Consider the function f : R → R given by f(x) = {. e-1/42 f if x0 if x = 0 a) Prove that f has derivatives of all orders at x = 0 and f(0) = 0 * b) Can f be written as a series f(x) = Xaxxk, ax ER k=0 convergent on some interval (-R,R), R > 0?

To have a horizontal asymptote at y = 3 and vertical asymptotes at x = 4 and x = -3, the rational function should have the following form:

f(x) = (a polynomial in x) / ((x - 4)(x + 3))

The polynomial in the numerator can have any degree, but it must be of lower degree than the denominator.

Therefore, among the given rational functions, the one that satisfies these conditions would be the one in the form:

f(x) = (a polynomial) / ((x - 4)(x + 3))

Please provide the specific options you have, and I can help you determine which of those options matches this form.

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Answer:

50 units

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Equality Properties

Trigonometry

[Right Triangles Only] Pythagorean Theorem: a² + b² = c²

Step-by-step explanation:

Step 1: Define

a = 48

b = 14

c = ?

Step 2: Solve for c

Answer:

12.6

Step-by-step explanation:

divide the length value by 3

Ok. So 14% of 18.00 can be figured out without a calculator. 18.00 / 10 = 1.80 = 10%.

(18.00 / 100) x 4 = 0.72

1.80 + 0.72 = $2.52 is the tip.

$18 + $2.52 = $20.52

⭐ Answered by Foxzy0⭐

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A class interval refers to option b) a division used for grouping a set of observations.

The correct answer is (b) a division used for grouping a set of observations. In statistics, when dealing with a large set of data, it is often helpful to group the data into intervals or classes to better understand the distribution. A class interval represents a range of values that are grouped together. It is defined by specifying the lower and upper boundaries of each interval.

For example, if we are analyzing the heights of individuals, we may create class intervals such as 150-160 cm, 160-170 cm, and so on. The purpose of using class intervals is to simplify the data and provide a clearer picture of the distribution. It allows us to summarize the data and identify patterns or trends within specific ranges. Therefore, option (b) is the correct description of a class interval.

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The expected value of the given random variable is 8.3. This means that, on average, if we repeatedly sample from this random variable, we can expect the resulting values to be around 8.3.

To find the expected value of a random variable, you multiply each possible value by its corresponding probability and sum them up. In this case, we have a combination of probabilities and numerical values. Let's calculate the expected value:

Multiply each numerical value by its corresponding probability:

(0.35 * 8) + (0.3 * 9) + (0.05 * 10) + (0.1 * 11) + (0.05 * 11) + (0.15 * 11)

Perform the calculations:

2.8 + 2.7 + 0.5 + 1.1 + 0.55 + 1.65

Sum up the results:

8.3

Therefore, the expected value of the given random variable is 8.3. This means that, on average, if we repeatedly sample from this random variable, we can expect the resulting values to be around 8.3. The expected value provides a measure of central tendency for the random variable.

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Answer:

5.318 * 10 to power of 8

Step-by-step explanation:

Answer:

5.318 × [tex]10^{8}[/tex]

Step-by-step explanation:

Answer:

16%

Step-by-step explanation:

(174-150)/150= 0.16

0.16*100 = 16%

The quantified statement (∃x)¬D(x) can be translated into simple, everyday English as "There exists a penguin that is not dangerous."

The quantified statement (∃x)¬D(x) can be further explained in the context of penguins. Let's break it down:

The symbol (∃x) denotes the existence of an object or entity that satisfies a certain condition. In this case, it refers to a penguin that meets the condition specified afterward.

The predicate ¬D(x) can be understood as the negation of the predicate D(x), where D(x) represents the statement "x is dangerous." The negation symbol (¬) in front of D(x) indicates the opposite or negation of the statement.

Combining these elements, the quantified statement (∃x)¬D(x) asserts that there is at least one penguin for which the predicate "is not dangerous" holds true. In everyday English, this statement can be translated as "There exists a penguin that is not dangerous."

Essentially, it implies that within the domain of all penguins being considered, at least one penguin can be found that is not considered dangerous. This quantified statement allows for the possibility that not all penguins are dangerous and acknowledges the existence of non-threatening penguins.

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Answer:

60 mins

Step-by-step explanation:

15+45=60

The time needed to make 138 pancakes if both Mark and Charlotte work together is 227.142 minutes.

A fraction is a way to describe a part of a whole. such as the fraction 1/4 can be described as 0.25.

