Question

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y=\sqrt[3]{x},(1,1) $$

Short answer

The curvature at the point (1,1) is approximately 0.074, and the radius of curvature is approximately 13.51.

Step-by-step solution

Analyze the Function

The given function is \( y = \sqrt[3]{x} \). This is a cube root function. It has a unique feature where it stretches along both axes, displaying symmetry about the origin.

Sketch the Curve

To sketch \( y = \sqrt[3]{x} \), note that the curve is symmetric about the origin. As \( x \) increases, \( y \) increases, and as \( x \) decreases, \( y \) decreases. The curve passes through the origin and the point (1,1), continuously rising as x moves away from zero in both directions.

Compute the Derivatives

First, calculate the first derivative \( y' \) of the function, which represents the slope:\[y' = \frac{d}{dx} (x^{1/3}) = \frac{1}{3}x^{-2/3} = \frac{1}{3\sqrt[3]{x^2}}\]Then calculate the second derivative \( y'' \) for curvature:\[y'' = \frac{d^2}{dx^2} (x^{1/3}) = -\frac{2}{9}x^{-5/3} = -\frac{2}{9\sqrt[3]{x^5}}\]

Calculate the Curvature at (1,1)

The curvature \( \kappa \) at a given point \((x, y)\) is given by:\[ \kappa = \frac{|y''|}{(1+(y')^2)^{3/2}} \]Substitute \( x = 1 \) into your derivatives:\[ y' = \frac{1}{3}, \quad y'' = -\frac{2}{9} \]Then, substitute these into the formula for curvature:\[ \kappa = \frac{|-\frac{2}{9}|}{(1+\left( \frac{1}{3} \right)^2)^{3/2}} = \frac{2/9}{(1+1/9)^{3/2}} \]\[ = \frac{2/9}{(10/9)^{3/2}} \approx 0.074 \]

Compute the Radius of Curvature

The radius of curvature \( R \) is the reciprocal of the curvature:\[R = \frac{1}{\kappa}\]Given \( \kappa \approx 0.074 \), compute \( R \):\[ R = \frac{1}{0.074} \approx 13.51 \]

Key concepts

Cube Root Function

The cube root function, represented by \( y = \sqrt[3]{x} \), is a fascinating mathematical expression. It falls under the category of radical functions, characterized by the root operations. Unlike square root functions, the cube root function spans all real numbers. This special property makes it unique because it maintains symmetry about the origin.
This means when you reflect one side of the graph across both the x-axis and y-axis, you’ll notice that the function mirrors itself.
For example, if you examine the point \((1, 1)\), you’ll find that \((-1, -1)\) also lies on this curve.
The cube root extends both positive and negative directions along the x-axis and the y-axis, providing an S-shaped curve that smoothly passes through the origin \((0,0)\).

• Key features: Passes through the origin
• Symmetrical about the origin
• Increases as \( x \) increases, and vice versa
• Smooth and continuous curve

Slope Derivative

The slope of the curve at any given point is described by its derivative. For the function \( y = \sqrt[3]{x} \), the first derivative helps us understand how steep the curve is at a particular point.
Mathematically, the first derivative \( y' \) is calculated as: \[ y' = \frac{1}{3x^{2/3}} \] This derivative tells us how the function rises or falls as \( x \) changes.
At the point \((1,1)\), you can substitute \( x = 1 \) into the derivative, resulting in \( y' = \frac{1}{3} \).
This shows that the slope at this point is \( \frac{1}{3} \), meaning the function is rising gently.

• Derivatives indicate slope
• Positive derivative indicates increasing function
• At \( x = 1 \), \( y' = \frac{1}{3} \), indicating a gentle rise

Second Derivative

The second derivative of a function provides insights into the curve’s concavity or how it bends. For the function \( y = \sqrt[3]{x} \), the second derivative \( y'' \) is critical in determining the curvature.
The formula for the second derivative is: \[ y'' = -\frac{2}{9}x^{-5/3} \] At the point \((1,1)\), by substituting \( x = 1 \), we get \( y'' = -\frac{2}{9} \).
The negative sign indicates that the curve is concave down, or bending downwards, at this point.
This curvature tells us about the changing rate of the slope of the curve, a key aspect in analyzing the path of the function.

• Second derivative addresses curve bending or concavity
• Negative result indicates the curve is concave down
• Helps in calculating curvature of the function at a point

Radius of Curvature

Curvature measures how much a curve deviates from being a straight line. For this, we use the concept of the radius of curvature, which gives an idea of the "tightness" of the curve.
A smaller radius indicates a sharper curve. Mathematically, the radius of curvature \( R \) at a point is the reciprocal of the curvature \( \kappa \).
For the function \( y = \sqrt[3]{x} \) at the point \((1,1)\), the computed curvature \( \kappa \) is approximately 0.074.
Thus, the radius of curvature is:\[ R = \frac{1}{\kappa} \approx 13.51 \] This means at \((1, 1)\), the curve is quite broad or not very tight, associated with a gentle swing or curvature.

• Indicates curve tightness
• Reciprocal of curvature
• Provides a sense of the curve's flow