Problem 93 - Thomas Calculus

Question

a. Graph \(y=x^{2 / 3}\left(x^{2}-2\right)\) for \(-3 \leq x \leq 3 .\) Then use calculus to confirm what the screen shows about concavity, rise, and fall. (Depending on your grapher, you may have to enter \(x^{2 / 3}\) as \(\left(x^{2}\right)^{1 / 3}\) to obtain a plot for negative values of \(x .\) b. Does the curve have a cusp at \(x=0,\) or does it just have a corner with different right-hand and left-hand derivatives?

Step-by-Step Solution

Answer

The function has a cusp at \(x=0\), as indicated by the derivative's behavior there.

Step 1: Understanding the Function

The function given is \(y = x^{2/3}(x^2 - 2)\). Before proceeding with calculus, we need to visualize how this function looks by graphing it over the range \(-3 \leq x \leq 3\). This is because the function involves the fractional power \(x^{2/3}\), which can affect the curve’s shape at \(x=0\) and other points.

Step 2: Graphing the Function

Graph the function \(y = x^{2/3}(x^2 - 2)\) using a graphing tool. To graph this, make sure your grapher correctly interprets \(x^{2/3}\) for negative values, possibly by entering it as \((x^2)^{1/3}\). Observe the overall shape, noting specifically any apparent points of inflection, peaks, valleys, and behavior near \(x=0\).

Step 3: Finding the Derivative

To analyze concavity, local maxima, and minima, compute the first derivative of \(y = x^{2/3}(x^2 - 2)\) using the product rule. Let \(u = x^{2/3}\) and \(v = x^2 - 2\), then \(u' = \frac{2}{3}x^{-1/3}\) and \(v' = 2x\). The derivative is:\[ y' = u'v + uv' = \frac{2}{3}x^{-1/3}(x^2 - 2) + x^{2/3}(2x) \]

Step 4: Analyzing Critical Points and Concavity

Set the first derivative \(y'\) to zero to find critical points and use them to determine rises and falls. Check the sign of \(y'\) across intervals defined by the critical points to identify where the function is increasing or decreasing. Then, find the second derivative to analyze concavity. The behavior of these derivatives around \(x=0\) will also indicate if there is inflection or a cusp.

Step 5: Investigating x=0 for Cusp or Corner

Evaluate whether the function has a cusp or corner at \(x=0\) by comparing the one-sided limits of the derivative \(y'\) as \(x\) approaches 0 from the left and right. If the one-sided limits of the derivative diverge or result in discontinuity, it suggests a corner; if they converge but are undefined due to \(x^{2/3}\) or transition abruptly, it might indicate a cusp.

Key Concepts

Concavity Analysis

When we talk about concavity in calculus, we're looking at how the curve of a function bends or arches. A function's graph is concave up when it looks like a "U" shape. It's concave down when it's more of an "n" shape. To figure out where our function is concave up or down, we use the second derivative.

For the function we are working with, after finding the second derivative, you check where it is positive (indicating concave up) or negative (indicating concave down). This information tells us the curvature of our function's graph.

• Concave up: Second derivative > 0
• Concave down: Second derivative < 0

By examining concavity, you'll understand where the graph bends up or down, enhancing the accuracy of your graph reading or sketch.

Critical Points

Critical points are where a function changes direction, either from increasing to decreasing (or vice versa), or possibly flattens out, such as at a peak or a trough. These points occur where the first derivative is zero or undefined.

To find these points for our function, find the derivative and set it equal to zero. Also check where the derivative does not exist. Solving these equations will give the critical points. Once you have these, test small intervals around each critical point to see if the function value moves from increasing to decreasing or the opposite.

• Derivative = 0: Potential extrema (maxima or minima)
• Derivative undefined: Check for special points like cusps or corners

These points help us understand the key turning points in our function's graph.

Derivatives

A derivative tells us the rate of change of a function. In simpler terms, it's like the speedometer of a function, showing how fast or slow the function is moving at any given point.

For our function, you use the product rule to find the derivative since it is a product of two different expressions. The derivative helps us determine many characteristics:

• Where the function is increasing or decreasing
• Identifying critical points
• Investigating the behavior at singularities

Understanding derivatives is crucial because they provide insightful information about the slopes and tangents to the graph at various points, aiding in the complete picture of the graph's behavior.

Function Behavior at Singularities

Singularities in functions, such as cusps, corners, or discontinuities, can be tricky when analyzing a graph. They occur where the mathematical expression defining the function becomes undefined or behaves atypically.

For this function, particularly at the point where the value of the expression inside the fractional power changes its behavior drastically, it is essential to understand whether it forms a cusp or a corner. A cusp is a pointed end that smoothly transitions the graph's direction, whereas a corner shares a similar look but is identified by the derivative behavior.

• Cusp: Smooth transition but often with undefined derivative
• Corner: Different left-hand and right-hand derivatives, causing a sharp change

By knowing the behavior of the function at singularities, you can better anticipate and describe the overall behavior of the function in instances where traditional analysis, through derivatives alone, might be subject to limitations.