Problem 30 - Essential Calculus Early Transcendentals

Question

Use a graphing calculator or computer to graph both the curve and its curvature function \(\kappa(x)\) on the same screen. Is the graph of \(\kappa\) what you would expect? $$y=x^{4}-2 x^{2}$$

Step-by-Step Solution

Answer

Yes, the graph of \( \kappa(x) \) reflects changes in the slope of the curve.

Step 1: Input the Function into the Graphing Tool

Open your graphing calculator or a computer graphing program. Input the function \( y = x^4 - 2x^2 \) to graph it. Ensure that the viewing window encompasses essential features, typically using a range like \( -3 \leq x \leq 3 \) on both axes.

Step 2: Find the First and Second Derivatives

Calculate the first derivative of the function \( y = x^4 - 2x^2 \). This gives \( y' = 4x^3 - 4x \). Then, calculate the second derivative: \( y'' = 12x^2 - 4 \). These derivatives are necessary for computing the curvature function \( \kappa(x) \).

Step 3: Calculate the Curvature Function

The curvature function \( \kappa(x) \) is determined by the formula \( \kappa(x) = \frac{|y''|}{(1 + (y')^2)^{3/2}} \). Substituting the derivatives, you get \( \kappa(x) = \frac{|12x^2 - 4|}{(1 + (4x^3 - 4x)^2)^{3/2}} \).

Step 4: Graph the Curvature Function

Input the curvature function \( \kappa(x) = \frac{|12x^2 - 4|}{(1 + (4x^3 - 4x)^2)^{3/2}} \) into the graphing tool. Use the same viewing window as before to observe both graphs.

Step 5: Analyze the Curvature Graph

Observe how the curvature function \( \kappa(x) \) relates to the graph of \( y = x^4 - 2x^2 \). Expected characteristics include sharper peaks and bends at points where the function's slope changes rapidly, corresponding to the behavior of \( \kappa(x) \).

Key Concepts

First Derivative

The first derivative of a function gives us important information about the rate of change of the function at any point. For the function \( y = x^4 - 2x^2 \), the first derivative is calculated as \( y' = 4x^3 - 4x \).

Calculating the first derivative involves applying the power rule of differentiation. By doing this, we can determine how the function's slope changes along the curve. These slopes help us understand whether the original function is increasing or decreasing in specific intervals.

• For \( x < -1 \) and \( x > 1 \), \( y' > 0 \) (the function is increasing).
• For \( -1 < x < 0 \) and \( 0 < x < 1 \), \( y' < 0 \) (the function is decreasing).

Examining the slope also helps identify critical points, which occur where \( y' = 0 \) or where \( y' \) does not exist. These points in this function are \( x = -1, 0, \text{and} 1 \), and they play a crucial role in graphing and analyzing the curvature.

Second Derivative

The second derivative of a function provides insight into the curvature or concavity of the graph. For \( y = x^4 - 2x^2 \), the second derivative is \( y'' = 12x^2 - 4 \).

The second derivative helps us understand how the slope itself is changing along the curve. When \( y'' > 0 \), the function is concave up, and when \( y'' < 0 \), it's concave down.

• The function is concave up when \( x < -\sqrt{\frac{1}{3}} \) and \( x > \sqrt{\frac{1}{3}} \).
• It is concave down when \( -\sqrt{\frac{1}{3}} < x < \sqrt{\frac{1}{3}} \).

The points where the concavity changes are called inflection points. For this function, the inflection points occur at \( x = -\sqrt{\frac{1}{3}} \) and \( x = \sqrt{\frac{1}{3}} \). Understanding these changes in curvature is vital for accurately sketching the graph.

Graphing Calculator

A graphing calculator is a powerful tool that helps visualize the function and its derivatives. To graph both the function \( y = x^4 - 2x^2 \) and its curvature function \( \kappa(x) \), you can follow these straightforward steps.

• Enter the primary function and ensure the initial window captures essential points.
• Graph the first derivative to analyze where the function is increasing or decreasing.
• Similarly, graph the second derivative to observe concavity changes and identify inflection points.
• Input the curvature function \( \kappa(x) = \frac{|12x^2 - 4|}{(1 + (4x^3 - 4x)^2)^{3/2}} \) using the same range.

By using the graphing calculator, you can immediately see the relationship between the curvature function \( \kappa(x) \) and the primary function. Peaks on the graph of \( \kappa(x) \) suggest rapid changes in slope on the original function, providing a visual insight into how the curve bends at different points. This visual representation is a great way to deepen understanding of calculus concepts such as derivatives and curvature.