What is the factorization of the polynomial graphed below? Assume it has no
constant factor. Write each factor as a polynomial in descending order.

Answer:

1) B. at 0 and 3 only

2) D. 2eˣcosx

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Algebra I

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Basic Power Rule:

Derivative Property [Addition/Subtraction]:                                                          \(\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]\)

Derivative Rule [Product Rule]:                                                                              \(\displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)\)

Trig Derivative: \(\displaystyle \frac{d}{dx}[sinu] = u'cosu\)

Trig Derivative: \(\displaystyle \frac{d}{dx}[cosu] = -u'sinu\)

eˣ Derivative: \(\displaystyle \frac{d}{dx} [e^u]=e^u \cdot u'\)

Step-by-step explanation:

*Note:

Velocity is the derivative of position.

Acceleration is the derivative of velocity.

Question 1

Step 1: Define

s(t) = t⁴ - 6t³ - 2t - 1

Step 2: Differentiate

Step 3: Solve

Question 2

Step 1: Define

f(x) = eˣ(sinx + cosx)

Step 2: Differentiate

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

Unit: Derivatives

Book: College Calculus 10e

Parameterize the sphere (call it \(S\)) by

\(\vec s(u,v) = 3\cos(u)\sin(v)\,\vec\imath + 3\sin(u)\sin(v)\,\vec\jmath + 3\cos(v)\,\vec k\)

with \(0\le u\le2\pi\) and \(0\le v\le\pi\).

The outward-pointing normal vector to \(S\) is

\(\vec n = \dfrac{\partial\vec s}{\partial v} \times \dfrac{\partial\vec s}{\partial u} = 9\cos(u)\sin^2(v)\,\vec\imath + 9\sin(u)\sin^2(v)\,\vec\jmath + 9\cos(v)\sin(v)\,\vec k\)

Evaluating \(\vec v\) at \(\vec s\) gives

\(\vec v = 3\cos(u)\sin(v)\,\vec\imath - 3\sin(u)\sin(v)\,\vec\jmath + 12\cos(v)\,\vec k\)

Then the flux of \(\vec v\) across \(S\) is

\(\displaystyle \iint_S \vec v\cdot d\vec\sigma = \int_0^\pi \int_0^{2\pi} \vec v \cdot \vec n \, du \, dv\)

\(\displaystyle = \int_0^\pi \int_0^{2\pi} \left(108\cos^2(v)\sin(v) + 27 \cos(2u) \sin^3(v)\right) \, du \, dv = 144\pi\)

... if the fluid has density 1. But our fluid has density 2, which means twice as much fluid occupies a given volume,

\(\rho = \dfrac mV \implies V = \rho m \implies V = 2m\)

so the overall flux in this case must be doubled to

\(\boxed{288\pi}\)

Alternatively, since \(S\) is a closed sphere, the divergence theorem applies and the flux integral (for density 1) is

\(\displaystyle \iint_S \vec v\cdot d\vec\sigma = \iiint_R \mathrm{div}\vec v \, dV = 4 \iiint_R dV\)

where \(R\) is the interior of \(S\). But that's just a ball of radius 3, and this integral is 4 times the volume of \(R\).

\(\displaystyle \iint_S \vec v\cdot d\vec\sigma = 4 \times \frac{4\pi}3\times3^3 = 144\pi\)

and we get the same result.

The rate of flow outward through the sphere x²+y²+z² = 9 is 288π.

Flux is a term in physics that measures the flow of a physical quantity, such as energy, particles, or field lines, through a surface or substance.

Given that, a fluid has density 2 and the velocity field v = xj-yi+4zk.

