18.07.2021

what describes the inflection point of a bell shaped curve

. 8

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English
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P Answered by Master

And when these days were expired, the king made a feast unto all the people that were present in Shushan the palace, both unto great and small, seven days, in the court of the garden of the king's palace; Where were white, green, and blue, hangings, fastened with cords of fine linen and purple to silver rings and pillars of marble: the beds were of gold and silver, upon a pavement of red, and blue, and white, and black, marble.

English
Step-by-step answer
P Answered by Master

And when these days were expired, the king made a feast unto all the people that were present in Shushan the palace, both unto great and small, seven days, in the court of the garden of the king's palace; Where were white, green, and blue, hangings, fastened with cords of fine linen and purple to silver rings and pillars of marble: the beds were of gold and silver, upon a pavement of red, and blue, and white, and black, marble.

StudenGPT
Step-by-step answer
P Answered by Studen AI
To determine the concavity of the polynomial equation \(y=x^4-10x^2+16\), we will need to find the second derivative of the equation. The second derivative tells us whether the curve is concave up or concave down. Here's how we can solve the problem step by step using math laws: Step 1: Find the first derivative of the equation: To find the first derivative, we differentiate each term of the equation with respect to \(x\) using the power rule. The power rule states that if we have a term of the form \(x^n\), the derivative will be \(n \cdot x^{n-1}\). Applying this rule to each term: \(\frac{dy}{dx} = \frac{d}{dx} (x^4) - \frac{d}{dx} (10x^2) + \frac{d}{dx} (16)\) Simplifying, we get: \(\frac{dy}{dx} = 4x^3 - 20x\) Step 2: Find the second derivative of the equation: To find the second derivative, we differentiate the first derivative obtained in step 1 with respect to \(x\) using the power rule again: \(\frac{d^2y}{dx^2} = \frac{d}{dx}(4x^3) - \frac{d}{dx}(20x)\) Simplifying, we get: \(\frac{d^2y}{dx^2} = 12x^2 - 20\) Step 3: Determine the concavity: Now that we have the second derivative, we can analyze its sign to determine the concavity. For concave up (U-shaped curve), the second derivative must be positive. For concave down (∩-shaped curve), the second derivative must be negative. For points of inflection, where the curve changes concavity, the second derivative must change sign. In our case, the second derivative is \(12x^2 - 20\). To find the points of inflection, we need to solve the equation \(12x^2 - 20 = 0\) for \(x\). Adding 20 to both sides: \(12x^2 = 20\) Dividing by 12: \(x^2 = \frac{20}{12} = \frac{5}{3}\) Taking the square root of both sides: \(x = \pm\sqrt{\frac{5}{3}}\) So, we have found the possible x-values of the points of inflection: \(\sqrt{\frac{5}{3}}\) and \(-\sqrt{\frac{5}{3}}\). Step 4: Analyze concavity To analyze the concavity, we need to see if the second derivative is positive or negative for different values of \(x\). For \(x > \sqrt{\frac{5}{3}}\), the second derivative \(12x^2 - 20\) is positive, indicating that the curve is concave up. For \(x < -\sqrt{\frac{5}{3}}\), the second derivative \(12x^2 - 20\) is positive as well, indicating that the curve is concave up here too. Between \(-\sqrt{\frac{5}{3}}\) and \(\sqrt{\frac{5}{3}}\), the second derivative \(12x^2 - 20\) is negative. This is where the curve changes concavity, so these are the points of inflection. In summary, the curve described by the polynomial equation \(y=x^4-10x^2+16\) is concave up for \(x > \sqrt{\frac{5}{3}}\), concave up for \(x < -\sqrt{\frac{5}{3}}\), and has two points of inflection at \(\sqrt{\frac{5}{3}}\) and \(-\sqrt{\frac{5}{3}}\).
Mathematics
Step-by-step answer
P Answered by Master