As it is given that Mark can make 9 pancakes in 15 minutes, therefore, the number of pancakes that Mark can make in one minute,

[tex]\text{Number of Pancake in one minute} = \dfrac{9}{15}[/tex]

Now, for Charlotte, it is given that he makes 42 pancakes in 45 minutes, therefore, the number of pancakes that Charlotte can make in one minute,

[tex]\text{Number of Pancake in one minute} = \dfrac{42}{45} = \dfrac{12}{15}[/tex]

Further, the total pancakes that can be made in one minute,

[tex]\text{Total Number of Pancake in one minute} = \dfrac{9}{15} +\dfrac{12}{15} = \dfrac{21}{15}[/tex]

As they both need to make 138 pancakes together, therefore, the time they need is,

[tex]\rm Time\ Needed = \dfrac{\text{Total number of pancakes}}{\text{Total number of pancakes in one minute}}[/tex]

[tex]\rm Time\ Needed = \dfrac{138}{\frac{21}{15}} = \dfrac{138\times 15}{21} = 227.142[/tex]

Hence, the time needed to make 138 pancakes if both Mark and Charlotte work together is 227.142 minutes.

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Answer:

9/2 = 4 1/2

Step-by-step explanation:

Answer:

Tens digit, x = 3

Unit digit, y = 8

Number = 38

Step-by-step explanation:

Let the 2 digit number = xy

y = x + 5 - - - (1)

10x + y = 4(x + y) - 6

10x + x + 5 = 4(x + x + 5) - 6

11x + 5 = 4(2x + 5) - 6

11x + 5 = 8x + 20 - 6

11x + 5 = 8x + 14

11x - 8x = 14 - 5

3x = 9

x = 9/3

x = 3

From (1)

y = x + 5

y = 3 + 5

y = 8

Answer:

Where is the question so that I can help you with it

I hope the answer is helpful

Answer:

(x - 3)^2 + (y - 6)^2 = 17

Step-by-step explanation:

The standard form of a circle's equation is (x-h)² + (y-k)² = r² where (h,k) is the center and r is the radius.

(x - 3)^2 + (y - 6)^2 = 17

Answer:

B. 9

Step-by-step explanation:

Proportions are just like fractions and to figure them out sometimes you can use simplification in different forms.

[tex]\frac{x}{6}[/tex] = [tex]\frac{36}{24}[/tex]

Now to get to 6, 24 had to be divided by 4...in proportions and fractions usually, the top and bottom are both simplified or proportioned to the same number or scale

36 ÷ 4 = 9

Check your answer by inserting it there to see if it works

9*4 = 36            6*4 = 24

Answer:

[tex]\boxed {\boxed {\sf B. \ 9}}[/tex]

Step-by-step explanation:

We are given this proportion:

[tex]\frac {x}{6}=\frac{36}{24}[/tex]

We want to solve for x, so we must isolate the variable using inverse operations.

It is being divided by 6. The inverse of division is multiplication, so we multiply both sides of the proportion by 6.

[tex]6*\frac {x}{6}=\frac{36}{24}*6[/tex]

[tex]x=\frac{36}{24}*6[/tex]

[tex]x=1.5*6 \\x=9[/tex]

Another way to solve is with cross multiplication. Multiply the first numerator by the second denominator, then the first denominator by the second numerator.

[tex]\frac { x}{6}=\frac{36}{24}[/tex]

[tex]24*x=6*36[/tex]

[tex]24x=216[/tex]

The variable is being multiplied by 24. The inverse of multiplication is division, so we divide both sides of the equation by 24.

[tex]24x/24=216/24\\x=9[/tex]

The value of x that makes this proportion true is 9.

Answer:

50

Step-by-step explanation:

this can be solved by ratio   15/x = 30/100.  Cross multiply and solve for x.  The answer is 50.

The main effects of the relevant variables are 3.5x and -1.6x, while the interaction effect is 0.9x*x.

The given model for manufacturing yield, expressed as a percentage of good products, is represented by the equation ỹ = 95 + 3.5x - 1.6x + 0.9x*x. In this equation, x represents the reactor temperature, and X2 represents the air pressure.

The coefficient 3.5 corresponds to the main effect of the reactor temperature, indicating that for each unit increase in temperature, the yield is expected to increase by 3.5 percentage points.

Similarly, the coefficient -1.6 represents the main effect of air pressure, implying that for each unit increase in pressure, the yield is expected to decrease by 1.6 percentage points.

Additionally, the term 0.9x*x accounts for the interaction effect between temperature and pressure. This suggests that the combined influence of temperature and pressure on the yield is not solely determined by the sum of their individual effects. Instead, the interaction effect captures the nonlinear relationship between these variables.

To estimate the probability of the yield being higher than 96% for a temperature of 315K and pressure of 130kPa, we need to evaluate the model equation for these specific values.

Substituting x = 315 and X2 = 130 into the equation, we can calculate the corresponding yield. If the yield exceeds 96%, the estimated probability would be 100%; otherwise, it would be 0%.

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The correct statement is : M spans P2.

The set M = {[tex]x+1, x^2-2, 3x[/tex]} consists of three polynomials in the variable x.

