Problem 45 - Essential Calculus

Question

In Exercises 45 and \(46,\) use a computer algebra system to find the maximum value of \(\left|f^{4}(x)\right|\) on the closed interval. (This value is used in the error estimate for Simpson's Rule, as discussed in Section 4.6.) \(\frac{\text { Function }}{f(x)=(x+1)^{2 / 3}} \frac{\text { Interval }}{[0,2]}\)

Step-by-Step Solution

Answer

The maximum value of \(\left|f^{4}(x)\right|\) on the interval [0,2] is \(0.00205761\).

Step 1 - Find the first derivative of the function

The first step will be to find the first derivative of the function \(f(x)=(x+1)^{2 / 3}\). This can be done using the power rule for differentiation, resulting in \(f'(x) = \frac{2}{3}(x+1)^{-1/3}\).

Step 2 - Find the second, third, and fourth derivatives

Continue differentiating to find the second, third, and fourth derivatives. Using successive applications of the chain rule and simplifying each result, you may find that \(f''(x) = -\frac{2}{9}(x+1)^{-4/3}\), \(f'''(x) = \frac{8}{27}(x+1)^{-7/3}\), and \(f^{(4)}(x) = -\frac{32}{81}(x+1)^{-10/3}\).

Step 3 - Find maximum value of the fourth derivative

To find the maximum value of \(\left|f^{4}(x)\right|\) within the closed interval [0,2], substitute the endpoints and any critical points of \(f^{(4)}(x)\) in the absolute value of the fourth derivative and compare the results. Critical points are obtained where \(f^{(4)}(x) = 0\) or \(f^{(4)}(x)\) is undefined. In this case, as there are no real numbers where \(f^{(4)}(x) = 0\), consider only the endpoints. Calculation will show that \(\left|f^{4}(2)\right|\) is the maximum, equals to \(0.00205761\).

Key Concepts

Fourth Derivative

In calculus, the fourth derivative of a function is related to the shape and smoothness of the graph of a function. The fourth derivative, denoted as \( f^{(4)}(x) \), is the derivative of the third derivative and so forth. For the given function \( f(x) = (x+1)^{2/3} \), the fourth derivative is:

• \( f^{(4)}(x) = -\frac{32}{81}(x+1)^{-10/3} \)

This derivative helps in understanding the behavior of curves, particularly in determining concavity and inflection points. In practical terms, the fourth derivative is essential when honing in on the finer details of a curve's behavior over an interval.
In the context of Simpson's Rule, knowing the maximum of the absolute value of this derivative over a specified interval allows us to approximate the error bound of the numerical integration. The smoothness of a curve impacts integration accuracy, as a smoother curve generally ensures more accurate approximations.

Error Estimate

Simpson's Rule is a method for numerically approximating definite integrals. While it provides a good balance between complexity and accuracy, it’s crucial to estimate the potential error when using this method. The error estimate relies heavily on the maximum value of the fourth derivative of the function over the integration interval.

• The error bound for Simpson's Rule is given by:\[ E = -\frac{(b-a)^5}{180n^4} \max_{x \in [a,b]} \left| f^{(4)}(x) \right| \]where \( a \) and \( b \) are the endpoints of the interval and \( n \) is the number of subintervals.
• In our function, the maximum value of the fourth derivative on the interval \([0, 2]\) was found to be approximately \(0.00205761\).

This error estimation helps us understand how the approximation compares to the exact value of the integral. If the maximum of \( \left| f^{(4)}(x) \right| \) is small, we can expect the error in the estimation to be correspondingly small. Thus, knowing the fourth derivative's behavior is crucial for numerical accuracy.

Critical Points

In calculus, a critical point of a function is a point where the function's derivative is zero or undefined. Finding critical points is important because they are potential locations of local maxima and minima or points of inflection.

• For our fourth derivative \( f^{(4)}(x) = -\frac{32}{81}(x+1)^{-10/3} \), there are technically no critical points as the derivative does not equal zero.
• Instead, attention is shifted to the endpoints of the interval \([0, 2]\) where the maximum or minimum values might occur due to boundary conditions.

Critical points are typically considered along with boundaries when analyzing derivative behavior over an interval, especially when determining error estimates or maximum values. This thorough analysis ensures that all potential maximum values of the derivate are identified, which supports accurate mathematical solutions.

Absolute Value

In mathematics, the absolute value of a number or expression refers to its magnitude regardless of its sign. In this context, \( |x| \) reflects the distance of \( x \) from zero on the number line, ensuring it's always a non-negative value.

• For Simpson's Rule and calculus problems, the absolute value is crucial in error estimation. It ensures that negative or positive fluctuations in the derivative are considered equally when evaluating their impact on calculations.
• Using the fourth derivative \( f^{(4)}(x) \), we calculate \( \left| f^{(4)}(x) \right| \) to find the maximum magnitude over the interval. Thus, for \( x = 2 \), \( \left| f^{(4)}(2) \right| \approx 0.00205761 \).

The role of absolute value here is to encompass all possible deviations in function behavior across an interval, ensuring a comprehensive error analysis. These insights allow for more reliable approximations in numerical methods, as users can confidently apply these values knowing that both positive and negative extremities have been catered for.