Question

\(f(x)=x^{4}, \quad a=1, \quad d x=0.01\)

Short answer

The derivative of the function \(f(x) = x^{4}\) at the point \(a = 1\) is approximately 4.060401.

Step-by-step solution

Substitute The Values Into The Function

Substitute the given \(a\) and \(dx\) values into the function \(f(x)\). For \(f(a)\), substitute \(x\) with \(a = 1\), resulting in \(f(a) = 1^{4} = 1\). For \(f(a + dx)\), substitute \(x\) with \(a + dx = 1.01\), resulting in \(f(a + dx) = (1.01)^{4}\).

Substitute The Function Results Into The Derivative Formula

Now, substitute the \(f(a)\) and \(f(a + dx)\) values into the formula for the derivative. This yields \[f'(a) = \frac{(1.01)^{4} - 1}{0.01}\].

Simplify The Numerator

Compute the value of \((1.01)^{4}\) and subtract 1. This results in approximately \[f'(a) = \frac{1.04060401 - 1}{0.01}\].

Complete The Division

Divide the difference by 0.01 to get the final result for the derivative at the point \(a = 1\). The result is approximately 4.060401, rounding to seven decimal places. This is the approximate rate of change of the function \(f(x) = x^{4}\) at the point \(a = 1\).

Key concepts

Numerical Differentiation

Numerical differentiation is a method of finding an approximation to the derivative of a function at a particular point using discrete values of the function. Unlike analytical differentiation where we find the exact derivative expression, in numerical differentiation, we often employ a difference quotient which incorporates function evaluations at specific points. This method is especially useful when dealing with functions that may not be easy to differentiate analytically or when working with data points rather than functions.

An example of numerical differentiation is the forward difference method, which uses the formula \[f'(a) = \frac{f(a + dx) - f(a)}{dx}\] to estimate the slope of the curve, or the rate of change, at a point. The term \(dx\) represents a small change in the input value which, in practical applications, should be chosen carefully as too large a value can decrease the accuracy of the differentiation, while a value too small can amplify computational errors.

Function Evaluation

The process of function evaluation involves computing the output of a function given specific input values. This is a fundamental aspect of both algebra and calculus, used for everything from graphing curves to solving equations. When using numerical methods, precise function evaluation at specific points is crucial to obtaining accurate results.

For instance, in the context of numerical differentiation, we first evaluate the function at the points \(a\) and \(a + dx\) and then use these function values to estimate the derivative. This step is critical because any mistakes in evaluation can lead to erroneous results in subsequent calculations like the derivative approximation.

Substitution Method

The substitution method involves replacing variables with their corresponding values or expressions. This technique is used extensively in calculus, particularly in solving integrals and differentials. In numerical differentiation, for example, we substitute the known values of \(a\) and \(dx\) into the function and into the formula for the derivative.

It is essential to perform these substitutions accurately. A common exercise for students is to substitute with care and then simplify the resulting expression before proceeding further, as shown in the step-by-step example with the function \(f(x) = x^{4}\) and values \(a = 1\) and \(dx = 0.01\).

Rate of Change

The rate of change is a fundamental concept in calculus, which, in its simplest form, can be understood as the speed at which a function's output is changing at a particular point with respect to its input. In everyday language, this is akin to how quickly or slowly a car is traveling at a specific instant. When it comes to functions, this translates to the concept of the derivative, which is the exact rate of change at any given point on a function's curve.

This translates directly to the slope of the tangent line to the function at a particular point and is exactly what we approximate through numerical differentiation. The exercise given illustrates an approximation of the rate of change for the function \(f(x) = x^{4}\) at the point \(a = 1\) using a very small increment \(dx\). Calculating and understanding the rate of change help not only in mathematics but also in various practical fields such as physics, economics, and engineering.