# S8.5.1.64 values of m for region

• MHB
• karush
In summary, for values of m between 0 and 1, the line y=mx and the curve y=x/(x^2+1) enclose two symmetrical regions in quadrants I and III. To find the area of these regions, we can integrate using the formula A=2 \int_0^{\sqrt{\frac{1}{m}-1}} \dfrac{x}{x^2+1} - mx \, dx. The slope of the line must be between 0 and 1 in order to intersect the curve and enclose these regions.
karush
Gold Member
MHB
$\tiny{s8.5.1.64\p364}$

For what values of m do the line $y=mx$ and the curve $y=\dfrac{x}{x^2+1}$ enclose a region.
Find the area of the regionok I could only estimate this by observation but it looks $m\ne 1$
not sure how you solve by calculation

both functions are odd (symmetric to the origin) and pass through the origin. the curve is also asymptotic to $y=0$.

$\dfrac{d}{dx} \bigg[\dfrac{x}{x^2+1} \bigg] = \dfrac{1-x^2}{(x^2+1)^2}$

at the origin, the slope of the curve equals 1, therefore $y=x$ would be tangent to the curve.

so, to intersect the curve to form closed symmetrical regions in quadrants I and III, the slope of the line must be $0 < m < 1$

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ok that helped a lot

karush said:
$\tiny{s8.5.1.64\p364}$

For what values of m do the line $y=mx$ and the curve $y=\dfrac{x}{x^2+1}$ enclose a region.
Find the area of the regionok I could only estimate this by observation but it looks $m\ne 1$
not sure how you solve by calculation
Since the problem says "enclose a region" I would just see where they intersect:
$mx= \frac{x}{x^2+ 1}$
An obvious solution is x= 0. If x is not 0 we can divide by x to get $m= \frac{1}{x^2+ 1}$. Multiply both sides by $x^2+ 1$ to get $m(x^2+ 1)= 1$. If m= 0 that is impossible and as long as m is not 0, divide both sides by m to get $x^2+ 1= \frac{1}{m}$. Subtract $1$: $x^2= \frac{1}{m}- 1$. If m> 1 there is no such x. If m< 1 $x= \pm\sqrt{\frac{1}{m}- 1}= \pm\sqrt{\frac{1- m}{m}}$ so the lines cross in three points and enclose two regions. The answer is "all m less than 1" not "all m except 1".

Last edited:
karush said:
$\tiny{s8.5.1.64\p364}$

For what values of m do the line $y=mx$ and the curve $y=\dfrac{x}{x^2+1}$ enclose a region.

Find the area of the region

for $x > 0$ and $0 < m < 1$ ...

$mx = \dfrac{x}{x^2+1} \implies x^2+1 = \dfrac{1}{m} \implies x = \sqrt{\dfrac{1}{m} - 1}$

Sum of the areas of both regions in quads I and III ...

$$\displaystyle A = 2 \int_0^{\sqrt{\frac{1}{m}-1}} \dfrac{x}{x^2+1} - mx \, dx = m - (1+\ln{m})$$

skeeter said:
both functions are odd (symmetric to the origin) and pass through the origin. the curve is also asymptotic to $y=0$.

$\dfrac{d}{dx} \bigg[\dfrac{x}{x^2+1} \bigg] = \dfrac{1-x^2}{(x^2+1)^2}$

at the origin, the slope of the curve equals 1, therefore $y=x$ would be tangent to the curve.

so, to intersect the curve to form closed symmetrical regions in quadrants I and III, the slope of the line must be $0 < m < 1$

View attachment 10706

well pretty valuable to know that...

## What is the significance of S8.5.1.64 values of m for region?

S8.5.1.64 values of m for region refer to the specific values of the quantum number m that are allowed for a particular region of an atom. This quantum number describes the orientation of an atomic orbital in space and can affect the energy and behavior of an electron within that region.

## How do S8.5.1.64 values of m for region impact atomic structure?

The values of m for a given region determine the shape and orientation of the atomic orbitals within that region, which in turn affects the overall structure and properties of the atom. Different values of m can also lead to different energy levels and electron configurations within the atom.

## What is the range of S8.5.1.64 values of m for region?

The range of S8.5.1.64 values of m for region depends on the specific region and type of atom. For example, in the s subshell, m can have values of -1, 0, or 1. In the p subshell, m can have values of -2, -1, 0, 1, or 2. The range of m values for each subshell follows a pattern based on the angular momentum quantum number, l.

## How are S8.5.1.64 values of m for region determined experimentally?

S8.5.1.64 values of m for region can be determined experimentally through techniques such as spectroscopy, which involves studying the energy levels and transitions of electrons within atoms. By analyzing the wavelengths of light emitted or absorbed by atoms, scientists can determine the allowed values of m for different regions.

## What are the implications of S8.5.1.64 values of m for region in chemistry?

The specific values of m for a given region can affect the chemical properties and reactivity of an atom. This is because the orientation and energy of electrons within that region can influence how they interact with other atoms and molecules. Understanding the values of m for different regions is important in predicting and explaining chemical behavior.

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