Problem 62 - Calculus Early Transcendentals

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

Varying \( b \) changes the curve's width and height; it becomes narrower and taller as \( b \) increases. The maximum is always at \( x = 0 \), and inflection points are at \( x = \pm\sqrt{\frac{1}{2b}} \).

Step 1: Understand the Function

We have the function given by the family of curves: \( y = e^{-b x^2} \) where \( b > 0 \). This is a Gaussian function, commonly known for its bell-shaped curve.

Step 2: Generate Graphs for Various Values of b

Using a graphing tool, plot the function for different values of \( b \). Start by considering \( b = 0.5, 1, 2, 3 \) to observe how the graph changes.- For \( b = 0.5 \), observe a wider and flatter bell curve.- For \( b = 1 \), the bell curve becomes narrower.- As \( b \) increases to 2 and then 3, the bell curve continues to get narrower and taller.

Step 3: Analyze the Effect of b on Curve Shape

As \( b \) increases, the graph becomes narrower and taller. This is because the width of the curve along the x-axis decreases, making the peak more pronounced.

Step 4: Determine the Location of Relative Extrema

To find the extremum, take the derivative \( \frac{dy}{dx} = -2bx e^{-bx^2} \) and set it to zero.- Since \( e^{-bx^2} eq 0 \), solve: \(-2bx = 0\), which gives \( x = 0 \).- So, the relative extremum is at \( x = 0 \) for all values of \( b \). This is a maximum because the function always peaks at this point.

Step 5: Determine the Inflection Points

The second derivative is \( \frac{d^2y}{dx^2} = (4b^2x^2 - 2b) e^{-bx^2} \). Set this to zero to find inflection points:\( 4b^2x^2 - 2b = 0 \) leads to \( 4b(bx^2 - 0.5) = 0 \), or \( x^2 = \frac{1}{2b} \).- Thus, inflection points are at \( x = \pm\sqrt{\frac{1}{2b}} \). The inflection points move closer to the origin as \( b \) increases.

Key Concepts

Family of Curves

The family of curves described by the function \( y = e^{-b x^2} \) showcases an intriguing set of Gaussian functions. A Gaussian function is often known for its characteristic bell-shaped curve. This family is generated by varying the parameter \( b \), where \( b \) is always greater than zero. The changes in the value of \( b \) significantly impact the shape of the curve.
When you graph different members of this family, you can clearly see how \( b \) acts as a shape-modifier. As \( b \) values start smaller, such as \( b = 0.5 \), you get a wider and flatter bell. Conversely, when \( b \) increases to values like 2 or 3, an evident transition into a narrower, taller shape occurs. This transformation happens because of the exponent \(-bx^2\), which scales the curve along the x-axis.
By visualizing different \( b \) values in a graphing utility, students can gain a better understanding of how this parameter influences the general form. It effectively determines how concentrated the bell curve is around its central peak.

Relative Extrema

In mathematics, a relative extremum refers to a point on a curve where the function reaches a local maximum or minimum. For the Gaussian function \( y = e^{-b x^2} \), the relative extremum is particularly interesting due to its unique, consistent behavior. Here, the function achieves a maximum across all \( b \) values. To pinpoint this turning point, we need the derivative: \( \frac{dy}{dx} = -2bx e^{-bx^2} \).
Setting the derivative equal to zero helps locate possible extremum points: \(-2bx = 0\). Solving this equation reveals that the peak of the curve is at \( x = 0 \). Regardless of the \( b \) value, the Gaussian curve reaches its maximum here. Hence, the relative maximum is really an absolute maximum in the context of this function.

• The function peaks at \( x = 0 \), which means the curve hits its highest point right at the origin on the x-axis.
• This consistency adds simplicity to the analysis, as no further variations in \( x \) could result in another peak.

Users can easily predict and leverage this peak information when analyzing Gaussian functions, making this concept a neat example of mathematical predictability.

Inflection Points

Inflection points mark where a curve changes its concavity, from curving upwards to downwards or vice versa. For the Gaussian function \( y = e^{-b x^2} \), determining these points unveils a graceful dance of the function’s structure. Inflection points appear symmetrically relative to the y-axis because of the curve's nature.
To find these points, we require the second derivative: \( \frac{d^2y}{dx^2} = (4b^2x^2 - 2b) e^{-bx^2} \). Setting this equal to zero yields the condition for inflection: \( 4b(bx^2 - 0.5) = 0 \) implies \( x^2 = \frac{1}{2b} \).
The inflection points are located at \( x = \pm\sqrt{\frac{1}{2b}} \), meaning:

• As \( b \) increases, these points move closer to the origin (0,0). A higher \( b \) compacts the curve, causing inflection points to shrink towards the center.
• This shift illustrates how changes in \( b \) alter the balance between widening and narrowing of the curve, giving each member of the curve family a distinct character.

This analysis of inflection points adds depth to understanding how the Gaussian family of curves behave dynamically with varying parameters, offering insightful lessons on symmetry and mathematical beauty.