Problem 92 - 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 curve has a cusp at \(x=0\) due to the nonexistence of \(y'\) at that point.

Step 1: Understanding the Function

The function we need to analyze is \(y = x^{2/3}(x^2 - 2)\). The graphing task will provide a general idea of how the function behaves within the specified domain \([-3, 3]\).

Step 2: Set Up the Graph

When graphing \(y = x^{2/3}(x^2 - 2)\), remember that \(x^{2/3}\) can be written as \((x^2)^{1/3}\) to include both positive and negative x-values. This setup allows for visualization directly in graphing software.

Step 3: Observe the Graph

By plotting the function, observe the key features such as where it increases or decreases (indicating rise or fall), and any observable points of inflection to assess concavity.

Step 4: Calculate the Derivative

Compute the derivative of \(y = x^{2/3}(x^2 - 2)\) using the product rule: \( y' = \dfrac{d}{dx}[x^{2/3}](x^2 - 2) + x^{2/3}(\dfrac{d}{dx}[x^2 - 2]) \). This results in:\[ y' = \dfrac{2}{3}x^{-1/3}(x^2 - 2) + x^{2/3}(2x) = \dfrac{2(x^2 - 2)}{3x^{1/3}} + 2x^{5/3} \]

Step 5: Determine the Critical Points

Find where \(y' = 0\) or is undefined to determine where the function changes from increasing to decreasing and vice versa. These points help us understand the function's rise and fall.

Step 6: Analyze Concavity

To analyze concavity, find the second derivative \(y''\). This involves differentiating \(y'\) again. After finding \(y''\), determine where it is positive (concave up) or negative (concave down).

Step 7: Examine Behavior at x=0

Check the behavior of \(y'\) at \(x=0\). Since \(y'\) for \(x=0\) involves division by \(x^{1/3}\), the derivative does not exist at this point, indicating a potential cusp or sharp turn rather than a smooth corner with continuous derivatives.

Key Concepts

Concavity

Concavity is a property of a curve that tells us about its bending direction. A portion of a graph is said to be "concave up" if it resembles a cup facing upwards. This means that the second derivative, denoted as \(y''\), is positive in this interval.On the other hand, the graph is "concave down" when it looks like a frown. Here, the second derivative is negative. Understanding concavity helps us predict how a function behaves over different intervals. For example, if a function is concave up, we anticipate a local minimum; if concave down, we may expect a local maximum.To evaluate concavity, we compute the second derivative \(y''\). By checking values where \(y''\) changes sign, known as points of inflection, we can determine the concavity's shifts. This assessment is crucial to get a visual understanding of a function's graph.

Derivative

The derivative of a function at a point gives us the slope of the tangent line at that point on the curve. This is a measure of how the function is changing. To grasp its significance, think of the derivative as the rate of change or the function's velocity.Calculating the derivative involves applying rules such as the product rule and the chain rule. For our specific function, we use the product rule to find: \[y' = \frac{2}{3}x^{-1/3}(x^2 - 2) + x^{2/3}(2x) = \frac{2(x^2 - 2)}{3x^{1/3}} + 2x^{5/3}\]The derivative exceeds being just a mathematical operation. It helps us determine where the function is increasing or decreasing by finding where \(y' = 0\) (zero slope) or where it does not exist. Such points can indicate peaks, troughs, or important transitions in the graph's shape.

Critical Points

Critical points are values of \(x\) where the derivative \(y'\) is zero or undefined. These points are crucial because they can signal where the function switches from increasing to decreasing, or vice versa.Finding these points involves solving \(y' = 0\) or checking for undefined points in the derivative. Solving these equations helps identify potential maximums, minimums, and points of inflection. Once identified, these points allow us to better understand the behavior and shape of the function's graph.After finding critical points, plotting them can provide a clear picture of where changes in the function's rise and fall occur. This aids in constructing an accurate graph and understanding the overall behavior of the function.

Cusp vs Corner

In graphing, a cusp and a corner are types of non-smooth points, but they differ in terms of the function's derivative behavior.- **Cusp:** A cusp occurs when the slope approaches infinity in opposite directions on either side of the point, leading to a sharp point. For function \(y = x^{2/3}(x^2 - 2)\), the derivative does not exist at \(x = 0\) due to division by \(x^{1/3}\), indicating a cusp.- **Corner:** In contrast, a corner arises when different one-sided derivatives at a point equate to finite but distinct values. The derivatives from left and right differ, suggesting a change in direction without smoothing out.Understanding whether a function has a cusp or corner helps in graphing and predicting the behavior around the point. This distinction allows for a clearer interpretation of what type of irregular shape or change is present in the function's graph.