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. $$ f(x)=\frac{4 x}{x-3} $$

Short answer

The function \(f(x)=\frac{4 x}{x-3}\) is differentiable for all \(x\) except for \(x=3\).

Step-by-step solution

Plot the Graph

First, we need to plot the function \(f(x)=\frac{4 x}{x-3}\) using a graphing utility. It's important to note that our function will be undefined when \(x=3\) as it would result in division by zero.

Identify the points of discontinuity

Examine the graph and identify the points where the graph has breaks or holes. These points are where the function is not differentiable. In our case, this occurs at \(x=3\).

Identify the points of differentiability

The points of differentiability are all other x-values except for \(x=3\).

Key concepts

Understanding Graphing Utilities

Graphing utilities are powerful tools used to visualize mathematical functions, making them an essential resource in the study of calculus. They enable students and mathematicians to quickly and accurately plot the behavior of a function across different intervals and identify crucial features like intercepts, asymptotes, and inflection points.

By entering the function such as \( f(x)=\frac{4x}{x-3} \) into a graphing utility, one can see the curve that represents all the possible points. This visual representation aids in comprehending how the function behaves, which is not always intuitive just from the equation itself. For instance, it can reveal symmetry or periodicity and how the function grows or decays as the variable \(x\) increases or decreases in value. Graphing utilities are especially useful when considering complex functions that are difficult to graph by hand.

In educational contexts, graphing utilities help students to bridge the gap between abstract concepts and tangible understanding. They serve as a practical means to experiment with the effects of changing parameters and to predict the nature of a function before delving into analytical methods. Thus, they are invaluable in the toolkit of anyone studying functions in a mathematical setting.

Interpreting Points of Discontinuity

Points of discontinuity are where a function is not well-behaved, meaning it lacks a defined value or does not follow a predictable pattern. These problematic points disrupt the graph and are often where the function cannot be differentiable. Identifying these points is crucial when analyzing a function's differentiability.

Discontinuities come in various forms, such as holes (removable discontinuities), vertical asymptotes (infinite discontinuities), and jump discontinuities. The graph of a function will illustrate these by showing breaks or sudden changes in direction. For instance, with the function \( f(x)=\frac{4x}{x-3} \), we observe a vertical asymptote at \(x=3\), which indicates that the function 'jumps' to infinity and the value of the function is not defined at this point.

Understanding discontinuities is important because they highlight the limitations and bounds within which a function operates. Differentiability requires a function to be continuous and smooth, so points of discontinuity automatically disqualify the function from being differentiable at those specific values. By graphing the function, students can visually identify and understand these critical points more easily, furthering their comprehension of calculus concepts.

Division by Zero and Its Implications

Division by zero is a significant concept in mathematics as it points to undefined or unbounded behavior within a function. In calculus, division by zero within a function like \( f(x)=\frac{4x}{x-3} \) explains why certain values are excluded from the domain and why specific points of discontinuity exist.

When the denominator in a function equals zero, the function does not have a real-number output at that point; this is why graphing utilities often display an asymptote or a break in the graph. Taking the current function as an example, we cannot define \( f(x) \) when \(x=3\) because it would require dividing 4 by zero which is mathematically impermissible. This leads to an infinite discontinuity at this specific value.

In this context, division by zero doesn't just signal a problem in evaluation, but also informs us about the boundaries of a function's behavior and where it becomes non-differentiable. Recognizing and understanding the concept of division by zero is a foundational element for students as it touches upon the limits and potential infinity inherent in mathematical functions, a key stepping stone in mastering calculus.