Question

Suppose that \(f\) is differentiable for all \(x\) and consider the function $$D(x)=\frac{f(x+0.01)-f(x)}{0.01}.$$ For the following functions, graph \(D\) on the given interval, and explain why the graph appears as it does. What is the relationship between the functions \(f\) and \(D ?\) $$f(x)=\frac{x^{3}}{3}+1 \text { on }[-2,2]$$

Short answer

Answer: The function \(D(x)\) approximates the actual derivative of the function \(f(x)\) by using a finite difference with \(\Delta x = 0.01\). As \(\Delta x\) gets smaller, the approximation becomes more accurate. In this exercise, \(D(x)\) is an approximation of the derivative of the given function \(f(x) = \frac{x^3}{3} + 1\), which is \(f'(x) = x^2\).

Step-by-step solution

Calculate the expression for D(x) by using the given formula

We are given the fomula for \(D(x)\) as: $$D(x)=\frac{f(x+0.01)-f(x)}{0.01}.$$ Let \(f(x) =\frac{x^{3}}{3}+1\). Then we can write \(D(x)\) as: $$D(x)=\frac{f(x+0.01)-f(x)}{0.01} = \frac{\frac{(x+0.01)^{3}}{3}+1-\left(\frac{x^{3}}{3}+1\right)}{0.01}.$$ Now simplify the expression: $$D(x) = 100\left(\frac{(x+0.01)^3}{3} - \frac{x^3}{3}\right).$$

Graph D(x) on the given interval [-2, 2]

Now that we have the expression for \(D(x)\), you can use any graphing calculator to plot it on the interval [-2, 2]. Observe the general shape and behavior of the graph.

Find the actual derivative of f(x) and compare it to D(x)

To figure out the relationship between the functions \(f\) and \(D\), let's find the actual derivative of the function \(f(x)\), \(f'(x)\): $$ f(x) = \frac{x^3}{3} + 1. $$ Differentiate this function with respect to \(x\): $$ f'(x) = \frac{d}{dx}\left(\frac{x^3}{3} + 1\right) = \frac{d}{dx}\left(\frac{x^3}{3}\right) + \frac{d}{dx}(1) = x^2. $$ Now compare the expression for \(D(x)\) with the actual derivative \(f'(x)\). Note that \(D(x)\) is an approximation of \(f'(x)\) that uses a finite difference with \(\Delta x = 0.01\). As \(\Delta x\) approaches 0, \(D(x)\) will approach \(f'(x) = x^2\).

Explain the relationship between the functions f and D

Based on our findings, the function \(D(x)\) approximates the actual derivative of the function \(f(x)\) by using a finite difference with \(\Delta x = 0.01\). As the size of \(\Delta x\) gets smaller, the approximation becomes more accurate. In this case, \(D(x)\) is an approximation of the derivative of the given function \(f(x) = \frac{x^3}{3} + 1\), which is \(f'(x) = x^2\).

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function changes at any given point. It is used to calculate derivatives, which describe how a function behaves and changes its value. In simpler terms, the derivative of a function at a particular input value characterizes how rapidly the function's output value is changing at that point.

The derivative is represented mathematically by the notation \( f'(x) \) for a function \( f(x) \). In our exercise, we differentiated \( f(x) = \frac{x^3}{3} + 1 \). Applying basic differentiation rules, we found that its derivative, \( f'(x) \), is \( x^2 \).

This computed derivative represents the slope of the tangent line to the curve of the function at any point \( x \). In essence, it quantifies the instantaneous rate of change. Differentiation can often involve algebraic manipulation and the application of rules such as the power rule, product rule, or chain rule to find the derivative of complex functions.

Approximation

Approximation in mathematics often comes into play when exact solutions are difficult to find. One especially common form of approximation in calculus is estimating a derivative using a method like the finite difference formula. In the given exercise, we approximated the derivative of \( f(x) = \frac{x^3}{3} + 1 \) using the formula \( D(x) = \frac{f(x+0.01) - f(x)}{0.01} \).

Here, the approximation involves calculating the slope between two points on the function that are very close together. This gives us a value for \( D(x) \), which provides an estimate for \( f'(x) \), the true derivative. This concept allows us to get a feel for how a function behaves without the need for complex calculations or when exact derivatives are hard to compute.

As the interval \( \Delta x = 0.01 \) becomes smaller (i.e., approaches 0), \( D(x) \) becomes a more accurate approximation of \( f'(x) \). This is due to the fact that we are bringing the two points we are using to calculate our slope closer together, which in turn provides a more reliable estimate of the tangent slope.

Finite Difference

The finite difference method provides a way to approximate derivatives using differences between function values at specific intervals. This method is widely used in numerical analysis to solve differential equations and estimate derivatives.

In our exercise, the finite difference formula \( D(x) = \frac{f(x+0.01) - f(x)}{0.01} \) was utilized. Here, the interval between the points, given as \( 0.01 \), is termed the finite difference step size. It is this finite step that allows us to calculate an approximate slope of the function \( f(x) \) over this small range.

Choosing an appropriate step size is crucial: a smaller step size generally improves accuracy but may require more calculations. Conversely, a larger step size might simplify computations but sacrifices precision. Finite differences are especially practical for computing derivatives when analytic methods are unfeasible or too labor-intensive. This approximation technique efficiently bridges the gap between theoretical calculus concepts and real-world computational applications.