Use a graphing utility to plot the curve and the tangent line.
y = cos x / 1−cos x; x = π/3

Verified step by step guidance

1. First, understand the function y = \( \frac{\cos x}{1 - \cos x} \). This is a rational function where the numerator is \( \cos x \) and the denominator is \( 1 - \cos x \).
2. Next, find the derivative of the function to determine the slope of the tangent line at \( x = \frac{\pi}{3} \). Use the quotient rule: \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \), where \( u = \cos x \) and \( v = 1 - \cos x \).
3. Calculate \( u' \) and \( v' \). For \( u = \cos x \), \( u' = -\sin x \). For \( v = 1 - \cos x \), \( v' = \sin x \). Substitute these into the quotient rule formula.
4. Evaluate the derivative at \( x = \frac{\pi}{3} \) to find the slope of the tangent line. Substitute \( x = \frac{\pi}{3} \) into the derivative expression you obtained.
5. Finally, use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope found in the previous step, and \( (x_1, y_1) \) is the point on the curve at \( x = \frac{\pi}{3} \). Plot the curve and the tangent line using a graphing utility.

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. It represents the instantaneous rate of change of the function at that specific location. To find the equation of the tangent line, one typically needs the derivative of the function evaluated at the point of tangency.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In practical terms, the derivative provides the slope of the tangent line to the curve at any given point, which is essential for analyzing the behavior of the function.

Graphing Utility

A graphing utility is a software tool or calculator that allows users to visualize mathematical functions and their properties. It can plot curves, compute derivatives, and display tangent lines, making it easier to analyze complex functions. Using a graphing utility helps in understanding the relationship between a function and its tangent line, especially at specific points like x = π/3.