\displaystyle  \lim_{x \to 3^-} f(x) = - \infty

\displaystyle  \lim_{x \to 3^+} f(x) = \infty

\displaystyle  \lim_{x \to -3^-} f(x) = - \infty

\displaystyle  \lim_{x \to -3^+} f(x) = \infty

General Formulas and Concepts:

Algebra I

Functions

Algebra II

Analyzing Cartesian Planes and Graphs

Calculus

Limits

Left Limit:                                                                                                          \displaystyle  \lim_{x \to a^-} f(x)Right Limit:                                                                                                        \displaystyle  \lim_{x \to a^+} f(x)

Graphical Limits

Step-by-step explanation:

We approach this question by analyzing the graph. We notice we have asymptotes at x = -3 and x = 3.

Question 1

\displaystyle  \lim_{x \to 3^-} f(x) = \ ?

Essentially, the question is asking what the value is for f(x) when x approaches 3 from the left. We see from the graph f(x) that if we approach 3 from the left, we would be going towards the x = 3 asymptote, specifically -∞.

\displaystyle  \lim_{x \to 3^-} f(x) = - \infty

Question 2

\displaystyle  \lim_{x \to 3^+} f(x) = \ ?

Essentially, the question is asking what the value is for f(x) when x approaches 3 from the right. We see from the graph f(x) that if we approach 3 from the right, we would be going towards the x = 3 asymptote, specifically ∞.

\displaystyle  \lim_{x \to 3^+} f(x) = \infty

Question 3

\displaystyle  \lim_{x \to -3^-} f(x) = \ ?

Essentially, the question is asking what the value is for f(x) when x approaches -3 from the left. We see from the graph f(x) that if we approach -3 from the left, we would be going towards the x = -3 asymptote, specifically -∞.

\displaystyle  \lim_{x \to -3^-} f(x) = - \infty

Question 4

\displaystyle  \lim_{x \to -3^+} f(x) = \ ?

Essentially, the question is asking what the value is for f(x) when x approaches -3 from the right. We see from the graph f(x) that if we approach -3 from the right, we would be going towards the x = -3 asymptote, specifically ∞.

\displaystyle  \lim_{x \to -3^+} f(x) = \infty

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Limits

Book: College Calculus 10e

English
Step-by-step answer
P Answered by Specialist

It's all about controlling your tone, sound, and more.

Explanation:

These are the starting points. But here is a guide. Try saying the lines out loud.

italics lines = dramatic

bold lines= powerful

"The outlook wasn’t brilliant for the Mudville nine that day;

The score stood four to two with but one inning more to play.

And then when Cooney died at first, and Barrows did the same,

A sickly silence fell upon the patrons of the game.

A straggling few got up to go in deep despair. The rest

Clung to that hope which springs eternal in the human breast;

They thought if only Casey could but get a whack at that—

We’d put up even money now with Casey at the bat.

But Flynn preceded Casey, as did also Jimmy Blake,

And the former was a lulu and the latter was a cake;

So upon that stricken multitude grim melancholy sat,

For there seemed but little chance of Casey’s getting to the bat.

But Flynn let drive a single, to the wonderment of all,

And Blake, the much despised, tore the cover off the ball;

And when the dust had lifted, and men saw what had occurred,

There was Jimmy safe at second and Flynn a-hugging third.

Then from 5,000 throats and more there rose a lusty yell;

It rumbled through the valley, it rattled in the dell;

It knocked upon the mountain and recoiled upon the flat,

For Casey, mighty Casey, was advancing to the bat.

There was ease in Casey’s manner as he stepped into his place;

There was pride in Casey’s bearing and a smile on Casey’s face.

And when, responding to the cheers, he lightly doffed his hat,

No stranger in the crowd could doubt ’twas Casey at the bat.

Ten thousand eyes were on him as he rubbed his hands with dirt;

Five thousand tongues applauded when he wiped them on his shirt.

Then while the writhing pitcher ground the ball into his hip,

Defiance gleamed in Casey’s eye, a sneer curled Casey’s lip.

And now the leather-covered sphere came hurtling through the air,

And Casey stood a-watching it in haughty grandeur there.

