Problem 54 - Calculus A New Horizon

Question

Consider the family of curves \(y=e^{-b x^{2}}(b>0)\) (a) Use a graphing utility to generate some members of this family. (b) Discuss the effect of varying \(b\) on the shape of the graph, and discuss the locations of the relative extrema and inflection points.

Step-by-Step Solution

Answer

Higher \( b \) narrows curves, concentrates maxima at the origin, and shifts inflection points closer to zero.

Step 1: Understand the Exercise

We're tasked with analyzing how the parameter \( b \) in the function \( y = e^{-bx^2} \) affects the graph's shape, focusing on the locations of relative extrema and inflection points.

Step 2: Graph the Curves

Using a graphing utility, plot several graphs of \( y = e^{-bx^2} \) for different values of \( b \). Try values such as \( b = 0.5, 1, 2, \) and \( 3 \), capturing how the graph changes as \( b \) increases. Notice that as \( b \) increases, the curves become narrower.

Step 3: Analyze the Shape Changes

Observe that the graphs become steeper and more concentrated near \( x = 0 \) as \( b \) increases. This effect is due to the exponent \(-bx^2\) growing larger more quickly, causing \( y \) to approach zero faster when \( x \) deviates from zero.

Step 4: Determine Relative Extrema

The relative extrema occur where the derivative \( \frac{dy}{dx} \) equals zero. Compute \( \frac{dy}{dx} = -2bx e^{-bx^2} \), which gives a critical point at \( x = 0 \). There is a relative maximum at \( x = 0 \) for any \( b > 0 \).

Step 5: Analyze Inflection Points

To find inflection points, calculate the second derivative \( \frac{d^2y}{dx^2} = 2be^{-bx^2} (2b x^2 - 1) \). Set this to zero to find \( x = \pm \sqrt{\frac{1}{2b}} \) as potential inflection points. The sign of the second derivative changes as \( x \) passes through these points, confirming them as inflection points.

Step 6: Discuss Effects of Varying \( b \)

With higher \( b \) values, inflection points get closer to the origin, making the graph narrower and the changes in curvature more pronounced near \( x = 0 \). This results in the curve having more pronounced peaks and steeper slopes.

Key Concepts

Curve Sketching

Curve sketching is a valuable tool in calculus that lets us visualize the graph of a function and understand its behavior. When sketching curves, especially for functions like \( y = e^{-bx^2} \), the graph's shape dramatically changes with variations in the parameter \( b \).

• As \( b \) increases, the curve appears to "squeeze" tighter around the y-axis.
• The height of the curve at \( x = 0 \) remains 1, since \( e^0 = 1 \), even as \( b \) changes.
• The tails of the graph approach zero more rapidly as \( b \) increases, reflecting the faster decline of the exponential factor.

Using a graphing utility, test different values of \( b \), such as 0.5, 1, 2, and 3. Observe how the graph's width and steepness alter. These sketches offer insights into more advanced concepts, such as extrema and inflection points, which are crucial for analyzing function behavior.

Relative Extrema

Relative extrema refer to points on a graph where the function reaches a local maximum or minimum. For the function \( y = e^{-bx^2} \), we focus on finding where the derivative \( \frac{dy}{dx}\) equals zero. This determines the extrema's location.
By computing the derivative, \( \frac{dy}{dx} = -2bx \,e^{-bx^2} \), we find a critical point at \( x = 0 \). For any \( b > 0 \), \( x = 0 \) is a relative maximum since the derivative changes sign and the function has a hill-like shape.
This reflects the curve's peak at the origin, where the slope is zero and the graph transitions from increasing to decreasing. Identifying these points helps us understand more about the graph's overall character, making the study of relative extrema essential.

Inflection Points

Inflection points occur where a graph changes its curvature from concave up to concave down or vice versa. For \( y = e^{-bx^2} \), these points are determined by where the second derivative, \( \frac{d^2y}{dx^2} \), equals zero.
The calculation gives us \( \frac{d^2y}{dx^2} = 2b \,e^{-bx^2} (2bx^2 - 1) \). Setting this to zero helps find potential inflection points at \( x = \pm \sqrt{\frac{1}{2b}} \).
As \( x \) crosses these points, the second derivative's sign changes, confirming these as inflection points. This interplay of curvature plays a crucial role in the graph's appearance. When \( b \) increases, these points draw closer to the origin, making the graph's curves sharper and more pronounced around \( x = 0 \). Understanding inflection points aids in grasping the graph's subtler changes and transitions.

Parameter Variation

In calculus, parameter variation involves changing a parameter within a function to observe how the graph's traits adjust. With \( y = e^{-bx^2} \), the parameter \( b \) significantly influences the curve's form.

• As \( b \) increases, the graph becomes narrower around the y-axis, emphasizing the rapid fall-off on the sides.
• Higher values of \( b \) push inflection points toward the origin, making changes in curvature more noticeable.
• Relative extrema remain at \( x = 0 \) for any \( b \), but the shape of the maximum becomes steeper.
By varying \( b \), we can control the graph's appearance, observing how these shifts affect overall structure and critical points. Parameter variation shows the dynamic interplay between a function and its parameters, enhancing comprehension of how each component influences the whole picture.