Problem 79 - Thomas Calculus

Question

In Exercises \(75-80,\) you will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps. $$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$ $$ f(x)=\sqrt{x}+\cos x, \quad[0,2 \pi] $$

Step-by-Step Solution

Answer

The absolute minimum value occurs at \( x = 0 \), and the maximum value occurs at \( x = \frac{3\pi}{2} \).

Step 1: Plot the Function

Using a graphing tool or CAS (Computer Algebra System), plot the function \( f(x) = \sqrt{x} + \cos x \) over the interval \([0, 2\pi]\). This will help visualize the behavior and shape of the function within this interval.

Step 2: Find Critical Points where f' = 0

First, compute the derivative: \( f'(x) = \frac{1}{2\sqrt{x}} - \sin x \). Using a CAS, solve the equation \( f'(x) = 0 \) within the interval \([0, 2\pi]\) to find the critical points where the derivative equals zero.

Step 3: Identify Points where f' Does Not Exist

Examine the derivative \( f'(x) = \frac{1}{2\sqrt{x}} - \sin x \) and determine where it is undefined. The derivative has a term \( \frac{1}{2\sqrt{x}} \), which is undefined at \( x = 0 \). Thus, \( x = 0 \) is a point where the derivative does not exist.

Step 4: Evaluate the Function at Key Points

Calculate \( f(x) \) at the critical points found in Step 2, at points where \( f' \) does not exist found in Step 3 (i.e., \( x = 0 \)), and at the endpoints of the interval: \( x = 0 \), \( x = 2\pi \). Evaluate \( f(x) \) at each point.

Step 5: Determine Absolute Extrema

From the evaluations in Step 4, compare the resulting values of \( f(x) \) to find the absolute minimum and maximum values over the interval \([0, 2\pi]\). Identify which points correspond to these extreme values.

Key Concepts

Critical points

In calculus, critical points are significant markers on the graph of a function. These are the points where the derivative of the function is zero, indicating possible maxima, minima, or points of inflection.
For a function \( f(x) \), finding critical points involves calculating \( f'(x) \) and solving \( f'(x) = 0 \). These solutions give us the values where the slope of the tangent to the curve is horizontal.
Critical points can be:

• Local maxima - peaks in the graph.
• Local minima - troughs in the graph.
• Points of inflection - where the curve changes concavity.

While critical points give crucial insights, they do not immediately reveal if these points are indeed absolute or local extrema. Further analysis is needed, often involving second derivatives or testing endpoints.

Derivative

The derivative of a function is a fundamental concept in calculus, representing the rate of change or slope of the curve at any point.
The derivative, denoted as \( f'(x) \), is calculated to understand how the function behaves locally. It helps identify increasing or decreasing trends and critical points.
For instance, in our exercise, the derivative of the function \( f(x) = \sqrt{x} + \cos x \) is \( f'(x) = \frac{1}{2\sqrt{x}} - \sin x \). This derivative helps in finding where the slope of the function is zero (i.e., critical points) or where the slope becomes undefined.
Calculus relies heavily on derivatives for:

• Graphing function shape and slope.
• Optimizing functions by finding maxima and minima.
• Analyzing motion and changes in physics and engineering.

Absolute extrema

Absolute extrema refer to the highest or lowest values a function can achieve on a given interval. These can be either absolute maximum or absolute minimum values.
Unlike local extrema, which only considers nearby points, absolute extrema are the definitive highest or lowest points over the entire interval.
To find the absolute extrema:

• Calculate \( f(x) \) at critical points.
• Check values at all endpoints of the interval.

The absolute extrema occur where these values are the largest or smallest.
Absolute extrema are vital in applications like:

• Determining maximum profits or losses.
• Finding optimal dimensions for minimal material use in engineering.

Closed interval

In mathematics, a closed interval is a range of numbers that includes its endpoints. It's generally written as \([a, b]\), meaning the interval includes both \(a\) and \(b\).
Solving problems on closed intervals often means evaluating function behavior at both endpoints, alongside any critical points identified within the interval. This approach ensures the comprehensive analysis necessary to locate maximum or minimum values, ensuring the function’s absolute extrema are addressed.
Closed intervals resemble a summary of all values the function can take within the specified domain, offering:

• A complete picture of a function’s behavior.
• Precise calculations by including boundaries.

Recognizing closed intervals is crucial for correctly applying theorems like the Extreme Value Theorem.

Plotting functions

Plotting is a visual technique in mathematics, offering a way to explore the shape and behavior of functions. By graphing functions, students can see changes and trends, making abstract concepts more concrete.
In our exercise, plotting \( f(x) = \sqrt{x} + \cos x \) over the interval \([0, 2\pi]\) helps visualize how the function behaves, offering insights into where it reaches peaks or valleys.
Benefits of plotting functions include:

• Immediate visual understanding of critical and extrema points.
• Aid in identifying periodic behavior, symmetry, or asymmetry.

Regularly plotting functions helps students develop intuition about function behavior, enhancing learning and retention.