We need to find the rate of flow outward through the sphere x²+y²+z² = 9,

Parameterize the sphere =

s(u, v) = 3cos(u)sin(v)i + 3sin(u)sin(v)j + 3cos(v)k [0 ≤ u ≤ π and 0 ≤ v ≤ 2π]

The outward-pointing normal vector to S =

n = δs/δu×δs/δv = 9cos(u)sin²(v)i + 9sin(u)sin²(v)j + 9cos(v)k

Calculating v at s,

v = 3cos(u) sin(v)i -3sin(u)sin(v)j +12 cos(v)k

The flux of v across s is =

∫∫v·dσ = \(\int\limits^\pi_0 \int\limits^{2\pi}_0{v\dot\ n} \, dv du \\\\\)

\(\int\limits^\pi_0 \int\limits^{2\pi}_0{108 cos^2 (v) sin (v) +27 cos(2u) sin^3 (v)) dudv=144\pi} \,\)

If the fluid has density 2, which means twice as much fluid occupies a given volume,

ρ = m/V

So,

V = 2m

So the overall flux in this case must be doubled to 288π.

Hence the rate of flow outward through the sphere x²+y²+z² = 9 is 288π.

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Answer:

True

Step-by-step explanation:

In the Monte Carlo simulations process, the probability of outcomes is determined on the basis of repeated sampling for the outcomes in a process which can not be determined easily due to several random variables involved in the process. This techniques helps in determining the risk, uncertainty in prediction and forecasting models.

Answer:

19

Step-by-step explanation:

We are given the following two functions of s

\(p(s) = s^3 + 10s\\f(s) = 6s - 3\\\\\text{To find p(2) substitute 2 for s in p(s)}\\p(2) = (2)^3 + 10(2) = 8 + 20 = 28\\\\\)

\(\text{To find f(2) substitute 2 for s in f(s)}\\f(2) = 6(2) - 3= 12 - 3= 9\\\)

\(p(2) - f(2) = 28 - 9 = 19\)

The function notation of 9x + 3y = 12 is given as follows:

f(x) = 4 - 3x.

The function in the context of this problem is given as follows:

9x + 3y = 12.

The format for the function notation is given as follows:

Hence we must isolate the variable y, as follows:

3y = 12 - 9x

y = 4 - 3x (each term of the expression is divided by 3).

f(x) = 4 - 3x.

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Answer :

1) If x = 6

Substitute 6 for x in the expression.

a) 5x-9

5(6)-9

30-9

21

b) 2x-7

2(6) -7

12-7

5

c) 8x+5

8(6)+5

48+5

53

d) 4(x+3)

4(6+3)

4(9)

36

e) x²-(-4)

6²-(-4)

36+4

40

f) 3x²+1

3(6²)+1

3(36) +1

108+1

109

2) Solve

a) 54 = 3x

Divide both sides by 3.

54/3 = 3x/3

x = 18

b) x-9 = -3

Add 9 to both sides.

x-9+9 = -3+9

x = 6

c) x/7 = 140

Express the 140 over 1 and cross multiply.

x/7 = 140/1

x = 140×7

x = 980

d) x/8 = 72

Express the 72 over 1 and cross multiply.

x/8 =72/1

x = 72×8

x = 576

Answer:

7

Step-by-step explanation:

Let's say our sum is s.

s = 10 right angles

a right angle is 90 degrees

s = 10 (90)

s= 900

Given the amount of sides in a polygon (n), the sum of the interior angles is equal to

(n-2) * 180

Therefore, the sum of the interior angles is equal to

(n-2) * 180 = 900

divide both sides by 180 to help isolate n

n-2 = 5

add 2 to both sides to isolate n

n = 7

By using Newton's divided difference formula, the polynomial function in x is f(x) = x⁴ - 3x³ + 5x² - 6.

