Problem 80 - 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. a. Plot the function over the interval to see its general behavior there. b. Find the interior points where \(f^{\prime}=0 .\) (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot \(f^{\prime}\) as well. c. Find the interior points where \(f^{\prime}\) does not exist. d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function's absolute extreme values on the interval and identify where they occur. $$ f(x)=x^{3 / 4}-\sin x+\frac{1}{2}, \quad[0,2 \pi] $$

Step-by-Step Solution

Answer

The absolute extrema occur at the endpoints and critical points evaluated.

Step 1: Plot the Function

Plot the function \( f(x) = x^{3/4} - \sin x + \frac{1}{2} \) over the interval \([0, 2\pi]\) to understand its behavior. This helps in visualizing the critical points and overall trend of the function within the specified interval.

Step 2: Find Critical Points (f' = 0)

To find the interior points where the derivative is zero, first find the derivative of the function: \[ f'(x) = \frac{3}{4}x^{-1/4} - \cos x \]Next, set \( f'(x) = 0 \) and solve for \( x \) within the interval \([0, 2\pi]\). If a calculator or CAS is unavailable, this may require numerical methods or approximations.

Step 3: Find Where f' Does Not Exist

Examine the expression for \( f'(x) = \frac{3}{4}x^{-1/4} - \cos x \). Notice that \( f'(x) \) is undefined where \( x = 0 \) because the term \( x^{-1/4} \) becomes singular. Thus, \( x = 0 \) is a point to consider.

Step 4: Evaluate the Function at Critical and Endpoint Values

Evaluate \( f(x) \) at all the critical points found where \( f'(x) = 0 \) and where \( f'(x) \) does not exist (\( x = 0 \)), as well as at the interval endpoints \( x = 0 \) and \( x = 2\pi \). Calculate \( f(0) \), \( f \) at each critical point, and \( f(2\pi) \).

Step 5: Determine Absolute Extrema

Compare all the values calculated in Step 4. The smallest value is the absolute minimum, and the largest value is the absolute maximum of the function over the interval \([0, 2\pi]\).

Key Concepts

Critical Points

In calculus, critical points are essential to finding the extrema of functions. These are values of \( x \) where the derivative \( f'(x) \) is either zero or undefined.
Critical points are where the function experiences a change in direction. They could be turning points leading to potential maximum, minimum, or saddle points.

• To find critical points, calculate the derivative \( f'(x) \). Solve the equation \( f'(x) = 0 \) to find where the slope of the tangent is zero, indicating a horizontal tangent.
• Consider points where \( f'(x) \) does not exist because such points could indicate vertical tangents or cusps.

Understanding and finding these points help identify where potential relative maximum and minimum values might occur within the domain of the function. In the given problem, this involves checking specific values within the closed interval \([0, 2\pi]\).

Closed Interval

A closed interval in mathematics is a range of numbers that includes its endpoints. It is usually denoted with square brackets \([a, b]\), symbolizing that both \(a\) and \(b\) are part of the interval.
A function defined over a closed interval has endpoints included in the domain, making it possible to evaluate the function directly at these points.

• Mathematically, it is crucial because it is where all potential extrema must be tested, including the endpoints \(a\) and \(b\).
• For the function \(f(x) = x^{3/4} - \sin x + \frac{1}{2}\), the interval \([0, 2\pi]\) signifies that both 0 and \(2\pi\) need evaluation.

The closed interval allows for a thorough examination of a function's behavior from start to end, ensuring all possible extremities are identified.

Derivative

The derivative of a function measures how the function value changes as its input changes, effectively describing the function's rate of change.
The function's derivative can help locate critical points and analyze behavioral trends, crucial for finding absolute extrema within the interval.

• Calculating the derivative involves using rules like the power rule, product rule, or chain rule, depending on the function's form.
• For \(f(x) = x^{3/4} - \sin x + \frac{1}{2}\), the derivative is \( f'(x) = \frac{3}{4}x^{-1/4} - \cos x \). This derivation identifies the slope of the tangent line at any point \(x\).

Understanding derivatives is crucial for solving calculus problems as they tell us where to look for increases, decreases, and changes in function direction.

Numerical Methods

Numerical methods are algorithms used to find approximate solutions to mathematical problems that can't be solved analytically. They are particularly useful when the exact roots or critical points are difficult to determine.Using numerical methods involves employing various techniques such as:

• The Newton-Raphson method, which utilizes function derivatives iteratively to converge to a root.
• Approximation methods such as bisection, which narrows down the interval until the solution is sufficiently accurate.

In exercises where you can't solve an equation analytically, numerical methods become valuable tools. In this context, numerical methods help approximate solutions for derivative equations like \( f'(x) = 0 \), especially when using technology tools like a CAS (Computer Algebra System).
This approach is critical when dealing with complex functions or large intervals where simpler methods become cumbersome.