Problem 44 - Calculus of a Single Variable

Question

Finding Absolute Extrema In Exercises \(41-44,\) use a graphing utility to graph the function and find the absolute extrema of the function on the given interval. $$ f(x)=\sqrt{x}+\cos \frac{x}{2}, \quad[0,2 \pi] $$

Step-by-Step Solution

Answer

The absolute extrema values will be the maximum and minimum \(y\)-values obtained after evaluation of the critical points and the endpoints in step 3. The corresponding \(x\)-values will be where these maximum and minimum values occur. Due to the task's dependence on a calculator, these values can't be exactly specified without an actual calculation.

Step 1: Graph the Function

Using a graphing utility, input the function \(f(x) = \sqrt{x} + \cos(\frac{x}{2})\). The graph will help visually identify approximations of the maximum and minimum points within the interval [0, 2π].

Step 2: Find the Critical Points

A critical point occurs where the first derivative of the function is zero or undefined. To find this, compute \(f'(x)\), set it to zero and solve for \(x\). However, due to the complexity of the derivative of this function, we can use the calculator's numerical derivative feature to find approximate values for the critical points.

Step 3: Evaluate the Function at Critical Points and Endpoints

Now, substitute the critical points and the interval endpoints (0 and 2π) back into the original function \(f(x)\). These will give the \(y\)-values at these points.

Step 4: Find the Maximum and Minimum Values

Compare the \(y\)-values obtained in the previous step. The highest value will correspond to the absolute maximum of the function on the interval [0, 2π], and the lowest value will correspond to the absolute minimum.

Key Concepts

Graphing Utility

Graphing utilities are powerful tools that help us visualize mathematical functions. When dealing with complex functions like \( f(x) = \sqrt{x} + \cos \left(\frac{x}{2}\right) \), a graphing utility makes it easier to identify patterns and potential extrema. By inputting the function into a graphing software or calculator, you can see the curve that represents the function over a given interval, such as \([0, 2\pi]\). This visual representation allows you to estimate where the maximum and minimum values, also known as extrema, might occur. These estimated points are crucial as they help us focus our numerical calculations on these areas. A graphing utility not only shows the function's behavior but also provides a way to quickly verify the results from more detailed analytical methods.

Critical Points

Critical points of a function are where the function's derivative is zero or undefined. These points are important because they can indicate potential maxima, minima, or inflection points of the function. When dealing with the function \( f(x) = \sqrt{x} + \cos \left(\frac{x}{2}\right) \), finding the critical points involves:

• Taking the derivative \( f'(x) \).
• Setting the derivative equation \( f'(x) = 0 \) to solve for \( x \).

However, because the derivative of our function is complex, using a numerical derivative calculator can greatly simplify finding these critical points. Understanding the critical points allows us to further explore the function's behavior and is a crucial step in determining its absolute maximum and minimum on the interval.

Numerical Derivative

Numerical derivatives offer an alternative to analytically finding derivatives, especially when dealing with complicated functions where algebraic derivatives are difficult to compute. The numerical derivative gives us an approximation of the slope of the tangent line to the function at a particular point. For the function \( f(x) = \sqrt{x} + \cos \left(\frac{x}{2}\right) \), this approximation:

• Helps identify critical points more easily.
• Aids in evaluating the function's rate of change at specific points.
• Simplifies the process when the derivative is too complex to solve by hand.

Numerical methods are implemented in most graphing calculators and software, which calculate these derivatives by using small changes around a point to estimate the slope. This makes understanding and using derivatives accessible to students without advanced calculus knowledge, making it a practical approach in finding extrema.

Endpoints Evaluation

When looking for absolute extrema of a function on a closed interval, it is essential to evaluate the endpoints. For a function like \( f(x) = \sqrt{x} + \cos \left(\frac{x}{2}\right) \) over the interval \([0, 2\pi]\), we calculate the function's value at both \( x = 0 \) and \( x = 2\pi \).

• Endpoints can be locations for absolute minima or maxima.
• Checking these values ensures that no potential extrema are overlooked just because they occur at the boundary of the interval.
• After evaluating the function at these points, compare them with the values at the critical points to determine the absolute maximum and minimum.

This step is crucial because endpoints often hold the key answers, particularly when the critical points do not yield the extrema. Evaluating the function at the endpoints complements the analysis of critical points to achieve a comprehensive understanding of the function's behavior over the interval.