For what values of x does g(x) = x−sin x have a slope of 1?

Verified step by step guidance

1. First, understand that the slope of a function at a given point is represented by its derivative. So, we need to find the derivative of g(x) = x - sin(x).
2. Calculate the derivative of g(x). The derivative of x is 1, and the derivative of sin(x) is cos(x). Therefore, the derivative g'(x) = 1 - cos(x).
3. Set the derivative equal to the desired slope, which is 1. This gives us the equation: 1 - cos(x) = 1.
4. Solve the equation 1 - cos(x) = 1 for x. Simplifying, we find that cos(x) must be equal to 0.
5. Determine the values of x for which cos(x) = 0. These values occur at x = (2n+1)π/2, where n is an integer.

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is often interpreted as the slope of the tangent line to the graph of the function at a given point. In this context, finding where the slope of g(x) equals 1 involves calculating the derivative g'(x) and setting it equal to 1.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, especially when dealing with periodic phenomena. The function sin(x) oscillates between -1 and 1, and its behavior influences the overall shape and slope of the function g(x) = x - sin(x). Understanding how these functions behave is crucial for analyzing the derivative.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are essential for determining where the function's slope changes, which can indicate local maxima, minima, or points of inflection. In this problem, identifying critical points of g'(x) will help locate the values of x where the slope is equal to 1.