n\) is any integer.

Step-by-step solution

Understand Horizontal Tangents

For a tangent line to be horizontal, its slope must be zero. This means we need to find points where the derivative of the function is zero.

Differentiate the Function

Find the derivative of the function \(y = \tan^2 x\). Use the chain rule: \(\frac{d}{dx}[u^2] = 2u \cdot \frac{du}{dx}\), where \(u = \tan x\) and \(\frac{du}{dx} = \sec^2 x\). This gives us: \(y' = 2 \tan x \cdot \sec^2 x\).

Set the Derivative to Zero

We set the derivative \(y' = 2 \tan x \cdot \sec^2 x = 0\). Since \( \sec^2 x \) is never zero, the equation simplifies to finding where \(\tan x = 0\).

Solve for \(x\)

The equation \(\tan x = 0\) is satisfied where \(x = n\pi\), where \(n\) is an integer, because the tangent function is zero at integer multiples of \(\pi\).

Identify Points on the Graph

Substitute \(x = n\pi\) back into the original function to find the points. Since \(y = \tan^2 x = 0^2\), all points have the form \((n\pi, 0)\), where \(n\) is an integer.

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus that deals with finding a rate of change. It involves computing the derivative of a function, which represents how the function's value changes as its input changes. In simpler terms, differentiation helps us find the slope of a curve at any given point, which is essential in understanding the behavior of functions.

When we differentiate a function, we're essentially determining an equation that gives us the slope of the tangent line to the curve at any point. In our case, we worked with the function \(y = \tan^2 x\) and found its derivative to be \(y' = 2 \tan x \cdot \sec^2 x\). Here’s how it goes step by step:

• Identify the function you need the derivative for, in this case, \(y = \tan^2 x\).
• Apply the chain rule, which is a technique used when differentiating composite functions, to find \(y' = 2 \tan x \cdot \sec^2 x\).
• Analyze the derivative equation to understand the slope behavior of the function.

Understanding differentiation is key to solving problems involving rates of change and understanding the geometry of curves.

Calculus

Calculus is the mathematical study of continuous change and is divided into two main branches: differential calculus and integral calculus. In essence, it provides us with tools to model and solve problems involving change and motion.

Differential calculus, which includes differentiation, focuses on finding how functions change. It's all about breaking down a problem into infinitely small pieces and analyzing them.

• In problems like finding horizontal tangents, we use differentiation to understand where a function's rate of change is zero.
• This involves computing derivatives to determine where slopes of tangents are horizontal, or zero.

In our example, calculus allows us to analyze the function \(y = \tan^2 x\) to determine where its derivative is zero, hence finding points where the tangent is horizontal.

Calculus is applied across various fields such as physics, engineering, economics, statistics, and many more, illustrating its importance and versatility.

Tangent Line

A tangent line to a curve is a straight line that just "touches" the curve at a particular point without crossing it. The concept of the tangent line is essential in understanding the slopes of curves at particular points. The goal is usually to determine where this line is horizontal, which means it has a slope of zero.

To find horizontal tangents:

• First, find the derivative of the function representing the curve.
• Set this derivative equal to zero to identify the points where the slope is zero, indicating a horizontal tangent line.

In the problem with the function \(y = \tan^2 x\), setting the derivative \(y' = 2 \tan x \cdot \sec^2 x\) equal to zero simplifies to \(\tan x = 0\). This equation tells us that the values of \(x\) where the tangent is horizontal are those for which \(x = n\pi\), where \(n\) is an integer. These points give us the coordinates \((n\pi, 0)\).

Understanding tangent lines help us analyze the nature and behavior of curves in various mathematical and practical applications.