Problem 123 - Calculus Volume 1

Question

Find the local and/or absolute maxima for the functions over the specified domain. \(y=x+\sin (x)\) over \([0,2 \pi]\)

Step-by-Step Solution

Answer

The absolute maximum is \(y = 2\pi\) at \(x = 2\pi\).

Step 1: Find the derivative

To find the critical points where maxima might occur, first find the derivative of the function. The function given is \(y = x + \sin(x)\). Its derivative is \(y' = 1 + \cos(x)\).

Step 2: Set the derivative to zero

To find critical points, set the derivative equal to zero and solve for \(x\): \(1 + \cos(x) = 0\). This gives \(\cos(x) = -1\).

Step 3: Solve for x

Solve \(\cos(x) = -1\) in the interval \([0, 2\pi]\). This occurs at \(x = \pi\).

Step 4: Evaluate function at critical point and endpoints

Calculate the value of \(y\) at the critical point and the endpoints of the interval. At \(x=0\), \(y = 0 + \sin(0) = 0\). At \(x=\pi\), \(y = \pi + \sin(\pi) = \pi\). At \(x=2\pi\), \(y = 2\pi + \sin(2\pi) = 2\pi\).

Step 5: Compare values to find maxima

Compare the function values: \(y(0) = 0\), \(y(\pi) = \pi\), and \(y(2\pi) = 2\pi\). The highest value is \(y(2\pi)=2\pi\).

Key Concepts

Understanding Derivative

A derivative measures how a function changes as its input changes. In simple terms, it represents the "instantaneous rate of change" of a function. If you imagine driving a car, the derivative would indicate your speed at any given moment, showing how your position changes with respect to time.
To find a derivative, you take the original function and apply differentiation rules. For our function, \(y = x + \sin(x)\), the derivative is \(y' = 1 + \cos(x)\). This tells us how the function \(y\) changes with respect to \(x\).

• The derivative \(1\) shows the constant change from the \(x\) term.
• The \(\cos(x)\) part indicates the change rate of the sine component.

Having the derivative is crucial as it sets the stage for finding critical points and understanding where a function might have maxima or minima.

Critical Points Explained

Critical points are where the function's derivative is zero or undefined, suggesting potential maximum or minimum points. Think of them as places where the function "changes direction".
To find these, look at the derivative, \(y' = 1 + \cos(x)\), and set it equal to zero: \(1 + \cos(x) = 0\). Solving \(\cos(x) = -1\) gives the critical point \(x = \pi\) within the interval \([0, 2\pi]\).

• At a critical point, the curve might go from increasing to decreasing or vice versa.
• This can indicate where local maximum or minimum points occur.

Understanding these points is essential for identifying where the function might level out or peak.

Identifying Absolute Maxima

Absolute maxima are the highest points in the entire interval of the function. Think of them as the tallest peaks in a landscape. They are not confined to a small neighborhood but are the highest value of the function over the whole domain.
To find these points, evaluate the function at critical points and endpoints. For \(y = x + \sin(x)\), we compute the values at \(x = 0\), \(x = \pi\), and \(x = 2\pi\):

• \(y(0) = 0\)
• \(y(\pi) = \pi\)
• \(y(2\pi) = 2\pi\)

The highest value among these is at \(x = 2\pi\), where \(y = 2\pi\). In this context, \(x=2\pi\) is the absolute maximum for \([0, 2\pi]\).

Understanding Local Maxima

A local maximum is a point where a function reaches a peak within a small neighborhood but is not necessarily the highest point overall. Imagine a hilly landscape where each hilltop represents a local maximum.
For the function \(y = x + \sin(x)\) within the interval \([0, 2\pi]\), we identify possible local maxima by examining around our critical point and endpoints. By checking the values of the function at these points,

• The critical point \(x = \pi\) gives a value \(y(\pi) = \pi\), which is greater than \(y(0) = 0\) but less than \(y(2\pi) = 2\pi\).
• This indicates \(x = \pi\) is a local maximum within its vicinity.

A local maximum is relative to nearby values and helps understand the function's behavior along the domain.