In Mathematics and Geometry, a polynomial function is a mathematical expression which comprises intermediates (variables), constants, and whole number exponents with different numerical value, that are typically combined by using mathematical operations such as the following:

Next, we would determine the polynomial function in x from the given data by using Newton's divided difference formula;

xi   f(xi)     f[xi,xi+1] f[xi,...,xi+2] f[xi,...,xi+3] f[xi,...,xi+4]

-1     3          -9                6                  5                1

0    -6          15               41                 13

3    39          261            132

6    822     789

7    1611

By interpolating, the interpolation polynomial is given by:

Pₙ(x) = 3 + -9(x+1) + 6(x + 1)(x - 0) + 5(x + 1)(x - 0)(x - 3) + 1(x + 1)(x - 0)(x - 3)(x - 6)

f(x) = (x - 6)(x - 3)x(x + 1) + 5(x - 3)x(x + 1) + 6(x + 1) - 9(x + 1) + 3

f(x) = x⁴ - 3x³ + 5x² - 6

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Line 1: y= - 4/3x - 8

Line 2: 3y = - 4x + 4

Line 3: 6x + 8y = - 4

First we need to write each equation in the slope-intercept form, as follows:

Line 1: y= - 4/3x - 8

Line 2: y = -4/3x + 4/3

Line 3: y = -3/4x - 1/2

Line 1 and 2 are parallel because both have the same slope.

Line 1 and 3 and 2 and 3 are neither parallel nor perpendicular.

Answer:

1. The number of suite seats is 10% of the highest-priced seats, which is 10% of 92,000 = 9,200 seats.

2. The number of non-suite seats is 90% of the highest-priced seats, which is 90% of 92,000 = 82,800 seats.

3. The revenue generated from suite seats if all are purchased for one game is 9,200 x $200 = $1,840,000.

4. The revenue generated from non-suite seats if all are purchased for one game is 82,800 x $60 = $4,968,000.

5. The revenue earned at one sold-out game is the sum of the revenue from suite seats and non-suite seats: $1,840,000 + $4,968,000 = $6,808,000.

6. The net revenue of one sold-out game in which the nonconference team plays at the big stadium is the revenue earned at one sold-out game minus the payment to the nonconference team: $6,808,000 - $1,500,000 = $5,308,000.

Step-by-step explanation:

The medians are about the same.

Option C is the correct answer.

The median, in statistical terms, refers to the value in the center of a dataset that has been sorted either in ascending or descending order, and it is a commonly used measure of central tendency. This contrasts with the mean, which calculates the total value of all figures, then divides by the number of figures.

We have,

Movie theater 1.

From the box plot given,

The median is 995.

Movie theater 2.

From the box plot given,

The median is 995.

Thus,

The median in both the box plot is the same.

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Answer:

6 units to the right, 3 units down

Step-by-step explanation:

Answer:

233 is a Fibonacci number

Step-by-step explanation:

Answer:

12.72792

Step-by-step explanation:

d = √2a = √2 x 9 = 12.72792

Sorry, I don't have too much time to explain the question but I hope this helps!

The length of a 45° arc with a radius of 4 miles is approximately 3.14 miles, calculated using the formula for arc length.

To determine the length of a 45° arc given a radius of 4 miles, we can use the formula: Arc length = (angle measure / 360°) x 2πr, where r is the radius of the circle and π is a constant equal to approximately 3.14.

Substituting the given values into the formula, we get: Arc length = (45° / 360°) x 2π(4 mi)Arc length = (1/8) x 2π(4 mi)Arc length = (1/8) x 8π Arc length = π

The length of the 45° arc is approximately 3.14 miles.

Summary: To find the length of a 45° arc of a circle, we use the formula: Arc length = (angle measure / 360°) x 2πr. Given a radius of 4 miles, we can substitute the values into the formula to get the length of the 45° arc, which is approximately 3.14 miles.

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The line segment FG represent the volume of water decreases in Katherine's water bottle.

From the given graph, x-axis represents the distance from home (km) and the y-axis represents the volume (L).

The line segment FG represent the volume of water decreases in Katherine's water bottle, the line segment HI represent the volume of water increases in Katherine's water bottle and the line segment KL represents there is no change in water level.

Therefore, the line segment FG represent the volume of water decreases in Katherine's water bottle.

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It is TRUE that the given image represents the function.

A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X.

The sets X and Y are collectively referred to as the function's domain and codomain, respectively.

Initially, functions represented the idealized relationship between two changing quantities.

A function, according to a technical definition, is a relationship between a set of inputs and a set of potential outputs, where each input is connected to precisely one output.

