Question

Let \(f(x)=x-\sin x .\) Find all points on the graph of \(y=f(x)\) where the tangent line is horizontal. Find all points on the graph of \(y=f(x)\) where the tangent line has slope 2 .

Short answer

Horizontal: points \( (2k\pi, 2k\pi) \); slope 2: points \( ((2k+1)\pi, (2k+1)\pi + 1) \).

Step-by-step solution

Understanding the Problem

We need to find all points on the graph of the function where the tangent line is horizontal and where it has a slope of 2. This requires finding the derivative of the function and setting it equal to 0 and 2, respectively.

Taking the Derivative

We find the derivative of the function: Given the function \( f(x) = x - \sin x \), the derivative is \( f'(x) = 1 - \cos x \).

Finding Horizontal Tangents

For the tangent to be horizontal, the derivative must be equal to 0: \( f'(x) = 1 - \cos x = 0 \).This simplifies to \( \cos x = 1 \). The solutions to this equation are at \( x = 2k\pi \), where \( k \) is any integer.

Finding Tangents with Slope 2

For the tangent to have a slope of 2, set the derivative equal to 2:\( f'(x) = 1 - \cos x = 2 \). This simplifies to \( \cos x = -1 \). The solutions to this equation are at \( x = (2k+1)\pi \), where \( k \) is any integer.

Identifying Points on the Graph

For horizontal tangents when \( x = 2k\pi \), the points on the graph are \( (2k\pi, 2k\pi) \).For tangents with slope 2 at \( x = (2k+1)\pi \), the points on the graph are \( ((2k+1)\pi, (2k+1)\pi + 1) \), since \( \sin((2k+1)\pi) = 0 \).

Key concepts

Derivative

In calculus, the derivative of a function provides critical information about rates of change. In other words, it tells how a function's output value changes as its input changes. To find the derivative of the function \( f(x) = x - \sin x \), we use differentiation rules. One essential rule is that the derivative of \( x \) is 1, and the derivative of \( \sin x \) is \( \cos x \). By applying these rules, we find that the derivative is \( f'(x) = 1 - \cos x \). This derivative expression gives the slope of the tangent line to the curve at any given point on the function's graph.

• If \( f'(x) = 0 \), the tangent line is horizontal because there's no change in the height of the curve at that point.
• In other cases, where \( f'(x) \) equals other values, like 2, it provides the slope of the tangent line at that specific point.

Understanding how to compute and interpret derivatives forms the foundation for more complex calculus concepts and is crucial for solving problems involving rates of change.

Tangent Line

A tangent line to a curve at a given point is a straight line that just "touches" the curve at that point. This line has the same slope as the function at that point. Therefore, if we know the derivative of the function, we can determine the slope of the tangent line.
In our problem, the tangent line is horizontal when its slope is 0. For the function \( f(x) = x - \sin x \), setting the derivative \( f'(x) = 1 - \cos x \) equal to 0 gives \( \cos x = 1 \). These are the \( x \) values at which the tangent is horizontal, occurring at \( x = 2k\pi \), where \( k \) is an integer. At these points, the tangent doesn't rise or fall, indicating a peak or trough on the graph.
For a tangent line with a slope of 2, setting \( f'(x) = 2 \) leads to \( \cos x = -1 \). This occurs at \( x = (2k+1)\pi \), indicating points where the slope of the tangent is 2, meaning the line rises at a steady rate of 2 for every 1 unit increase in \( x \). These concepts help us understand how the graph behaves at various points.

Trigonometric Functions

Trigonometric functions like sine and cosine are crucial in understanding oscillatory and periodic behaviors in mathematics and science. In the function \( f(x) = x - \sin x \), the cosine function appears in its derivative \( f'(x) = 1 - \cos x \).

• The cosine function has specific point values: \( \cos x = 1 \) at \( x = 2k\pi \), where \( k \) is an integer, making the rate of change zero and the tangent horizontal.
• Meanwhile, \( \cos x = -1 \) at \( x = (2k+1)\pi \) provides a slope of 2, indicating a specific rate of change for the tangent line.

The periodic nature of sine and cosine impacts how often these values recur. Understanding these functions' periodicity is essential for predicting behavior of wave-like phenomena, a concept often leveraged in physics and engineering. Knowing these functions helps solve equations involving trigonometric expressions and enables us to describe complex cyclic phenomena effectively.