Problem 66 - University Calculus: Early Transcendentals

Question

A caution about centered difference quotients (Contimuation of Exercise \(65 .\) ) The quotient $$\frac{f(x+h)-f(x-h)}{2 h}$$ may have a limit as \(h \rightarrow 0\) when \(f\) has no derivative at \(x .\) As a case in point, take \(f(x)=|x|\) and calculate $$\lim _{h \rightarrow 0} \frac{|0+h|-|0-h|}{2 h}.$$ As you will see, the limit exists even though \(f(x)=|x|\) has no derivative at \(x=0 .\) Moral: Before using a centered difference quotient, be sure the derivative exists.

Step-by-Step Solution

Answer

The limit is 0, even though the derivative of \(|x|\) at \(x = 0\) doesn't exist.

Step 1: Substitute into the Formula

First, substitute \( f(x) = |x| \) into the centered difference quotient formula. For this problem, be sure to substitute \( x = 0 \), so it becomes: \[ \lim_{h \rightarrow 0} \frac{|0+h| - |0-h|}{2h}. \]

Step 2: Simplify the Absolute Value

Simplify the expression inside the limit. Given \(|0+h| = |h|\) and \(|0-h| = |-h| = |h|\), the expression becomes: \[ \lim_{h \rightarrow 0} \frac{|h| - |h|}{2h}. \]

Step 3: Simplify the Quotient

Recognize that \(|h| - |h| = 0\), so the expression inside the limit is \(\frac{0}{2h} = 0\). Thus, the limit simplifies to: \[ \lim_{h \rightarrow 0} 0 = 0. \]

Step 4: Conclude the Result

Since the simplified expression in Step 3 is a constant, the limit simply results in \(0\). Hence, \( \lim_{h \rightarrow 0} \frac{|h| - |h|}{2h} = 0. \)

Key Concepts

Centered Difference Quotient

The centered difference quotient is a mathematical expression used as an approximation of the derivative of a function at a specific point. It's defined as:\[ \frac{f(x+h) - f(x-h)}{2h} \]This method finds the average rate of change of the function between the points \(x+h\) and \(x-h\), essentially 'centering' the difference around \(x\). Unlike the forward or backward difference quotients, which consider the function behavior only in one direction, the centered difference quotient uses values from both sides of \(x\), potentially offering a more accurate approximation for derivatives. However, it's crucial to note that having a limit in the centered difference quotient as \(h\) approaches 0 does not always imply the existence of a derivative at that point. This can occur when functions are non-differentiable at certain points, such as the absolute value function at 0, yet the limit of the centered difference quotient exists.

Limits

Limits are foundational to understanding calculus. They describe the behavior of functions as inputs approach a specific point or infinity. Formally, the limit of a function \(f(x)\) as \(x\) approaches \(a\) is the value that \(f(x)\) gets closer to as \(x\) gets closer to \(a\).In symbolic terms, when\[ \lim_{x \to a} f(x) = L \]it means that \(f(x)\) tends towards \(L\) as \(x\) approaches \(a\). Limits are particularly important in the context of derivatives, as they help formalize the concept of 'instantaneous rate of change.' This is crucial in the solution of our exercise where we examined the limit of a centered difference quotient.It's vital to recognize that even though a limit may exist at a point, it doesn't necessarily confirm the existence of a derivative there. This concept is well illustrated in functions that are continuous but not smooth at certain points.

Differentiability

Differentiability is a concept that indicates whether a function has a derivative at each point in its domain. A function is differentiable at a point when its derivative exists at that point.A differentiable function will be smooth at that point, meaning there are no sharp turns or cusps. For a function \(f(x)\) to be differentiable at \(x = a\), it must be continuous at \(a\). However, continuity at a point doesn't guarantee differentiability.For instance, the absolute value function \(f(x) = |x|\) is not differentiable at \(x = 0\), even though it is continuous there. At \(x = 0\), the function has a sharp point, causing the slope of the tangent to be undefined. The differing slopes on either side of \(x = 0\) contribute to this discontinuity in the derivative.

Absolute Value Function

The absolute value function, denoted as \(f(x) = |x|\), is an important basic function in calculus. It gives the distance of a number from zero on the real number line, hence always yielding a non-negative value.Key properties of the absolute value function include:

• Non-negativity: For all \(x\), \(|x| \geq 0\).
• Symmetry: \(|x| = |-x|\).
• Triangle Inequality: \(|x+y| \leq |x| + |y|\).

This function is continuous everywhere, but not differentiable at \(x = 0\). At this point, the absolute value function has a 'corner' or cusp. This makes the derivative undefined at this specific point since the function does not have a single tangent line that can represent its slope here. Understanding these characteristics of absolute value functions aids in addressing calculus problems like the centered difference quotient.