Question

Writing to Learn Recall that the numerical derivative \((\) NDER \()\) can give meaningless values at points where a function is not differentiable. In this exercise, we consider the numerical derivatives of the functions 1\(/ x\) and 1\(/ x^{2}\) at \(x=0\) . (a) Explain why neither function is differentiable at \(x=0\) . (b) Find NDER at \(x=0\) for each function. (c) By analyzing the definition of the symmetric difference quotient, explain why NDER returns wrong responses that are so different from each other for these two functions.

Short answer

The functions \(1/x\) and \(1/x^2\) are not differentiable at \(x=0\) because they are not defined at that point. Consequently, the numerical derivative doesn't exist at \(x=0\). Using the symmetric difference quotient yields meaningless results because these functions are undefined and extremely volatile near \(x=0\).

Step-by-step solution

Understanding Differentiability

A function is said to be differentiable at a point if the derivative exists at that point. The functions \(1/x\) and \(1/x^2\) are not defined at \(x=0\), hence they are not differentiable at \(x=0\).

Calculating Numerical Derivative

The numerical derivative of a function f at a point x can be approximated using the symmetric difference quotient \((f(x+h)-f(x-h))/(2h)\). Here, h is a small value, usually taking a value close to zero. Given that the functions are undefined at \(x=0\), the derivatives also don't exist at \(x=0\).

Reasoning The Results

The symmetric difference quotient gives an approximation of the derivative based on finite differences. Since \(1/x\) and \(1/x^2\) are undefined at \(x=0\), the difference quotient will return meaningless values. The extreme nature of the functions near \(x=0\) makes the results even more meaningless, hence their numerical derivatives at \(x=0\) differ largely in terms of magnitude

Key concepts

Differentiability

Differentiability refers to the property of a function that allows it to have a derivative at a point. In simple terms, if you can find the slope of the tangent line at a point on a graph of a function, then the function is differentiable at that point. For a function to be differentiable at a point, the function itself must be defined there, and it should behave 'nicely' or 'smoothly' around that point.

For instance, imagine you're walking along a path that represents the graph of a function. If you can seamlessly continue walking without jumping or changing direction abruptly, the function is likely differentiable at that spot. However, if there's a sudden break in the path or a sharp corner, that's where differentiability is compromised. The functions \(1/x\) and \(1/x^2\) have breaks at \(x=0\), resembling a steep cliff that you can't just walk over. That's why they're not differentiable at \(x=0\).

Symmetric Difference Quotient

The symmetric difference quotient is a mathematical expression used as a method to approximate the derivative of a function at a certain point. Imagine it as using two points equally spaced from the point of interest to estimate the slope of the tangent line at that point rather than just evaluating at the point itself.

To visualize this, think about trying to estimate the steepness of a hill at the exact spot where you're standing. By looking at the slope between two points on either side of you, you get a pretty good idea of how steep the hill is right under your feet. Mathematically, you calculate this using the formula \[\frac{f(x+h)-f(x-h)}{2h}\], where \(h\) is a small distance away from \(x\). This formula essentially measures the average rate of change around \(x\) and is a good approximation when the function behaves 'nicely' in the vicinity of \(x\).

Undefined at a Point

What Does It Mean for a Function to Be Undefined?

For a function to be undefined at a point means that the function does not have an output for that input value. It's kind of like trying to call a friend, but there's no signal; you just can't get through.

Let's say you're dealing with the function \(1/x\). If you plug in \(x=0\), you're trying to divide 1 by 0, which is mathematically impossible (since division by zero is undefined). Similarly, for the function \(1/x^2\), plugging in \(x=0\) also results in division by zero. So, both \(1/x\) and \(1/x^2\) are undefined at the point \(x=0\), and it creates a sort of 'black hole' on the graph where the function just doesn't exist.

Consequences for Differentiability

Since a function must be defined at a point to be differentiable there, functions \(1/x\) and \(1/x^2\) are not differentiable at \(x=0\) due to this gap in their definition.

Approximation of the Derivative

Approximation of the derivative is a practical way to estimate the derivative when an exact calculation is too difficult or impossible. This is crucial in real-world scenarios where we often work with data that might be incomplete or too complex to analyze exactly.

Think of it as using a rough sketch to get the gist of an intricate painting. While you won't capture every detail, you'll still understand the main features. In calculus, one common method of approximation is the symmetric difference quotient. However, as we've seen with \(1/x\) and \(1/x^2\), when the function is undefined at a certain point, like \(x=0\), this method fails. The reason is that the approximation relies on the function's values at points around \(x=0\), but in our case, as you approach zero, the values explode to infinity for \(1/x\), and even more so for \(1/x^2\), leading to erratic and unreliable approximations. This is why the numerical derivatives, or NDER, for these functions give such different, meaningless results at \(x=0\).

It's important to remember that approximations are just that—estimates. They work beautifully when applied correctly but can give nonsensical results if the underlying assumptions are not met, as seen with functions undefined at a certain point.