Question

Graphical Reasoning In Exercises \(81-84,\) use a graphing utility to graph the function and find the \(x\) -values at which \(f\) is differentiable. $$ f(x)=|x-5| $$

Short answer

The function \(f(x)=|x-5|\) is differentiable at all real values of \(x\) except at \(x=5\).

Step-by-step solution

Function Graphing

Use a graphing utility, it could be a graphing calculator or an online tool, to plot the function \(f(x) = |x-5|\). The graph of this function is a V-shaped as it represents an absolute value function.

Understand Differentiability

Differentiability refers to the property of a function that allows it to have a derivative at a given point. Visually, a function is differentiable at a point if the graph of the function has a smooth turn at that point. Sharp corners, vertical tangents and discontinuities make a function non-differentiable at those points.

Identifying non-differentiability point

In the graph of \(f(x) = |x-5|\), you will notice a sharp turn at \(x=5\). This indicates that the function is not differentiable at \(x=5\). Hence, apart from \(x=5\), the function is differentiable for all other real values of \(x\).

Key concepts

Graphing Absolute Value Functions

When graphing absolute value functions, like the given function in the exercise, \(f(x) = |x-5|\), it's essential to understand the fundamental shape these functions take on. An absolute value function creates a 'V' or an inverted 'V' on the coordinate plane, depending on the coefficients and constants in the function.

Here's why: Absolute value functions are comprised of two linear pieces that connect at a point known as the vertex. The absolute value essentially mirrors the negative part of the function above the x-axis, ensuring all y-values are positive. Now, this has a major implication for graphing. Let's look at the equation step by step:

• It's set equal to \(f(x)\), meaning it's a function where 'x' inputs will create 'y' outputs.
• The value inside the absolute value bars, in this case, \(x-5\), is what dictates the shift of the vertex. Here, our vertex shifts right from the origin by 5 units, landing at \(x=5\).
• Since there's no coefficient in front of the absolute value bars, the 'V' shape will have a standard slope of 1 and -1 for its sides.

The key takeaway for graphing is knowing that the absolute value function will have a vertex acting as a turning point, and the rest of the function will take a linear path in both directions from that point. A graphing utility can accurately plot this shape, showing the clear 'V' form on the graph.

Non-Differentiability Points

Differentiability is one of the core concepts in calculus that describes the smoothness of a function's graph and, consequently, the existence of a derivative at a given point. However, certain points may render a function non-differentiable. These points typically manifest as sharp turns, cusps, vertical tangents, or discontinuities in the graph.

For the absolute value function in the exercise, \(f(x) = |x-5|\), the sharp vertex at \(x=5\) is a classic example of a non-differentiability point. At this vertex, the function's graph changes direction abruptly, forming a sharp corner. As a result, it is impossible to define a unique tangent line at that point since the slope of the function would change from positive to negative instantly.

To better understand, imagine trying to place a straight edge to represent the slope right at the corner point. You would find that two vastly different slopes could be drawn, which indicates that the derivative (an indication of slope) can't be consistent at that point. Hence, at \(x=5\), there's no single value that can represent the slope of the function, making \(f(x)\) non-differentiable there. Everywhere else, as the graph of \(f(x)\) consists of straight lines (which have constant slopes), the function is differentiable.

Using Graphing Utilities

Graphing utilities, such as calculators or online graphing tools, are powerful aids for visualizing and understanding the behavior of functions. While students might be tempted to try to sketch by hand, these tools offer precision and clarity that can greatly facilitate learning, especially when working with absolute value functions and determining points of non-differentiability.

When using a graphing utility, simply input the function formula. For our example \(f(x) = |x-5|\), most graphing utilities will require you to express the function in two parts, due to the absolute value. This means defining the function as \(f(x) = x-5\) for \(x\geq5\) and \(f(x) = -(x-5)\) for \(x<5\) to graph both halves of the 'V'.

After entering these pieces and setting an appropriate window for the x and y values, the utility provides a visual graph. You'll be able to zoom in on areas of interest, like the corner at \(x=5\), and better identify where the function isn't smooth—a feature that you just cannot emulate as effectively with paper and pencil.

These advanced features can also help when exploring more complex functions, or when you want to confirm your hand-drawn sketches. Always remember to double-check your inputs and understand how the tool interprets absolute value expressions; it's a great way to support your analytical understanding with a visual representation.