Problem 47 - Calculus of a Single Variable

Question

Finding Maximum Values Using Technology In Exercises 47 and \(48,\) use a computer algebra system to find the maximum value of \(\left|f^{\prime \prime}(x)\right|\) on the closed interval. (This value is used in the error estimate for the Trapezoidal Rule, as discussed in Section 4.6.) $$ f(x)=\sqrt{1+x^{3}}, \quad[0,2] $$

Step-by-Step Solution

Answer

The maximum value of the absolute value of the second derivative on the interval [0, 2] is 0.25.

Step 1: First Derivative

First we need to find the first derivative of the function \(f(x) = \sqrt{1 + x^{3}}\). We can do this by applying the chain rule for differentiation, which yields \(f'(x) = \frac{3x^2}{2(1+x^3)^{1/2}}\).

Step 2: Second Derivative

Next, we need to find the second derivative, which again involves applying the chain rule and also the quotient rule. The second derivative is a bit more complex, however by carefully applying these differentiation rules, we find \(f''(x) = \frac{-3x^{4}-3x^{2}+1}{4(1+x^{3})^{3/2}}\).

Step 3: Determine Maximum Value

Now, we need to find the maximum value of the absolute value of the second derivative over the interval [0, 2]. This can be achieved by evaluating the second derivative at the end points of the interval, that are \(x=0\) and \(x=2\), and finding the maximum value. After substituting the values we get \(|f''(0)| = 0.25\) and \(|f''(2)| = approx 0.022\).

Key Concepts

Maximum Value

Finding the maximum value of a function is a fundamental task in calculus. It involves identifying points where the function reaches its highest point in a given interval. In the context of the original exercise, we are interested in the maximum value of the absolute value of the second derivative, \(|f''(x)|\), over the interval \([0, 2]\).
This step is crucial for estimating the error in numerical methods, such as the Trapezoidal Rule. To determine the maximum value:

• Evaluate the second derivative at key points, such as the endpoints of the interval.
• Check any critical points within the interval by setting the first derivative of \(|f''(x)|\) to zero and solving for \(x\).
• Compare these values to find the largest one.

In this exercise, the evaluation at the endpoints provides a good estimate of the maximum value, which turned out to be \(|f''(0)| = 0.25\).

Second Derivative

The second derivative provides insights into the concavity of a function and is a key component in solving problems related to optimization.
To find the second derivative of a function like \(f(x) = \sqrt{1 + x^{3}}\), one must perform differentiation twice, often involving advanced rules such as the chain rule and the quotient rule.

• The **first derivative** gives the rate of change of the function.
• The **second derivative** helps determine if this rate is increasing or decreasing, showing if the graph is concave up or down.

In our specific case, careful application of these rules led to the complex expression \(f''(x) = \frac{-3x^{4}-3x^{2}+1}{4(1+x^{3})^{3/2}}\), which requires further analysis to identify concavity and calculate the maximum value.

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to approximate the definite integral of a function, often utilized when an exact integral is difficult to compute.
The accuracy of this approximation can be affected by the properties of the second derivative, specifically its maximum value on the interval. This forms part of the error estimation for the Trapezoidal Rule.
For the function \(f(x) = \sqrt{1 + x^{3}}\), finding \(|f''(x)|\)'s maximum helps understand how well the Trapezoidal Rule will approximate the integral over \([0, 2]\). The error is minimized when this maximum value is low, ensuring more accuracy. Hence, knowing that \(|f''(0)| = 0.25\) gives insight into the expected precision of this approximation method.

Chain Rule

The Chain Rule is an essential differentiation technique in calculus, often used to find derivatives of composite functions.
When handling functions like \(f(x) = \sqrt{1 + x^{3}}\), the Chain Rule allows us to simplify differentiation by breaking down complex expressions into manageable parts.
Here’s how it works:

• Identify the outer function and the inner function. For \(f(x)\), the outer function is \(\sqrt{u}\) and the inner function is \(u = 1 + x^3\).
• Differentiate the outer function with respect to the inner function, then multiply by the derivative of the inner function.

This essential calculus tool enabled us to correctly determine the first derivative \(f'(x) = \frac{3x^2}{2(1+x^3)^{1/2}}\), which was subsequently used to find the second derivative.