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Magazine Chapter 2: Problem 38
Zoom in toward the points \((1,0),(0,1),\) and \((-1,0)\) on the graph of the function \(g(x)=\left(x^{2}-1\right)^{2 / 3}\) . What do you notice? Account for what you see in terms of the differentiability of \(g .\)
Short Answer
Expert verified
The function has cusps at \((1,0)\) and \((-1,0)\), indicating non-differentiability at these points.
Step by step solution
01
Analyzing the Function
First, consider the function \( g(x) = (x^2 - 1)^{2/3} \). This function is defined for all real numbers since \( x^2 - 1 \) is a polynomial and can take both positive and negative values.
02
Identifying Critical Points
The points of interest are \((1,0)\), \((0,1)\), and \((-1,0)\). These are the points where the function might have notable characteristics or behavior, such as cusps or corners.
03
Investigate Behavior Around \( (1,0) \)
Near \( x = 1 \), the expression \( (x^2 - 1) \) becomes zero when \( x = 1 \), and the term becomes \( (0)^{2/3} = 0 \). Graphically, as \( x \to 1 \), \( g(x) \to 0 \). Look at the graph around \( x = 1 \) to see if there is a corner or cusp.
04
Investigate Behavior Around \( (0,1) \)
Since \( g(0) = ((0)^2 - 1)^{2/3} = (-1)^{2/3} = 1 \), this point is \( (0,1) \). As \( x \) varies through zero, observe the smoothness around this point or if there are any sharp changes in direction.
05
Investigate Behavior Around \( (-1,0) \)
Similarly to Step 3, when \( x = -1 \), \( g(-1) = ((-1)^2 - 1)^{2/3} = (0)^{2/3} = 0 \). Again, it is necessary to analyze the graph near \( x = -1 \) to check for differentiability or continuity.
06
Conclusion on Differentiability
A cusp or a corner indicates non-differentiability at that point. Since the derivative involves \( (-2/3) \) power, which translates to negative and fractional derivatives around these points \( x = 1 \) and \( x = -1 \), the function is non-differentiable at these points. At \( x = 0 \), the function appears smooth.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Critical Points
Critical points on a graph are where the function's derivative equals zero or is undefined. These points often indicate where a graph might change direction, have a peak or trough, or even a sharp turn known as a cusp or corner. For the function \( g(x) = (x^2 - 1)^{2/3} \), the critical points are at \((1,0)\), \((0,1)\), and \((-1,0)\).
- At \( x = 1 \) and \( x = -1 \), the expression \((x^2 - 1)\) equals zero, making the term \( (0)^{2/3} = 0 \), which are turning points on the graph.
- At \( x = 0 \), the function evaluates to a non-zero value \((-1)^{2/3} = 1\), representing another critical point \((0,1)\).
Polynomial Function
A polynomial function, like the one under discussion, is one that involves a sum of powers of \(x\) multiplied by coefficients. The function \( x^2 - 1 \) is a classic example of a polynomial function. It can take both positive and negative values as \(x\) varies, which becomes significant when further manipulated.
For \( g(x) = (x^2 - 1)^{2/3} \), the polynomial \(x^2 - 1\) is raised to a fractional power. This introduces interesting behavior in the function's graph, as discussed in the critical points section.
For \( g(x) = (x^2 - 1)^{2/3} \), the polynomial \(x^2 - 1\) is raised to a fractional power. This introduces interesting behavior in the function's graph, as discussed in the critical points section.
- It defines its domain since it is continuous and differentiable for many values except where it leads to \((x^2 - 1) = 0\).
- Behavior at these points is nuanced due to the non-integer exponent.
Graph Behavior
The graph behavior of a function provides insights into its continuity, smoothness, and points of inflection. For the function \( g(x) = (x^2 - 1)^{2/3} \), specific noteworthy behaviors are observed:
- At the critical points \( x = 1 \) and \( x = -1 \), the graph possesses cusps or sharp turns, indicating these are likely spots of non-differentiability.
- Near \( x = 0 \), the graph appears smooth and continuous, suggesting differentiability here.
Non-differentiability
Non-differentiability in a function occurs where the function's graph does not possess a well-defined tangent. This often includes points where there are cusps, corners, or vertical tangents. With \( g(x) = (x^2 - 1)^{2/3} \), the points \((1,0)\) and \((-1,0)\) display features of non-differentiability due to the sharpness in these regions of the graph.
Other indications include:
Other indications include:
- The change in slope at a rapid pace creating sharp turns which cause the derivative at these points to be undefined.
- Fractional exponents in the form \((-2/3)\) that affect the smoothness in derivatives around critical points.
