Problem 48 - Calculus - AP Edition

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 tangent line is \(y = -24x + 3 + 6\pi\).

Step 1: Calculate the derivative

To find the equation of the tangent line, we first need the derivative of the function. Given the function is \(y = 3 \cot^4 x\), we will use the chain rule. The derivative of \(\cot x\) is \(-\csc^2 x\). Thus, the derivative of \(y\) is \(y' = 4 \times 3 \cot^3 x \times (-\csc^2 x)\). So, \(y' = -12 \cot^3 x \csc^2 x\).

Step 2: Evaluate the derivative at the specified point

Now, substitute \(x = \frac{\pi}{4}\) into the derivative. First, calculate \(\cot \frac{\pi}{4}\) which is 1 and \(\csc \frac{\pi}{4}\) which is \(\sqrt{2}\). Substituting gives \(y'\left(\frac{\pi}{4}\right) = -12 \times 1^3 \times (\sqrt{2})^2 = -24\).

Step 3: Find the equation of the tangent line

The equation of the tangent line is given by the point-slope form \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1,y_1)\) is the point of tangency. We've found \(m = -24\). Calculate \(y(\frac{\pi}{4}) = 3 \times 1^4 = 3\). So, the point is \((\frac{\pi}{4}, 3)\). Substituting gives: \(y - 3 = -24(x - \frac{\pi}{4})\). Simplifying, the equation of the tangent line is \(y = -24x + 3 + 6 \pi\).

Key Concepts

Derivative of a Function

Understanding the derivative of a function is essential when tackling problems related to tangent lines. The derivative represents the instantaneous rate of change of a function concerning one of its variables. It's akin to determining the slope of a curve at a certain point, which is crucial for finding tangent lines.

For example, the function given in the exercise:

• \( y = 3 \cot^4 x \)

requires differentiation to find the tangent line at a specific point. By finding \( y' \), the derivative, we can understand how the function behaves around \( x = \pi/4 \). This behavior gives us the slope of the tangent line, which is the primary focus for determining its equation.

Chain Rule

The chain rule is a fundamental technique in calculus for finding the derivative of compositions of functions. It is vital when dealing with functions like \( y = 3 \cot^4 x \).

The chain rule states that if a function \( y \) is composed of a function within another function, we can find the derivative \( y' \) by taking the derivative of the outer function, multiplied by the derivative of the inner function.

In the given problem, we have:

• The outer function is \( 3 (\cdot)^4 \).
• The inner function is \( \cot x \).

Using the chain rule, the derivative \( y' \) is calculated as:

• \( y' = 4 \times 3 \cot^3 x \times (-\csc^2 x) \)
• This results in \( y' = -12 \cot^3 x \csc^2 x \).

This formula helps us prepare for calculating the exact slope of the tangent line at \( x = \pi/4 \).

Point-Slope Form

The point-slope form of a line is a straightforward method for writing the equation of a line when you know a point on the line and its slope. It is particularly useful for tangent lines because a tangent line always touches the curve at exactly one point.

The form is expressed as:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) is the point of tangency, and \(m\) is the slope, which we've previously derived indicating the rate of change of the function at that point.

Applying this to our problem:

• The slope \( m \) is \(-24\),
• The point \((x_1, y_1)\) is \( (\pi/4, 3) \).
• The tangent line equation becomes: \( y - 3 = -24(x - \pi/4)\).

Solving this gives us a clear representation of the tangent through:

• \( y = -24x + 3 + 6\pi \).

This equation describes the line that just grazes the curve at \( x = \pi/4 \), providing a linear approximation of the function at that point.