Problem 48 - Calculus Early Transcendentals

Question

Find an equation for the tangent line to the graph at the specified value of \(x .\) $$ y=3 \cot ^{4} x, x=\frac{\pi}{4} $$

Step-by-Step Solution

Answer

The equation of the tangent line is \( y - 3 = -24(x - \frac{\pi}{4}) \).

Step 1: Differentiate the Function

To find the equation of the tangent line, we need the derivative of the function to find the slope at the given point. The original function is \( y = 3 \cot^4 x \). We'll use the chain rule and the derivative of \( \cot x = -\csc^2 x \). The derivative is: \( \frac{dy}{dx} = 3 \cdot 4 \cdot \cot^3 x \cdot (-\csc^2 x) = -12 \cot^3 x \csc^2 x \).

Step 2: Evaluate the Derivative at \( x = \frac{\pi}{4} \)

Substitute \( x = \frac{\pi}{4} \) into the derivative to find the slope of the tangent line. At \( x = \frac{\pi}{4} \), \( \cot \frac{\pi}{4} = 1 \) and \( \csc^2 \frac{\pi}{4} = 2 \). So, the derivative becomes: \( -12 \cdot 1^3 \cdot 2 = -24 \). Thus, the slope \( m \) of the tangent line is \( -24 \).

Step 3: Find the Point on the Graph

Substitute \( x = \frac{\pi}{4} \) into the original function to find the y-coordinate. We have \( y = 3 \cot^4 \frac{\pi}{4} = 3 \). So the point on the graph is \( \left( \frac{\pi}{4}, 3 \right) \).

Step 4: Write the Equation of the Tangent Line

Using the point-slope form of the equation \( y - y_1 = m(x - x_1) \), where \( m = -24 \) and the point is \( \left( \frac{\pi}{4}, 3 \right) \), we get: \( y - 3 = -24\left( x - \frac{\pi}{4} \right) \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that deals with finding how a function changes as its input changes. This is termed as finding the derivative of a function, which serves as a tool to determine the slope of the tangent line to the graph of the function at any given point.
When you're tasked to find an equation for a tangent line, the derivative tells you how steep or flat the graph is at that point. For example, in our problem, to find the derivative of the function given by \( y = 3 \cot^4 x \), we need to follow the rules of differentiation. Thanks to differentiation, once we calculate the derivative, we're equipped to find out how rapidly \( y \) changes with \( x \), which directly gives us the slope of the tangent line at \( x = \frac{\pi}{4} \). The slope is crucial to crafting the equation of the tangent line.
Moreover, differentiation isn't confined only to algebraic functions. It finds its application widely across various types of functions, including trigonometric functions, exponentials, and logarithms.

Chain Rule

The chain rule is an essential differentiation technique especially useful when differentiating composite functions, those made by combining two or more simpler functions. When you see a function like \( y = 3 \cot^4 x \), it's essentially an application of an outer function \( u^4 \) (where \( u = \cot x \)) on an inner function \( \cot x \).
Here's where the chain rule makes its grandeur entrance. It helps you navigate through multiple layers of a function. You first differentiate the outer function treating the inner function as a variable, then multiply by the derivative of the inner function. For our specific case:

• The outer function \( u^4 \) differentiates to \( 4u^3 \).
• The inner function \( \cot x \) has a derivative of \( -\csc^2 x \).

When combined using the chain rule, the derivative becomes \( 4 \cot^3 x \cdot (-\csc^2 x) \), leading us to find \( \frac{dy}{dx} = -12 \cot^3 x \csc^2 x \). Therefore, the chain rule simplifies the differentiation process for complex compositions by structuring it into manageable parts.

Trigonometric Functions

Trigonometric functions are often encountered in calculus exercises. These functions — sine, cosine, tangent, cotangent, secant, and cosecant — are important because they describe smooth periodic rotations and oscillations.
In our exercise, the function \( y = 3 \cot^4 x \) revolves primarily around the cotangent function, which is the reciprocal of the tangent function. Understanding trigonometric identities can significantly ease the process of solving calculus problems.
For example:

• \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \)
• \( \csc x = \frac{1}{\sin x} \)

These identities can simplify your calculations, especially when evaluating trigonometric expressions at specific points. In the specific problem at point \( x = \frac{\pi}{4} \), understanding these functions allowed us to calculate \( \cot \frac{\pi}{4} = 1 \) and \( \csc^2 \frac{\pi}{4} = 2 \), thus accurately deciphering the slope and further solving for the tangent line equation. Understanding trigonometric functions is crucial for effectively applying calculus tools like differentiation to real-world problems.