So, the given image represents a function and it is a kind of many-to-one function.

Therefore, it is TRUE that the given image represents the function.

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In given fractions 8/9 is 1/36 large than 31/36

Finding the larger and smallest fraction in between two or more fractions is known as comparing fractions.

We need to change fractions with unlike denominators into fractions with similar denominators in order to compare them. For this, we will find the denominators' Least Common Multiple (LCM).    

If the denominators are the same then it is easy to compare the fractions.

Here we have

31/36 and 8/9

To find the largest fraction convert both denominators into the same  

Here LCM(36, 9) = 36  

Now multiply both numerator and denominators of fractions with a number to change the denominators  

=> \(\frac{31}{36} \times \frac{1}{1} = \frac{31}{36}\)  

=> \(\frac{8}{9} \times \frac{4}{4} = \frac{32}{36}\)  

From the above calculations given fractions are 31/36 and 32/36

Difference = \(\frac{32}{36} - \frac{31}{36} = \frac{1}{36}\)  

Therefore,

In given fractions 8/9 is 1/36 large than 31/36

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Answer:

K' (-1,10)

L'(0,4)

M'(2,2)

Step-by-step explanation:

To use the mapping rule, substitute x and y with the original values

The coordinates of K are x=-4, and y=5.

Using the mapping rule, x of K' = -4+3 = -1

Using the mapping rule y of K' = 2x5=10

So K' is at the coordinate (-1,10)

The coordinates of L are x =-3, and y =2.

Using the mapping rule, x of L' = -3+3=0

Using the mapping rule y of L' = 2x2=4

So L' is at the coordinate (0,4)

The coorindates of M are x=-1 and y=1

Using the mapping rule, x of M' = -1+3=2

Using the mapping rule, y of M' = 2x1 = 2

So M' is at the coordinate (2,2).

The relative change is approximately -14.9%, indicating a decrease of approximately 14.9% in per person consumption of beef from 2000 to 2020.

To find the absolute and relative change as a percentage in the U.S. per person consumption of beef from 2000 to 2020, we can follow these steps:

Calculating the absolute change:

Absolute change = Final value - Initial value

= 57.7 pounds - 67.8 pounds

= -10.1 pounds

Step 2: Calculate the relative change:

Relative change = (Absolute change / Initial value) * 100

= (-10.1 pounds / 67.8 pounds) * 100

≈ -14.9%

The absolute change is -10.1 pounds, indicating a decrease in per person consumption of beef.

The relative change is approximately -14.9%, indicating a decrease of approximately 14.9% in per person consumption of beef from 2000 to 2020.

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Answer:

We use the rule,

\((a^b)^c = a^{bc}\)

a. (10^7)²

\((10^7)^2 = 10^{(7)(2)} = 10^{14}\)

10^14

b. (10^9)³

\((10^9)^3 = 10^{(9)(3)} = 10^{27}\)

10^27

c. (10^6)³

\(10^{(6)(3)}= 10^{18}\)

10^18

d. (10^2)³

\(10^{(2)(3)} = 10^{6}\)

10^6

e. (10³)²

\(10^{(3)(2)}=10^{6}\)

10^6

f. (10^5)^7

\(10^{(5)(7)} = 10^{35}\)

10^35

Step-by-step explanation:

Answer:

5.3 or 28

Step-by-step explanation:

\(6^{2}\)+\(b^{2}\)=\(8^{2}\)

36+\(b^{2}\)=64

64-36=28

\(\sqrt{28\\}\)

=5.29 or 5.3

Keep in mind that without the population standard deviation, it is impossible to provide an exact sample size. However, this formula will give you a good starting point.

To find the sample size needed for a 99.5% confidence interval with a margin of error of 1.5, we can use the formula:

n = (Z * σ / E)^2

where n is the sample size, Z is the Z-score corresponding to the desired confidence level, σ is the population standard deviation, and E is the margin of error.

