Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

Curvature is approximately 0.212, and the radius of curvature is approximately 4.72.

Step 1: Sketch the Curve

The equation of the curve is given by \( y = \sqrt[3]{x} \). This equation represents a function where each x-value corresponds to one real y-value, and hence the graph is a smooth curve. Since this is a cube root function, it will pass through the origin, and the curve will be symmetric about the y-axis since \( y \) is an odd root of \( x \). The point (1,1) lies on this curve because substituting \( x=1 \) gives \( y=1 \). Sketch the curve considering these properties, showing it increasing from negative infinity through the origin to positive infinity.

Step 2: Find First Derivative

To find the curvature, we first need to find the derivative of \( y \) with respect to \( x \). Start by finding \( y' \) for \( y = x^{1/3} \). Using the power rule: \( y' = \frac{1}{3}x^{-2/3} \). Simplified, \( y' = \frac{1}{3}x^{-rac{2}{3}} = \frac{1}{3}\cdot \frac{1}{x^{2/3}} \). Substitute \( x = 1 \) to find the derivative at the given point, \( y'(1) = \frac{1}{3} \).

Step 3: Find Second Derivative

Next, find the second derivative \( y'' \). Differentiate \( y' = \frac{1}{3} x^{-2/3} \) using the power rule: \( y'' = -\frac{2}{9}x^{-5/3} \). Substitute \( x = 1 \) to find the second derivative at the point \( (1,1) \): \( y''(1) = -\frac{2}{9} \).

Step 4: Calculate Curvature

The formula for curvature \( k \) at any point on a curve is given by \( k = \frac{|y''|}{(1 + (y')^2)^{3/2}} \). Substituting \( y'(1) = \frac{1}{3} \) and \( y''(1) = -\frac{2}{9} \): \( k = \frac{|-\frac{2}{9}|}{(1 + (\frac{1}{3})^2)^{3/2}} = \frac{2/9}{(1 + 1/9)^{3/2}} = \frac{2/9}{(10/9)^{3/2}} \). Simplifying inside the parenthesis, \( (10/9)^{3/2} = (1.11)^{3/2} \approx 1.049 \). Hence \( k \approx \frac{2/9}{1.049} \approx 0.212 \).

Step 5: Calculate Radius of Curvature

The radius of curvature \( R \) is the reciprocal of curvature \( k \), thus \( R = \frac{1}{k} \). With \( k \approx 0.212 \), \( R \approx \frac{1}{0.212} \approx 4.72 \).

Key Concepts

Curve Sketching

Curve sketching is a fundamental skill in understanding the behavior of functions visually in the coordinate plane. For the equation given, which is \( y = \sqrt[3]{x} \), the graph of this function presents a distinct form, known as a cube root curve. This curve is smooth and continuous, extending from negative to positive infinity and passing through the origin \((0,0)\). The function is symmetric with respect to the origin because it is an odd function.To sketch the curve effectively:

• Identify key points: The origin \((0,0)\), and in this case, the point \((1,1)\) since substituting \(x=1\) results in \(y=1\).
• Recognize the curve's behavior: Notice how it gently curves through the origin from the third quadrant \((x, y < 0)\) to the first quadrant \((x, y > 0)\).
• Understand symmetry: Knowing that the function is odd informs us that the curve mirrors itself at the origin, aiding in sketch accuracy.

Do not strive for absolute perfection in freehand sketches, but instead aim for a clear representation of the curve’s general shape and behavior.

First Derivative

The first derivative of a function is crucial in determining the slope or rate of change at any point along the curve. For the function \(y = x^{1/3}\), finding the first derivative involves applying the power rule.To compute the first derivative:

• The derivative of \(y = x^{1/3}\) is \(y' = \frac{1}{3}x^{-2/3}\). This expression gives the slope at any x-value.
• Substitute \(x = 1\) into \(y'\) to find the slope at the point \((1,1)\). Here, \(y'(1) = \frac{1}{3}\).
• This result indicates a positive slope, reflecting the curve’s upward trend at this point.

The first derivative tells us whether the curve is increasing or decreasing, providing insights into the overall shape and behavior of the function around any given point. For \(x^{1/3}\), the positive slope at \(x=1\) is part of the increasing arc from negative to positive infinity.

Second Derivative

The second derivative provides information about the rate at which the slope of the curve changes, also known as concavity. To further understand the curve of \(y = x^{1/3}\), we find the second derivative by differentiating the first derivative.In calculating the second derivative:

• Start with \(y' = \frac{1}{3} x^{-2/3}\) and apply the power rule again to find \(y'' = -\frac{2}{9}x^{-5/3}\).
• Substitute \(x = 1\) into \(y''\) to evaluate the concavity at point \((1,1)\), resulting in \(y''(1) = -\frac{2}{9}\).
• The negative second derivative implies that the function is concave down at \(x=1\), informing us about the curvature direction at this point.

Understanding the concavity through the second derivative is essential in sketching curves more accurately and predicting the behavior at particular segments of the graph.

Radius of Curvature Calculation

The concept of radius of curvature relates to how sharply a curve bends at a given point. Here, we use both the first and second derivatives to compute the curvature and ultimately the radius of curvature.The process of calculating the radius of curvature includes:

• Determine the curvature \(k\) using the formula: \(k = \frac{|y''|}{(1 + (y')^2)^{3/2}}\).
• Plug in the specific derivative values at \(x=1\), \(y'(1) = \frac{1}{3}\), and \(y''(1) = -\frac{2}{9}\) into the formula.
• Computing this results in: \(k = \frac{2/9}{(1 + 1/9)^{3/2}} \approx 0.212\).
• The radius of curvature \(R\) is the reciprocal of \(k\), making \(R \approx \frac{1}{0.212} \approx 4.72\).

Thus, at the point \((1,1)\), the radius of curvature tells us about the size of the "circle" needed to match the curve’s bend at that point, providing insight into how tightly the curve turns at the segment.