Question

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y=e^{-x^{2}},(1,1 / e) $$

Short answer

Curvature: \( \kappa = \frac{2}{e(1+\frac{4}{e^2})^{3/2}} \). Radius of curvature: \( R = \frac{e(1+\frac{4}{e^2})^{3/2}}{2} \).

Step-by-step solution

Understanding the Curve

The given function is \( y = e^{-x^2} \). This is a bell-shaped curve symmetric about the y-axis, known as the Gaussian function without constants, peaking at \( y = 1 \) when \( x = 0 \), and tapering off to zero as \( x \) moves towards positive or negative infinity.

Calculating the First Derivative

The first derivative \( \frac{dy}{dx} \) of the function \( y = e^{-x^2} \) is determined using the chain rule. First, differentiate the outer function: \( \frac{d}{dx}e^{u} = e^u \), and the inner function: \( u = -x^2 \), \( \frac{du}{dx} = -2x \). Therefore, \( \frac{dy}{dx} = -2xe^{-x^2} \).

Calculating the Second Derivative

The second derivative \( \frac{d^2y}{dx^2} \) is found by differentiating \( \frac{dy}{dx} = -2xe^{-x^2} \). Use the product rule: \( u = -2x \) and \( v = e^{-x^2} \) where \( u' = -2 \) and \( v' = -2xe^{-x^2} \). Thus, \[ \frac{d^2y}{dx^2} = (-2)e^{-x^2} + (-2x)(-2xe^{-x^2}) = (-2)e^{-x^2} + 4x^2e^{-x^2} = (4x^2 - 2)e^{-x^2}. \]

Evaluating at the Point (1, 1/e)

With \( x = 1 \), evaluate the derivatives: \( \frac{dy}{dx} = -2(1)e^{-1} = -\frac{2}{e} \) and \( \frac{d^2y}{dx^2} = (4(1)^2 - 2)e^{-1} = \frac{2}{e} \).

Calculating Curvature

Curvature \( \kappa \) at a point \((x, y)\) is given by the formula:\[ \kappa = \frac{|y''|}{(1 + (y')^2)^{3/2}}. \] Substitute \( y'' = \frac{2}{e} \) and \( y' = -\frac{2}{e} \):\[ \kappa = \frac{\left| \frac{2}{e} \right|}{\left(1 + \left( -\frac{2}{e} \right)^2 \right)^{3/2}} = \frac{\frac{2}{e}}{\left(1 + \frac{4}{e^2} \right)^{3/2}}. \]

Calculating Radius of Curvature

The radius of curvature \( R \) is the reciprocal of the curvature: \[ R = \frac{1}{\kappa} = \frac{\left(1 + \frac{4}{e^2}\right)^{3/2}}{\frac{2}{e}} \] Simplify to get the numerical value or expression of the radius of curvature.

Key concepts

Gaussian Function

The Gaussian function is a mathematical expression represented by the formula \( y = e^{-x^2} \). This function describes a symmetrical bell-shaped curve centered around the y-axis.
This special function reaches its peak at \( y = 1 \) when \( x = 0 \), and it gradually approaches zero as \( x \) moves toward both positive and negative infinity.
Because of its shape, the Gaussian function is often associated with the concept of a normal distribution in statistics, but in this context, we're focusing on its geometric characteristics.
The symmetry and rapid decrease of the Gaussian function make it an interesting subject for exploring concepts in calculus such as derivatives and curvature.

First Derivative

Calculating the first derivative of a function gives us insight into its rate of change. For \( y = e^{-x^2} \), the first derivative \( \frac{dy}{dx} \) is derived using the chain rule.
The chain rule involves differentiating the outer function \( e^u \), with respect to the inner function \( u = -x^2 \). Thus, \( \frac{du}{dx} = -2x \).
Piping this back into our derivative calculation, we get \( \frac{dy}{dx} = -2xe^{-x^2} \).
This expression tells us how the slope of the function's tangent line changes as we move along the x-axis, giving us crucial information about the curve's behavior.

Second Derivative

The second derivative \( \frac{d^2y}{dx^2} \) of a function is particularly useful in analyzing the curvature of a graph. It is found by differentiating the first derivative.
For \( \frac{dy}{dx} = -2xe^{-x^2} \), we use the product rule to differentiate: consider \( u = -2x \) and \( v = e^{-x^2} \) where \( u' = -2 \) and \( v' = -2xe^{-x^2} \).
Applying the product rule gives us: \[ \frac{d^2y}{dx^2} = (-2)e^{-x^2} + 4x^2e^{-x^2} = (4x^2 - 2)e^{-x^2} \].
This function allows us to understand the concavity of the original curve. Positive values indicate a "smiling" or concave up shape, while negative values suggest a concave down shape.

Radius of Curvature

Curvature is a measure of how sharply a curve bends and is quantified by \( \kappa \). At any point \((x, y)\), curvature is defined by the formula: \[ \kappa = \frac{|y''|}{(1 + (y')^2)^{3/2}} \].
The radius of curvature \( R \) is the reciprocal of the curvature, offering a sense of the "radius of the osculating circle"—the circle that best fits the curve at a given point.
For point \((1, 1/e)\), the first derivative \( y' = -\frac{2}{e} \) and the second \( y'' = \frac{2}{e} \).
Using these values in our curvature formula, we find \( \kappa \) and its inverse yields \( R \), calculated as: \[ R = \frac{\left(1 + \left(\frac{4}{e^2}\right)\right)^{3/2}}{\frac{2}{e}} \].
The radius reflects how "tight" or "loose" the curve appears around that point, giving deeper insight into the curve's geometry.