To determine whether M spans P3 or P2, we need to consider the highest degree of the polynomials in M.

The highest degree of the polynomials in M is 2 (from [tex]x^2-2[/tex]), which means that M can span at most the space of polynomials of degree 2 or less, i.e., P2.

To check whether M spans P2 or not, we need to see if any polynomial of degree 2 or less can be expressed as a linear combination of the polynomials in M.

We can write any polynomial of degree 2 or less as [tex]ax^2 + bx + c[/tex], where a, b, and c are constants.

To express this polynomial as a linear combination of the polynomials in M, we need to solve the system of equations:

[tex]a(x^2-2) + b(x+1) + c(3x) = ax^2 + bx + c[/tex]

This can be written as:

[tex]ax^2 + (-2a+b+3c)x + (b+c) = ax^2 + bx + c[/tex]

Equating the coefficients of [tex]x^2, x,[/tex] and the constant term, we get:

[tex]a = a,\\-2a+b+3c = b,\\b+c = c.[/tex]

The first equation is always true, and the other two equations simplify to:

[tex]-2a+3c = 0,\\b = 0.[/tex]

Solving for a, b, and c, we get:

[tex]a = 3c/2,\\b = 0,\\c = c.[/tex]

Therefore, any polynomial of degree 2 or less can be expressed as a linear combination of the polynomials in M. This means that M spans P2.

However, M cannot span P3, because P3 includes polynomials of degree 3, which cannot be expressed as a linear combination of the polynomials in M (since the highest degree polynomial in M is [tex]x^2[/tex]).

Therefore, the correct statement is: M spans P2.

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Answer:

x=-2

Step-by-step explanation:

I think that the answer would be two triangles.

Answer:

angle 1 = 124

angle 2 = 56

angle 3 = 124

angle 4 = 56

Step-by-step explanation:

This problem is a bit like a puzzle. To make notation easier I'm going to do this:

angle 1 = a

angle 2 = b

angle 3 = c

angle 4 = d

Now, let's start with what we know from the image.

All angles added together form a circle or 360 degrees

a + b + c + d = 360

In the same regard a + d = 180, and b + c = 180

Also,

a = c

and

b = d

It also tells us angle 4 is 25 degrees greater than one fourth of angle 1. Which is written as.

d = 1/4(a) + 25

If we look at all the equations we have, we can see that two of the equations have two of the same variables:

a + d = 180

and

d = 1/4(a) + 25

Using substitution we can take the second equation substitute it for d in the first equation giving us:

a + (1/4(a) + 25) = 180

Now we just solve for a

[tex]\frac{4}{4} a+ \frac{1}{4} a + 25 = 180\\\\\frac{5}{4}a + 25 = 180 \\\\(\frac{5}{4}a + 25) - 25 = (180) -25\\\\\frac{5}{4} a = 155\\\\\frac{5}{4} a * \frac{4}{5} = 155 * \frac{4}{5}\\\\a = 124[/tex]

Therefore a, or angle 1, is 124

Since a = c, then c, or angle 3, is also 124

Since a + d = 180 and a = 124 then

d = 180 -124

d = 56

So, d, or angle 4, is 56

And because b = d then b, or angle 2, is also 56

a = 124

b = 56

c = 124

d = 56

For extra measure, we can check our work by using the first equation

a + b + c + d = 360

124 + 56 + 124 + 56 = 360

Answer:

x = 12 the correct answer

Answer:

$2500

Step-by-step explanation:

First, multiply the $15 for the 150 people attending, to get 2250. Next, add the last $250 to get a total cost of $2500

Hopefully this helps- let me know if you have any questions!

Answer:

i think it is 2500

Step-by-step explanation:

take 15 times the amount of people. then add the 250 fee

Answer:The diagram that models the ratio in the story is:

Mermaid book: Princess book = 3:2

To find out how many times Nolan has read the mermaid book, we can set up the following proportion:

3/2 = x/20

Cross-multiplying, we get:

2x = 3 * 20

2x = 60

Dividing both sides by 2, we find:

x = 60/2

x = 30

Therefore, Nolan has read the mermaid book 30 times this month.

Answer:

B = 48.7°

C = 61.3°

b = 12

Step-by-step explanation:

Given:

A = 70°

a = 15

c = 14

Required:

B, C, and b

Solution:

✔️Using the law of sines, let's find C:

Sin C/c = Sin A/a

Plug in the values

Sin C/14 = Sin 70/15

Cross multiply

Sin C × 15 = Sin 70 × 14

Divide both sides by 15

Sin C = (Sin 70 × 14)/15

Sin C = 0.8770

C = Sin^{-1}(0.8770)

C = 61.282566° = 61.3° (nearest tenth)

✔️Find B:

B = 180 - (70 + 61.3) (sum of triangle)

B = 48.7°

✔️Find b using the law of sines:

b/sinB = a/sinA

Plug in the values

b/sin 48.7 = 15/sin 70

Cross multiply

b*sin 70 = 15*sin 48.7

Divide both sides by sin 48.7

b = (15*sin 48.7)/sin 70

b = 11.9921789

b = 12.0 (nearest tenth)

The spherical coordinates for the point (0, -3, -3) are (3√2, -π/2, π/4).

