Problem 28 - Calculus for Biology and Medicine

Question

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=|x+2|-1 $$

Step-by-Step Solution

Answer

The function is not differentiable at \( x = -2 \).

Step 1: Define the Function

The given function is \( y = |x+2| - 1 \). This function is a transformation of the absolute value function \( |x| \). It shifts the graph of \( |x| \) to the left by 2 units and down by 1 unit.

Step 2: Understand Absolute Value Function

The absolute value function \( |x| \) is V-shaped, and it is defined as \( |x| = x \) if \( x \geq 0 \) and \( |x| = -x \) if \( x < 0 \). This feature creates a "corner" at \( x = 0 \), where the function is not differentiable.

Step 3: Transform the Critical Point

The function \( y = |x+2| - 1 \) has a critical point at \( x + 2 = 0 \), or \( x = -2 \). This is where the "corner" of the graph occurs, as this transformation introduces a vertex of the absolute value graph at \( x = -2 \).

Step 4: Graph the Function

To sketch the graph, shift the typical absolute value graph left by 2 units and down by 1 unit. The vertex of the graph will be at \( (-2, -1) \), with lines extending upward in both directions from this point. The function has a V-shape centered at this vertex.

Step 5: Identify Non-differentiable Points

Since the function has a sharp corner at \( x = -2 \), this is the point where the function is not differentiable. The sharp transition at this point prevents the existence of a unique tangent, indicating non-differentiability.

Key Concepts

Absolute Value Function

The absolute value function is one of the most fundamental pieces of mathematics you'll encounter and is expressed as \(|x|\). This function gives the distance of any number \(x\) from zero on a number line. It is always non-negative, meaning it will never give you a negative value. The formula for \(|x|\) is:

• \(|x| = x\) if \(x \geq 0\)
• \(|x| = -x\) if \(x < 0\)

This results in a graph that looks like the letter "V". It is symmetric about the y-axis and has a vertex, or a 'corner', at the origin (0,0). The defining characteristic of the absolute value function is this corner, where the function sharply changes direction. In the given exercise, the function \(y = |x+2| - 1\) is an absolute value function transformed by shifting it left 2 units and down 1 unit, creating the corner at the point \((-2, -1)\).
Understanding these shifts helps us to interpret how changes to the equation affect the graph and its properties.

Non-differentiable Points

Differentiability is a key concept in calculus referring to the ability to find a tangent line at any given point on a graph. However, when a graph has sharp corners, like those in an absolute value function, it becomes problematic. These points are known as non-differentiable points.
In mathematics, a function is differentiable at a point if it has a defined slope, or gradient, or a unique tangent line at that point. With the absolute value function, non-differentiability occurs at the corner because the direction of the function changes abruptly; thus, a single tangent cannot be drawn. Therefore, at \(x = -2\) in our exercise, the function is non-differentiable because the graph creates a sharp corner.
This lack of a smooth, consistent slope results in the non-existence of a derivative at that point. Recognizing these points is crucial for graph analysis and understanding the limitations of calculus.

Graphing Functions

Graphing functions involves plotting the equations to visualize their behavior. It is not just about connecting the dots; it involves understanding shifts and transformations.
Start with the parent function. For \(y = |x+2|-1\), begin with \(y = |x|\), which is the basic V-shaped graph with a vertex at the origin (0,0).

• To shift it left, adjust the inside of the absolute value to \(x+2\). This moves the graph 2 units to the left, setting the new vertex at \((-2,0)\).
• Next, subtract 1 from the function: \(y = |x+2| - 1\). This moves the graph 1 unit downwards, placing the final vertex at \((-2, -1)\).

The function \(y = |x+2| - 1\) is then graphed with the vertex at \((-2, -1)\), and lines extending symmetrically upwards. Accurately sketching these functions involves an understanding of each transformation's effect on their position and shape. This process is crucial for predicting the graph's appearance and for finding important points such as those where the function is non-differentiable.