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? $$f(x)=|x-3|, x=3$$

Short answer

The function \( f(x)=|x-3| \) is not differentiable at \( x=3 \). The left-hand limit of the derivative at \( x = 3 \) is -1 while the right-hand limit of the derivative is 1.

Step-by-step solution

Understanding the Function

The provided function is \( f(x) = |x-3| \). This function outputs the absolute difference between any number \( x \) and 3. Notice that this function has a kink or sharp turn at \( x=3 \).

Computing Left-Hand Limit of Derivative

The left-hand derivative of the function at \( x = 3 \) can be computed as \(\lim_{h\to0^-} [f(3+h)-f(3)]/h\). Substituting \( h = -0.001 \) (a value slightly less than zero) and \( f(x) = |x - 3| \) into the formula, we get \( [f(3-0.001)-f(3)]/-0.001 = [|2.999 - 3| - |3 - 3|]/-0.001 = -1 \).

Computing Right-Hand Limit of Derivative

The right-hand derivative at \( x = 3 \) is computed as \( \lim_{h\to0^+} [f(3+h)-f(3)]/h \). Substituting \( h = 0.001 \) (a value slightly greater than zero) and \( f(x) = |x - 3| \) into the formula, we obtain \( [f(3+0.001)-f(3)]/0.001 = [|3.001 - 3| - |3 - 3|]/0.001 = 1 \).

Checking Differentiability

A function is differentiable at a point if the left-hand limit and right-hand limit of the derivative at that point are equal. But here, the left-hand derivative at \( x = 3 \) is -1 while the right-hand derivative at \( x = 3 \) is 1. Since they are not equal, \( f(x) = |x-3| \) is not differentiable at \( x = 3 \).

Key concepts

Differentiability

Understanding the concept of differentiability is crucial when studying calculus and analyzing the behavior of functions. In essence, a function is said to be differentiable at a point if it has a derivative at that point. This means that the function is smooth and has no sharp turns or cusps at that specific location. For a function to be differentiable at a point 'x', the left-hand limit and right-hand limit of the derivative at 'x' must exist and they must be equal.

Take the example exercise with the absolute value function, defined as \( f(x) = |x - 3| \). The kink at \( x = 3 \) suggests a point of potential non-differentiability. The function must smoothly transition from one side of the point to the other without jumping or forming a sharp angle. Unfortunately, this is not the case with an absolute value function at a point where the argument of the absolute value transitions from negative to positive (or vice versa). As a result, the function in our exercise is not differentiable at \( x = 3 \), which we can also confirm by the differing left-hand and right-hand derivative limits.

Left-Hand Limit

The concept of a left-hand limit pertains to the behavior of a function as it approaches a given point from the left side (on a standard graph, this means from smaller values of 'x' towards larger ones). When looking at the numerical derivative from the left-hand side, we consider the limit as 'h' approaches zero from the negative side (\( h \to 0^- \)). This will give us the slope of the tangent to the curve just to the left of the point in question.

In our exercise, we calculate the left-hand limit for the derivative of \( f(x) = |x - 3| \) at \( x = 3 \), which involves evaluating the difference quotient with a very small negative 'h' value (such as \( h = -0.001 \)). The difference in function values over this interval divided by 'h' gives us the slope on the left side of \( x = 3 \). This computation reveals a slope of -1, indicating a downward slope just to the left of the point \( x = 3 \).

Right-Hand Limit

In contrast to the left-hand limit, the right-hand limit deals with approaching a certain point from the right side (meaning from larger values of 'x' toward smaller ones on a graph). We analyze the behavior of a function as 'h' approaches zero from the positive side (\( h \to 0^+ \)). This helps us understand the slope of the tangent line just to the right of our point of interest.

Applying this to our textbook exercise, the right-hand limit for the derivative at \( x = 3 \) for the function \( f(x) = |x - 3| \) requires evaluating the difference quotient with a small positive 'h' value, like \( h = 0.001 \). Upon calculation, we find the slope to be 1, which tells us the function is sloping upwards to the right of \( x = 3 \). This discrepancy between the left-hand limit and right-hand limit of the derivative reaffirms that the function is not differentiable at \( x = 3 \), as their equality is a necessary condition for differentiability.