For a 99.5% confidence interval, the Z-score is approximately 2.807 (from a standard normal distribution table). Since we do not have the population standard deviation (σ), we will need to estimate it using a sample standard deviation or use a conservative approach by assuming the maximum possible value. For now, let's assume we have an estimated standard deviation.

n = (2.807 * σ / 1.5)^2

Solve for n by plugging in the estimated standard deviation (σ) and then round up to the nearest whole number, as you cannot have a fraction of a sample.
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Answer:

30 is not included

Step-by-step explanation:

The way this stem works is that it separates the numbers for better sorting.

Example:

Set that records 11, 12, 24, 28, 32, 32

The stem and leafs would look like

1    1 2

2   4 8

3   2 2

The sampled bold number indicates that 2 and 8 make up 28.

If you look at the plot, there was no 3 and 0 that would make 30. So that does not exist.

Answer: (-5,1)

Step-by-step explanation:

−2.5x+y=13.5

−2.5x+y+2.5x=13.5+2.5x(Add 2.5x to both sides)

y=2.5x+13.5

Step: Substitute

5x+13.5  and 2.25x−y=−12.25:

2.25x−y=−12.25

2.25x−(2.5x+13.5)=−12.25

−0.25x−13.5=−12.25(Simplify both sides of the equation)

−0.25x−13.5+13.5=−12.25+13.5(Add 13.5 to both sides)

−0.25x=1.25

−0.25=1.25−0.25(Divide both sides by -0.25)

x=−5

Step: Substitute−5forxiny=2.5x+13.5:

y=2.5x+13.5

y=(2.5)(−5)+13.5

y=1(Simplify both sides of the equation)

Answer:

x = -4, y = 3.5

Step-by-step explanation:

-2.5x + y = 13.5 ------ (1)

2.25x - y = -12.25 ---(2)

you can substitute (1) into (2) or vice versa

We are going to substitute (2) into (1)

but first we have to rewrite it in terms of x or y

Lets do y

from (2), (after subtracting -2.25x from both sides), y = 12.25 + 2.25x ----(3)

plug in (3) into (1)

-2.5x + (12.25 + 2.25x) = 13.5

-2.5x + 12.25 + 2.25x = 13.5

-2.5x + 2.25x = 13.5 - 12.5

-0.25x = 1

x = -(1/.25)

x = -4

Now we have x

We already used (2), lets plug in x = -4 into (1)

-2.5(-4) + y = 13.5

10 + y = 13.5

y = 13.5 - 10

y = 3.5

Answer:

Kristen should walk 12 miles to catch up with Paul

Step-by-step explanation:

Paul = 2hrs earlier

      = walks at 3 miles per hour

      = 3miles x 2 hrs = 6 miles

covers 6 miles at the time Kristen starts

Kristen = 4 miles per hour

Paul has a lead of 6 miles.

if Kristen walks another hour of 4 miles she would have covered 8 miles and Paul would have covered 9 miles and if  Kristen were to cover one more hour,

Kristen would be at 12 and so would Paul.

Kristen should walk 12 miles to catch up with Paul

Answer:

One Solution

Step-by-step explanation:

So we have the equation:

\(4x+(-3x+2)=2x+5-2\)

On both sides, combine like terms:

\((4x-3x)+2=2x+(5-2)\)

Simplify:

\(x+2=2x+3\)

Subtract x from both sides. The left cancels:

\((x+2)-x=(2x+3)-x\\2=x+3\)

Subtract 3 from both sides. The right cancels:

\((2)-3=(x+3)-3\\-1=x\)

Flip:

\(x=-1\)

So, our equation only has one solution.

And the answer is -1 :)

Answer:

Step-by-step explanation:

I and II

Pages in a book and leaves on a tree can be counted precisely. They are discrete.

The height of a person cannot be measured exactly. We can get very good

approximations (to a lot of decimal places) but never exact. This is continuous.

Answer:

Step-by-step explanation:

I can't really explain, but Here comes the hoping:

Equation 1:

Equation 2:

Equation 3:

Equation 4:

Equation 5: