Problem 15 - The Calculus with Analytic Geometry

Question

Given \(f(x)=|x|\), draw a sketch of the graph of \(f\). Prove that \(f\) is continuous at 0 . Prove that \(f\) is not differentiable at 0 , but that \(f^{\prime}(x)=|x| / x\) for all \(x \neq 0\). (HINT: Let \(|x|=\sqrt{x^{2}}\).)

Step-by-Step Solution

Answer

Graph f(x) as two rays. f is continuous at 0. f is not differentiable at 0, but \(\frac{x}{|x|}\) for x ≠ 0.

Step 1: Sketch the Graph of f(x)

To graph the function f(x)=|x|, we need to plot points for different values of x. The function f(x) equals x when x is greater than or equal to 0 and -x when x is negative. So the graph consists of two rays: one starting at (0,0) and extending upwards to the right (y = x), and another starting at (0,0) and extending upwards to the left (y = -x).

Step 2: Continuity at x = 0

To prove f is continuous at 0, we use the definition of continuity. We have to show \(\text{lim}_{x \to 0} f(x) = f(0)\). Since f(0) = |0| = 0, we need to prove \(\text{lim}_{x \to 0} f(x) = 0\). As x approaches 0, whether from the positive side or the negative side, the value of f(x) = |x| approaches 0. Thus, f(x) is continuous at 0.

Step 3: Differentiability at x = 0

To determine if f(x) is differentiable at 0, we need to check \(\text{lim}_{h \to 0} \frac{f(h) - f(0)}{h}\). Since f(0)=0, the expression simplifies to \(\text{lim}_{h \to 0} \frac{|h|}{h}\). As h approaches 0 from the positive side, \(\frac{|h|}{h}=1\) and when h approaches 0 from the negative side, \(\frac{|h|}{h}=-1\). Since the left-hand limit and right-hand limit are not equal, \(\text{lim}_{h \to 0} \frac{|h|}{h}\) does not exist. Therefore, f is not differentiable at 0.

Step 4: Derivative for x ≠ 0

For all x ≠ 0, use the hint |x| = \(\text{lim}_{h \to 0} \frac{|x + h| - |x|}{h}\) which simplifies to \(\frac{|x|}{x} = \begin{cases} 1 & \text{if } x > 0 \ -1 & \text{if } x < 0 \end{cases}\). Thus, f^{\text{'}(x)} = \frac{x}{|x|} for all x ≠ 0.

Key Concepts

absolute value function

An absolute value function is a function that returns the absolute value of a number. The absolute value of any real number is its distance from zero on the number line, regardless of direction.
Mathematically, we denote the absolute value of a number \(x\) as \(|x|\). For any number \(x\):

• If \(x \geq 0\), then \(|x| = x\).
• If \(x < 0\), then \(|x| = -x\).

This results in a 'V'-shaped graph that touches the origin (0,0). To visualize, it consists of two linear parts:

• When \(x \geq 0\): the graph is \(y = x\), which is a ray starting from (0,0) and extending upwards to the right.
• When \(x < 0\): the graph is \(y = -x\), which is another ray starting from (0,0) and extending upwards to the left.

The combination forms the well-known 'V' shape, characteristic of the absolute value function.

continuity in calculus

Continuity at a point means that there is no sudden jump or break in the graph of the function at that point. A function \(f(x)\) is said to be continuous at \(x = c\) if three conditions are met:

• \(f(c)\) is defined.
• \(\text{lim}_{x \to c} f(x)\) exists.
• \(\text{lim}_{x \to c} f(x) = f(c)\).

For the absolute value function \(f(x) = |x|\), we can check these conditions at \(x = 0\):

• \(f(0) = |0| = 0\), so \(f(0)\) is defined.
• As \(x\) approaches 0 from either the positive or negative direction, \(f(x)\) approaches \(|x|\), which is 0. Thus, \(\text{lim}_{x \to 0} f(x) = 0\) exists.
• Since \(\text{lim}_{x \to 0} f(x) = 0\) and \(f(0) = 0\), all conditions for continuity are met. Thus, \(f(x)\) is continuous at \(x = 0\).

differentiability in calculus

A function is differentiable at a point if it has a defined derivative at that point. This means the function's graph has a well-defined tangent line at that point. To check differentiability at a point \(c\), the following limit must exist:
\[ \text{lim}_{h \to 0} \frac{f(c + h) - f(c)}{h}. \] For \(f(x) = |x|\) at \(x = 0\), this becomes:
\[ \text{lim}_{h \to 0} \frac{|h|}{h}. \] Evaluating this, we find:

• When \(h \rightarrow 0^+\) (approaches from right), \(\frac{|h|}{h} = 1\).
• When \(h \rightarrow 0^-\) (approaches from left), \(\frac{|h|}{h} = -1\).

Since the limits from the right and left are not equal, the derivative does not exist at \(x = 0\). Thus, \(f(x)\) is not differentiable at 0.
For \(x eq 0\), the function \(f(x)\) is differentiable. The derivative can be found using the definition of the absolute value function:

• For \(x > 0\), \(f'(x) = 1\).
• For \(x < 0\), \(f'(x) = -1\).

So, the derivative \(f'(x) = \frac{x}{|x|}\) for all \(x eq 0\).

graphing piecewise functions

Graphing piecewise functions involves plotting different parts of the function over specified intervals. The absolute value function \(f(x) = |x|\) is a classic example. To graph it:

• Determine the pieces: For \(x \geq 0\), the function is \(f(x) = x\). For \(x < 0\), the function is \(f(x) = -x\).
• Plot points for each piece to visualize: Start by plotting the origin (0,0).
• For \(x \geq 0\), plot points such as (1,1), (2,2), etc., and draw the ray extending upwards to the right.
• For \(x < 0\), plot points such as (-1,1), (-2,2), etc., and draw the ray extending upwards to the left.

Combining these pieces should show the characteristic 'V' shape.
By understanding the behavior of this piecewise function through its graph and specific values, students can grasp how the function behaves in different intervals.