(a) To change from rectangular coordinates to spherical coordinates, we use the following formulas:

rho = √(x² + y² + z²)

theta = atan2(y, x)

phi = acos(z / rho)

Given the rectangular coordinates (5, 5√3, 10√3), we can substitute the values into the formulas to find the corresponding spherical coordinates:

rho = √((5)² + (5√3)² + (10√3)²)

= √(25 + 75 + 300)

= √(400)

= 20

theta = atan2(5√3, 5)

= atan(√3)

≈ 1.0472 radians

phi = acos((10√3) / 20)

= acos(√3 / 2)

= π/6 radians

Therefore, the spherical coordinates for the point (5, 5√3, 10√3) are (20, 1.0472, π/6).

(b) Given the rectangular coordinates (0, -3, -3), we can apply the formulas for spherical coordinates:

rho = √((0)² + (-3)² + (-3)²)

= √(0 + 9 + 9)

= √(18)

= 3√2

theta = atan2(-3, 0)

= -π/2 radians

phi = acos((-3) / (3√2))

= acos(-1/√2)

= π/4 radians

Hence, the spherical coordinates for the point (0, -3, -3) are (3√2, -π/2, π/4).

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Answer:

3/10 inches apart

Step-by-step explanation:

total miles based on 2" equals 33 miles can be found by solving this proportion:

2/33 = 6/x

2x = 198

x = 99

99 miles divided by 11 rest stops means each stop is 9 miles apart

now use new ratio of 1" equals 30 miles

1/30 = x/9

30x = 9

x = 9/30 or 3/10 inches

a. For the function f(x) = (x + x)(log(x) + 3x) is O(x) with the witnesses C = 7 and k = 1.

b. It is proved that f(x) = (x + x)(log(x) + 3x) is (x) with the witnesses C = 1 and k = 1.

c. The obtained witnesses C = 1, C' = 10, and k =[tex]10^C.[/tex] such that g(x)log(x) < Cg(x) whenever x > k. And g(x) = 1.

a) To show that f(x) = (x + x)(log(x) + 3x) is O(x),

find witnesses C and k such that f(x) ≤ C × x for all x > k.

Let's simplify the expression for f(x):

f(x) = 2x × (log(x) + 3x)

= 2x × log(x) + 6x²

Now, find a witness C and a value k such that f(x) ≤ C × x for all x > k.

Let's choose C = 7 and k = 1.

This means show that f(x) ≤ 7x for all x > 1.

For x > 1,

f(x) = 2x × log(x) + 6x²

< 2x × log(x) + 6x² + 7x

= 2x × log(x) + 6x² + 7x

= x(2log(x) + 6x + 7)

≤ x(2log(x) + 13x)

≤ x × 7

= 7x

This implies,

f(x) = (x + x)(log(x) + 3x) is O(x) with the witnesses C = 7 and k = 1.

b) To show that f(x) = (x + x)(log(x) + 3x) is (x),

find witnesses C and k such that f(x) ≥ C × x for all x > k.

Let us simplify the expression for f(x):

f(x) = 2x × (log(x) + 3x)

= 2x × log(x) + 6x²

Now, find a witness C and a value k such that f(x) ≥ C × x for all x > k.

Let us choose C = 1 and k = 1.

This means show that f(x) ≥ x for all x > 1.

For x > 1,

f(x) = 2x × log(x) + 6x²

> x × log(x) + 6x²

= x(log(x) + 6x)

≥ x(log(x) + x)

≥ x × log(x)

≥ x

Therefore, we have shown that f(x) = (x + x)(log(x) + 3x) is (x) with the witnesses C = 1 and k = 1.

c) To find the obtained witnesses C, C', and k .

such that g(x)log(x) < Cg(x) whenever x > k,

Examine the expression f(x) = (x + x)(log(x) + 3x) and determine the function g(x).

Let us simplify the expression for f(x),

f(x) = 2x × (log(x) + 3x)

= 2x × log(x) + 6x²

From the given condition, we have g(x)log(x) < Cg(x) rewrite this as,

log(x) < C

Since log(x) is an increasing function, if log(x) < C, it means x < [tex]10^C.[/tex]Therefore, the witness k is [tex]10^C.[/tex]

Now let us determine g(x).

Since g(x)log(x) appears in the inequality, we can take g(x) = 1.

C = 1, C' = 10, and k = [tex]10^C.[/tex] for g(x)log(x) < Cg(x) whenever x > k. And g(x) = 1.

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