Close by the sturdy batsman the ball unheeded sped—

“That ain’t my style,” said Casey. “Strike one,” the umpire said.

From the benches, black with people, there went up a muffled roar,

Like the beating of the storm-waves on a stern and distant shore.

“Kill him! Kill the umpire!” shouted someone on the stand;

And it’s likely they’d have killed him had not Casey raised his hand.

With a smile of Christian charity great Casey’s visage shone;

He stilled the rising tumult; he bade the game go on;

He signaled to the pitcher, and once more the spheroid flew;

But Casey still ignored it, and the umpire said, “Strike two.”

“Fraud!” cried the maddened thousands, and echo answered fraud;

But one scornful look from Casey and the audience was awed.

They saw his face grow stern and cold, they saw his muscles strain,

And they knew that Casey wouldn’t let that ball go by again.

The sneer is gone from Casey’s lip, his teeth are clinched in hate;

He pounds with cruel violence his bat upon the plate.

And now the pitcher holds the ball, and now he lets it go,

And now the air is shattered by the force of Casey’s blow.

Oh, somewhere in this favored land the sun is shining bright;

The band is playing somewhere, and somewhere hearts are light,

And somewhere men are laughing, and somewhere children shout;

But there is no joy in Mudville—mighty Casey has struck out. "

This is the way that I would say it, but you can switch it up. Sounding dramatic when needed and sounding intense when needed are the best ways to say this poem.

Hopefully this helps you.

English
Step-by-step answer
P Answered by Master

The diagraph is a group of two letters that combined make just one phoneme or sound. This are some consonant digraphs: Bl, br, ch, cl, cr, dr, fl, fr, gl, gr, pl, pr, sc, sh, sk, sl, sm, sn, sp, st, sw, th, tr, tw, wh, wr.

In the following words you can find underlined the digraphs:

Characterization [kar-ik-ter-uh-zey-shuhn, -truh-zey-]

Diaphragm [dahy-uh-fram]

Metaphor [met-uh-fawr, -fer]

There also some words where a double consonant can also sound as one single phoneme, such as ss, mm, nn; however, for example, "misspell" which sound as [mis'spel] does not belong to this group, in this word the two “s” are pronounced separately.

In the following words you can find underlined the double consonants that make one sound:

Communication [kuh-myoo-ni-key-shuh n]

Rapport [ra-pawr, -pohr, ruh-]

English
Step-by-step answer
P Answered by Specialist

It's all about controlling your tone, sound, and more.

Explanation:

These are the starting points. But here is a guide. Try saying the lines out loud.

italics lines = dramatic

bold lines= powerful

"The outlook wasn’t brilliant for the Mudville nine that day;

The score stood four to two with but one inning more to play.

And then when Cooney died at first, and Barrows did the same,

A sickly silence fell upon the patrons of the game.

A straggling few got up to go in deep despair. The rest

Clung to that hope which springs eternal in the human breast;

They thought if only Casey could but get a whack at that—

We’d put up even money now with Casey at the bat.

But Flynn preceded Casey, as did also Jimmy Blake,

And the former was a lulu and the latter was a cake;

So upon that stricken multitude grim melancholy sat,

For there seemed but little chance of Casey’s getting to the bat.

But Flynn let drive a single, to the wonderment of all,

And Blake, the much despised, tore the cover off the ball;

And when the dust had lifted, and men saw what had occurred,

There was Jimmy safe at second and Flynn a-hugging third.

Then from 5,000 throats and more there rose a lusty yell;

It rumbled through the valley, it rattled in the dell;

It knocked upon the mountain and recoiled upon the flat,

For Casey, mighty Casey, was advancing to the bat.

There was ease in Casey’s manner as he stepped into his place;

There was pride in Casey’s bearing and a smile on Casey’s face.

And when, responding to the cheers, he lightly doffed his hat,

No stranger in the crowd could doubt ’twas Casey at the bat.

Ten thousand eyes were on him as he rubbed his hands with dirt;

Five thousand tongues applauded when he wiped them on his shirt.

