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? $$f(x)=x^{3}-4 x, x=-2$$

Short answer

The numerical derivative of \(f(x) = x^{3} - 4x\) at \(x = -2\) is 7.999999. Yes, the function is differentiable at \(x = -2\)

Step-by-step solution

Formula for Numerical Derivative

The numerical derivative of a function \(f(x)\) at a point \(x\) is often estimated using the difference quotient for some small number \(h\). The difference quotient formula is \[(f(x + h) - f(x))/h\]

Calculate f(x + h) and f(x)

For \(f(x) = x^{3} - 4x\), calculate \(f(x + h)\) and \(f(x)\) where \(x = -2\) and \(h = 0.001\). You will find that \(f(x + h) = (-2 + 0.001)^{3} - 4(-2 + 0.001) = -7.992006999\), and \(f(x) = (-2)^{3} - 4(-2) = -8\).

Substitute Values into the Formula

After substituting \(f(x + h)\) and \(f(x)\) into the difference quotient formula, you get \[(-7.992006999 - (-8))/0.001 = 0.007999999/0.001 = 7.999999\]

Check Differentiability

The function \(f(x) = x^{3} - 4x\) is a polynomial function. Polynomial functions are differentiable at any given point. Therefore, the function is differentiable at \(x = -2\)