Problem 147 - Calculus 3

Question

Find the radius of curvature of \(y=\ln (x+1)\) at point \((2, \ln 3)\)

Step-by-Step Solution

Answer

The radius of curvature is \(\frac{10\sqrt{10}}{3}.\)

Step 1: Understand the problem

We need to find the radius of curvature for the function \(y = \ln(x+1)\) at the point \((2, \ln 3)\). The formula for the radius of curvature \(R\) at a point is given by \[ R = \frac{(1 + (y')^2)^{3/2}}{|y''|}. \] First, we need to determine the first and second derivatives of the function.

Step 2: Find the first derivative

Calculate the derivative of \(y = \ln(x+1)\) with respect to \(x\). Using the chain rule, we have \(y' = \frac{d}{dx}\ln(x+1) = \frac{1}{x+1}\).

Step 3: Find the second derivative

Now, we find the second derivative by differentiating \(y' = \frac{1}{x+1}\) with respect to \(x\). Using the quotient rule, we find \(y'' = \frac{d}{dx} \left( \frac{1}{x+1} \right) = -\frac{1}{(x+1)^2}\).

Step 4: Evaluate the derivatives at the given point

We need to evaluate \(y'\) and \(y''\) at \(x = 2\). \[ y'(2) = \frac{1}{2+1} = \frac{1}{3}\] \[ y''(2) = -\frac{1}{(2+1)^2} = -\frac{1}{9} \]

Step 5: Calculate the radius of curvature

Substitute \(y'(2)\) and \(y''(2)\) into the radius of curvature formula: \[ R = \frac{(1 + (y'(2))^2)^{3/2}}{|y''(2)|} = \frac{(1 + \left(\frac{1}{3}\right)^2)^{3/2}}{\frac{1}{9}}. \] Simplify this expression: First, calculate \(\left(1 + \frac{1}{9}\right) = \frac{10}{9}\). Then, raise it to the power of \(\frac{3}{2}\): \(\left(\frac{10}{9}\right)^{3/2}\) approximately equals \(\frac{\sqrt{10}^3}{27} = \frac{10\sqrt{10}}{27}\). Finally, multiply by 9: \(R \approx \frac{90\sqrt{10}}{27}.\)

Step 6: Simplify the expression

Simplifying \(R = \frac{90\sqrt{10}}{27}\), divide both numerator and denominator by 9: \[ R = \frac{10\sqrt{10}}{3} \] Thus, the radius of curvature at the point \((2, \ln 3)\) is \(\frac{10\sqrt{10}}{3}.\)

Key Concepts

Derivatives

Derivatives are fundamental in calculus. To find the derivative of a function, we calculate the rate at which the function changes at any given point. In the context of curvature, derivatives help determine how a curve bends at different points.

For the function \( y = \ln(x+1) \), we use the differentiation rules to find the first derivative. By applying the chain rule, we find that the first derivative is \( y' = \frac{1}{x+1} \). This derivative helps us understand the slope of the tangent line at any point on the curve.

To determine how fast this slope is changing, we need the second derivative. Calculating the second derivative, \( y''\), involves using the quotient rule, which results in \( y'' = -\frac{1}{(x+1)^2} \). This second derivative indicates the level of curvature or bending at a particular point. In short:

• The first derivative \( y' \) tells us about the slope.
• The second derivative \( y'' \) tells us about the curvature.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions, commonly expressed in the form \( y = \ln(x) \). These functions are crucial in various scientific fields for modeling growth and decay processes.

In the specific problem of finding the radius of curvature, the function \( y = \ln(x+1) \) represents a natural logarithm with a horizontal shift. Understanding this function is key to applying the derivative formula correctly as it involves recognizing the transformations that occur due to shifting.

Some important characteristics of logarithmic functions include:

• They only work for positive values of \( x \).
• They increase, but at a decreasing rate, as \( x \) becomes larger.
• They have a vertical asymptote at \( x = -1 \) for the function \( y = \ln(x+1) \).

These properties help explain the behavior of the curve and its curvature, particularly how it flattens out as \( x \) increases.

Curvature Formula

The curvature of a curve at a particular point tells us how sharply the curve bends around that point. It's important to grasp this concept when dealing with geometric properties within calculus.

The radius of curvature \( R \) gives a measure of this bending, derived from both the first and the second derivatives, as shown in the formula:\[ R = \frac{(1 + (y')^2)^{3/2}}{|y''|} \]Here, the expression \( (1 + (y')^2)^{3/2} \) represents the stretch due to the slope, and \( |y''| \) accounts for the curvature influence.

In practical terms, a smaller radius means tighter bending, and a larger radius indicates a flatter curve. Calculating \( R \) involves evaluating:

• The first derivative \( y' \) to understand the slope.
• The second derivative \( y'' \) to understand the concavity.
• Substituting these into the formula to get \( R \).

For the function \( y = \ln(x+1) \), at the point \((2, \ln 3)\), the radius of curvature is calculated as \( \frac{10\sqrt{10}}{3} \), illustrating the curve's flattest point at this segment.