Problem 11 - Calculus with Concepts in Calculus

Question

Find all critical numbers of the given function. $$ f(x)=x+\sin x $$

Step-by-Step Solution

Answer

The critical numbers of the function are \( x = \\pi + 2k\pi \), where \( k \) is an integer.

Step 1: Understanding Critical Numbers

Critical numbers of a function occur where the derivative is zero or undefined. Our task is to solve for these values in the given function.

Step 2: Differentiate the Function

Differentiate the function with respect to \(x\): \[ f'(x) = 1 + rac{d}{dx}( ext{sin} \, x) = 1 + ext{cos} \, x \]

Step 3: Set the Derivative to Zero

To find the critical numbers, set the derivative equal to zero and solve for \(x\): \[ 1 + ext{cos} \, x = 0 \] Simplifying gives: \[ ext{cos} \, x = -1 \]

Step 4: Solve the Equation

Find the solutions for \( ext{cos} \, x = -1 \). The cosine of \(x\) is \(-1\) at \(x = \pi + 2k\pi\), where \(k\) is any integer.

Step 5: Identify and List the Critical Numbers

Thus, the critical numbers are all values \(x = \pi + 2k\pi\), where \(k\) is an integer.

Key Concepts

Derivative of a Function

The concept of the derivative is foundational in calculus. It provides a way to measure how a function changes as its input changes. In essence, the derivative of a function gives us the slope of the tangent line to the graph of the function at any given point. For a function \( f(x) \), the derivative is often denoted as \( f'(x) \) or \( \frac{df}{dx} \).

Differentiation involves applying specific rules to find the derivative of a function. One common rule involves basic differentiation of power functions: \( \frac{d}{dx}(x^n) = nx^{n-1} \). In our case, for the function \( f(x) = x + \sin x \), we need to differentiate both \( x \) and \( \sin x \).

• The derivative of \( x \) with respect to \( x \) is 1.
• The derivative of \( \sin x \) is \( \cos x \).

Therefore, the derivative of \( f(x) = x + \sin x \) is reached by adding these derivatives: \( f'(x) = 1 + \cos x \). Calculating derivatives accurately allows us to identify points of interest such as critical numbers. These critical numbers help in analyzing the behavior of the function beyond just slopes.

Trigonometric Functions

Trigonometric functions are a critical aspect of mathematics, especially in calculus and periodic phenomena modeling. These functions are based on angles and are most commonly applied using the sine, cosine, and tangent functions. They oscillate, reflecting angles in a unit circle.

In our task, we are dealing with the function \( f(x) = x + \sin x \), which includes the sine function. Here, knowing the corresponding cosine derivative is important.

• The \( \sin \) function is periodic with a period of \( 2\pi \).
• The \( \cos \) function, as the derivative, has a phase shift but the same period.

In the solution, we found the derivative to be \( 1 + \cos x \), which is a combination of a constant and a trigonometric function. The cosine function ranges from -1 to 1, allowing us to solve subsequent equations to find critical numbers. Understanding these properties of trigonometric functions is vital for solving equations and tackling calculus problems.

Solving Equations

Solving equations is about finding the values of the variable that make the equation true. In this instance, we're focusing on finding critical numbers by setting the derivative of a function equal to zero or finding where it's undefined.

For the function \( f(x) = x + \sin x \), the derivative is \( f'(x) = 1 + \cos x \). To find critical numbers, you solve \( 1 + \cos x = 0 \), leading to \( \cos x = -1 \).

• The solution \( \cos x = -1 \) occurs at specific points on the unit circle, namely when \( x = \pi + 2k\pi \), where \( k \) is an integer.
• This equation highlights the periodic nature of trigonometric solutions, reflecting the repeating pattern every \( 2\pi \), corresponding to full rotations around the circle.

Identifying solutions like \( x = \pi + 2k\pi \) enables understanding of the function's behavior at these points, referred to as critical numbers. This knowledge is crucial for deeper exploration of a function's increasing or decreasing intervals and turning points.