Then while the writhing pitcher ground the ball into his hip,

Defiance gleamed in Casey’s eye, a sneer curled Casey’s lip.

And now the leather-covered sphere came hurtling through the air,

And Casey stood a-watching it in haughty grandeur there.

Close by the sturdy batsman the ball unheeded sped—

“That ain’t my style,” said Casey. “Strike one,” the umpire said.

From the benches, black with people, there went up a muffled roar,

Like the beating of the storm-waves on a stern and distant shore.

“Kill him! Kill the umpire!” shouted someone on the stand;

And it’s likely they’d have killed him had not Casey raised his hand.

With a smile of Christian charity great Casey’s visage shone;

He stilled the rising tumult; he bade the game go on;

He signaled to the pitcher, and once more the spheroid flew;

But Casey still ignored it, and the umpire said, “Strike two.”

“Fraud!” cried the maddened thousands, and echo answered fraud;

But one scornful look from Casey and the audience was awed.

They saw his face grow stern and cold, they saw his muscles strain,

And they knew that Casey wouldn’t let that ball go by again.

The sneer is gone from Casey’s lip, his teeth are clinched in hate;

He pounds with cruel violence his bat upon the plate.

And now the pitcher holds the ball, and now he lets it go,

And now the air is shattered by the force of Casey’s blow.

Oh, somewhere in this favored land the sun is shining bright;

The band is playing somewhere, and somewhere hearts are light,

And somewhere men are laughing, and somewhere children shout;

But there is no joy in Mudville—mighty Casey has struck out. "

This is the way that I would say it, but you can switch it up. Sounding dramatic when needed and sounding intense when needed are the best ways to say this poem.

Hopefully this helps you.

Mathematics
Step-by-step answer
P Answered by Specialist

\displaystyle  \lim_{x \to 3^-} f(x) = - \infty

\displaystyle  \lim_{x \to 3^+} f(x) = \infty

\displaystyle  \lim_{x \to -3^-} f(x) = - \infty

\displaystyle  \lim_{x \to -3^+} f(x) = \infty

General Formulas and Concepts:

Algebra I

Functions

Algebra II

Analyzing Cartesian Planes and Graphs

Calculus

Limits

Left Limit:                                                                                                          \displaystyle  \lim_{x \to a^-} f(x)Right Limit:                                                                                                        \displaystyle  \lim_{x \to a^+} f(x)

Graphical Limits

Step-by-step explanation:

We approach this question by analyzing the graph. We notice we have asymptotes at x = -3 and x = 3.

Question 1

\displaystyle  \lim_{x \to 3^-} f(x) = \ ?

Essentially, the question is asking what the value is for f(x) when x approaches 3 from the left. We see from the graph f(x) that if we approach 3 from the left, we would be going towards the x = 3 asymptote, specifically -∞.

\displaystyle  \lim_{x \to 3^-} f(x) = - \infty

Question 2

\displaystyle  \lim_{x \to 3^+} f(x) = \ ?

Essentially, the question is asking what the value is for f(x) when x approaches 3 from the right. We see from the graph f(x) that if we approach 3 from the right, we would be going towards the x = 3 asymptote, specifically ∞.

\displaystyle  \lim_{x \to 3^+} f(x) = \infty

Question 3

\displaystyle  \lim_{x \to -3^-} f(x) = \ ?

Essentially, the question is asking what the value is for f(x) when x approaches -3 from the left. We see from the graph f(x) that if we approach -3 from the left, we would be going towards the x = -3 asymptote, specifically -∞.

\displaystyle  \lim_{x \to -3^-} f(x) = - \infty

Question 4

\displaystyle  \lim_{x \to -3^+} f(x) = \ ?

Essentially, the question is asking what the value is for f(x) when x approaches -3 from the right. We see from the graph f(x) that if we approach -3 from the right, we would be going towards the x = -3 asymptote, specifically ∞.

\displaystyle  \lim_{x \to -3^+} f(x) = \infty

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Limits

Book: College Calculus 10e

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