Question

In Exercises \(17-26\) , find the numerical derivative of the given function at the indicated point. Use \(h=0.001 .\) Is the function differentiable at the indicated point? $$f(x)=x^{2 / 3}, x=0$$

Short answer

The numerical derivative of the function \(f(x)=x^{2 / 3}\) at x = 0 using the \(h=0.001\) is obtainable using the numerical derivative formula. The value obtained will be finite. Therefore, the function is differentiable at the point x = 0.

Step-by-step solution

Understand the Function and the Point

The provided function is \(f(x)=x^{2 / 3}\) and we have to find the derivative at the point x = 0.

Apply the Numerical Derivative Formula

We are using \(h=0.001\) for our numerical derivative. The formula for numerical differentiation is \(f'(x) = \frac{f(x+h) - f(x-h)}{2h}\). Substituting the values into the formula, we have: \(f'(0) = \frac{f(0.001) - f(-0.001)}{2*0.001}\)

Calculate the Function Values

Now, it's needed to calculate the values of the function at \(x=0.001\) and \(x=-0.001\), which are \(f(0.001)=(0.001)^{2/3}\) and \(f(-0.001)=(-0.001)^{2/3}\) respectively.

Calculate the Numerical Derivative

By substituting the calculated function values into the formula for the numerical derivative, a suitable numerical value is obtained. This numeric value constitutes the slope of the function at the indicated point.

Check Differentiability

To check if the function is differentiable at the point x = 0, we have to ensure that the derivative exists and it is finite, which is already defended by the result of the numerical derivative. Therefore, the function is differentiable at x=0.