Problem 43 - Essential Calculus

Question

In Exercises 43 and \(44,\) 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 \(4.6 .)\) \(\frac{\text { Function }}{f(x)=e^{-x^{2} / 2}} \quad \frac{\text { Interval }}{[0,1]}\)

Step-by-Step Solution

Answer

The maximum absolute value of the second derivative of \(f(x) = e^{-x^{2} / 2}\) on the interval [0,1] is 0.

Step 1: Compute the Derivative

First, find the derivative, \(f'(x)\), of the function \(f(x) = e^{-x^2 / 2}\). Use the chain rule which states that \((f(g(x)))' = f'(g(x)) * g'(x)\) to obtain \(f'(x) = -x * e^{-x^2 / 2}\).

Step 2: Compute the Second Derivative

The second derivative, \(f''(x)\), is the derivative of the first derivative. Apply the chain rule again to obtain \(f''(x) = (1 - x^2) * e^{-x^2 / 2}\).

Step 3: Find the Maximum of the Absolute Value of the Second Derivative

The maximum value of \(\left|f''(x)\right|\) on the interval [0,1] is obtained by setting the derivative of \(\left|f''(x)\right|\) equal to zero, solving for x, and checking these x-values plus the end points of the interval. This gives \(x ~= 1\). Check that this is indeed maximum by confirming that the second derivative changes sign at \(x = 1\). Thus, the maximum value of \(\left|f''(x)\right|\) on [0,1] is \(\left|f''(1)\right| = (1 - 1^2) * e^{-1^2 / 2} = 0\).

Key Concepts

Trapezoidal Rule Error Estimate

The Trapezoidal Rule is a numerical method used to approximate the definite integral of a function. It is particularly useful when an analytical solution is difficult to obtain. One thing to be cautious about is the error associated with this approximation.

The error estimate for the Trapezoidal Rule depends on the second derivative of the function because it measures how far the function deviates from being linear – the Trapezoidal Rule is exact for linear functions. The error is given by the formula \[E_T = \frac{(b-a)^3}{12n^2}f''(\xi)\], where \([a, b]\) is the interval of integration, \(n\) is the number of trapezoids, and \(\xi\) is some number in the interval \([a, b]\). The aim is to find the maximum absolute value of the second derivative, \(|f''(x)|\), on the interval to ensure that the estimate is conservative and encompasses all possible errors within that interval.

Computer Algebra System

A computer algebra system (CAS) is a type of software specifically designed to manipulate mathematical expressions and perform symbolic calculations. Unlike standard calculators, which only process numerical values, CAS tools can handle symbols, variables, and mathematical expressions to solve equations, differentiate functions, and much more.

When using a CAS to estimate the maximum value of the second derivative, as in the given problem, it allows for quick calculation and simplification. It automates the process of applying the chain rule multiple times, setting derivatives equal to zero, and solving for critical points. A CAS can also handle the cumbersome task of finding the maximum of the absolute value of the second derivative through symbol manipulation and differentiation.

Chain Rule

The chain rule is fundamental in calculus for differentiating composite functions. Whenever you have a function that is made up of other functions, the chain rule provides a way to differentiate the entire composition.

The general form of the chain rule states \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\). In this case, for the function \(f(x) = e^{-x^2 / 2}\), we apply the chain rule to differentiate it with respect to \(x\). The outer function is the exponential function, and the inner function is \(-x^2 / 2\). Using the chain rule helps us find the extreme points of the function by enabling the calculation of the second derivative, which is crucial in both curvature estimation and error approximation for numerical integration.

Maximizing Functions

Maximizing a function involves finding the point at which it attains its highest value on a given interval. This process often requires finding the first derivative, setting it equal to zero to find critical points, and then analyzing the second derivative to determine whether these points are maxima, minima, or points of inflection.

In finding the maximum absolute value of the second derivative, it's important to consider both the critical points of \(f''(x)\) and the end points of the interval. Mathematically, we check the values of \(f''(x)\) at these points and determine where it reaches the greatest magnitude. The highest of these values yields the error estimate for the Trapezoidal Rule.