Problem 54 - Calculus Early Transcendentals

Question

The left-hand and right-hand derivatives of \(f\) at a are defined by $$f_{-}^{\prime}(a)=\lim _{h \rightarrow 0^{-}} \frac{f(a+h)-f(a)}{h}$$ and $$\quad f_{+}^{\prime}(a)=\lim _{n \rightarrow 0^{+}} \frac{f(a+h)-f(a)}{h}$$ if these limits exist. Then \(\mathrm{f}^{\prime}(\mathrm{a})\) exists if and only if these one-sided derivatives exist and are equal. (a) Find \(\mathrm{f}_{-}^{\prime}(4)\) and \(\mathrm{f}_{+}^{\prime}(4)\) for the function $$f(x)=\left\\{\begin{array}{ll}{0} & {\text { if } x \leqslant 0} \\ {5-x} & {\text { if } 0 < x < 4} \\ {\frac{1}{5-x}} & {\text { if } x \geq 4}\end{array}\right.$$ (b) Sketch the graph of f. (c) Where is f discontinuous? (d) Where is f not differentiable?

Step-by-Step Solution

Answer

f'(4) does not exist.

Step 1: Compute one-sided derivatives

Compute \( f'_-(4) = \lim_{h \to 0^-} \frac{f(4+h)-f(4)}{h} \) and \( f'_+(4) = \lim_{h \to 0^+} \frac{f(4+h)-f(4)}{h} \).

Step 2: Compare

If \( f'_-(4) \neq f'_+(4) \), then \( f'(4) \) does not exist.

Key Concepts

Left-Hand Derivative

The left-hand derivative focuses on the limit of the derivative from the left side at a particular point. In mathematical terms, it's expressed as \(f_{-}^{\prime}(a)=\lim _{h \rightarrow 0^{-}} \frac{f(a+h)-f(a)}{h}\). This formula calculates how the function behaves as we approach the point from values less than \(a\).

For the given function, to find \(f_{-}^{\prime}(4)\), the approach involves substituting 4 into the part of the piecewise function applicable just before x equals 4. Therefore, here, for values slightly less than 4, it's \(5-x\). Calculating gives \(-1\), showing a negative slope approaching from the left. Understanding these calculations can help determine whether the function is differentiable at the point of interest.

Right-Hand Derivative

The right-hand derivative uses the limit approaching the point from the right side. It's denoted as \(f_{+}^{\prime}(a)=\lim _{h \rightarrow 0^{+}} \frac{f(a+h)-f(a)}{h}\). This derivative helps examine how the function changes for values slightly greater than \(a\).

• For the function given, \(f_{+}^{\prime}(4)\) is obtained by looking at \(\frac{1}{5-x}\) since this is how the function is defined for values at and beyond 4.
• Calculating the limit results in \(-\frac{1}{1}\), which simplifies to \(-1\).

Both left-hand and right-hand derivatives matching indicates differentiability, confirming behavior consistency on both sides of the point.

Piecewise Function

Piecewise functions are made of different expressions over distinct intervals. Our function, for instance, is defined across three intervals:

• \(f(x) = 0\) whenever \(x \leq 0\).
• \(f(x) = 5 - x\) for the interval \(0 < x < 4\).
• \(f(x) = \frac{1}{5-x}\) when \(x \geq 4\).

These separate rules for different x intervals make it necessary to evaluate derivatives separately for each section. Each segment is treated as an individual function within its defined scope. Understanding piecewise functions involves identifying these segments and computing derivatives within each specified interval separately.

Discontinuity

Discontinuity in a function occurs when there are interruptions in the expected flow, such as jumps, gaps, or breaks. Identifying points of discontinuity is crucial when analyzing a function graph.

For the given piecewise function, notice the change between segments:

• The jump from \(5-x\) to \(\frac{1}{5-x}\) at \(x = 4\) creates a discontinuity because the function shifts from a linear function to a rational expression.

Discontinuities play a significant role in determining differentiability since a function must be continuous to be differentiable at a point.

Non-Differentiability

Non-differentiability occurs where the function does not have a derivative. This can happen at points of discontinuity or where the slope rapidly changes.

Our function lacks differentiability at \(x = 4\) despite left-hand and right-hand derivatives being equal. The piecewise construction causes a change in slope's nature, contributing to non-differentiability.

• Discontinuities imply non-differentiability unless the value matches limits from either side.
• Sharp corners or cusps are examples where differentiability breaks down.

Detecting where a function loses differentiability provides insights into its geometry and behavior over intervals.