1. Graph the function on the coordinate plane.

(a) What are the x-intercepts?
(b) What is the y-intercept?
(c) What is the maximum or minimum value?

The value of x in the algebraic equation is -1/2 and using system of linear equation to solve the problem, the cost of DVD and blue-ray are 84.3 and 178.6 respectively

This is a mathematical statement that represent an algebraic problem. It often two expression equating with one another.

8)

1/4 [(-2x - 7) / 8] = x / 2

Multiply through by 1/4

(-2x - 7) / 32 = x / 2

Cross multiply both sides

2(2x - 7) = 32x

4x - 14 = 32x

collect like terms

32x - 4x = - 14

28x = -14

x = -14 / 28

x = - 1/2

9)

To solve this problem, we should write a system of linear equations;

Let;

3x + 2y = 610 ...eq(i)

y - 10 = 2x ...eq(ii)

From equation (ii)

y = 2x + 10 ...eq(iii)

Put eq(iii) into equ(i)

3x + 2(2x + 10) = 610

3x + 4x + 20 = 610

x = 590/7 = 84.3

Put x = 84.3 into equ(ii)

y - 10 = 2(84.3)

y = 1250 / 7

y = 178.6

The cost of DVD is 84.3 and the cost of blue-ray is 178.6

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the total number of license plate is 17576000 and In this the number ways to place 'A' in step 1 is = '1'1, and the number of steps '0' in final steps id '1' thus the total answer is 67600

In mathematics, there are two alternative methods for dividing up a collection of items into subsets: combinations and permutations. Any order may be used by a combination to list the subset's elements. An ordered list of a subset's components is called a permutation.

There are [tex]26^3[/tex] = 17576, ways to perform step 1-3, and there are [tex]10^3[/tex] = 1000, ways to perform remaining steps .

the total number of license plate is 17576000

In this the number ways to place 'A' in step 1 is = '1'1, and the number of steps '0' in final steps id '1'

thus the total answer is 67600

26 X 25 X 24 X 10 X 9 X 8 = 11232000

24 X 10 X 9 X 8 = 17280

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Answer:

natural exponential

natural

quizlet

Answer:

b. natural exponential

c. natural

Step-by-step explanation:

Given function:

[tex]f(x)=e^x[/tex]

The given exponential function is called the:

The base e is called the:

The number "e" occurs naturally in math and the physical sciences.

It is the base rate of growth shared by all continually growing processes, and so is called the natural base.

It is an irrational number and named after the 18th century Swiss mathematician, Leonhard Euler, and so is often referred to as "Euler's number".

The cost of one chair is [tex]x=54.75[/tex] $ and the cost of one table is [tex]y=49.5[/tex] $.

What is the total cost?

The total cost formula is used to combine the variable and fixed costs of providing goods to determine a total. The formula is:

Total cost = (Average fixed cost x average variable cost)

Let cost of one chair is [tex]x[/tex]

Let cost of one table is [tex]y[/tex]

So,

[tex]5x+3y=31[/tex]       ...(1)

[tex]2x+6y=52[/tex]      ......(2)

Solving equation (1) and (2) we get

[tex]a_1x+b_1y+x_1=0\\\\a_2x+b_2y+c_2=0[/tex]

[tex]\frac{x}{b_1c_2-b_2c_1}=\frac{y}{c_1a_2-c_2a_1}=\frac{1}{a_1b_2-a_2b_1}\\\\\frac{x}{3(31)-6(52)}=\frac{y}{31(2)-52(5)}=\frac{1}{5(2)-2(3)}\\\\\frac{x}{93-312}=\frac{y}{62-260}=\frac{1}{10-6}\\\\\frac{x}{-219}=\frac{y}{-198}=\frac{1}{-4}\\\\x=54.75,y=49.5[/tex]

Hence, the cost of one chair is [tex]54.75[/tex] $ and the cost of one table is [tex]49.5[/tex] $.

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Answer:

sorry i messed up my first calculations, currently recalculating

Step-by-step explanation:

Answer:

Step-by-step explanation:

You want the number of $30 advance tickets and $35 same-day tickets sold, when a total of 55 tickets were sold for $1825.

Let s represent the number of same-day tickets sold. Then 55-s is the number of advance tickets sold, and total revenue is ...

  35s +30(55 -s) = 1825

  5s = 175 . . . . . . Simplify, subtract 1650

  s = 35

  55-s = 30

35 same-day and 30 advance tickets were sold.

the equilibrium constant for this reaction in terms of the equilibrium constants is Knet = √Ka/Kb

and for the reactions below:

2 S(s) + 3 O2(g) 2 SO3(g) is √Ka

S(s) + O2(g) SO2(g) is 1/Kb

A reversible reaction mechanism progresses from initial conditions to chemical equilibrium at a constant temperature, with all reactants and products present in constant non-zero amounts. At this temperature, an equilibrium constant "K" exists. It is frequently followed by a subscript to indicate the type of equilibrium constant that it considers depending on the phases of reaction species that it considers. The expression for each type of equilibrium constant differs with the direction and stoichiometry of the balanced reaction equation. This enables existing equations with known "K" values to be exploited in order to derive new "K" values for related systems.

SO2(g) + 1/2 O2(g)SO3 (g) is the desired reaction equation

We consider the first given reaction system at the bottom. We divide its coefficients by two to get the square root of the initial equilibrium constant: 2 S(s) + 3 O2(g) (g) 2 SO3(g) = √Ka

Then we reverse the second given reaction system at the bottom. Its current equilibrium constant is the reciprocal of the predecessor: S(s) + O2(g) SO2(g) is 1/Kb

The above two equations total to yield our target equation. The product of the two values obtained above yields the target equilibrium constant: Knet = √Ka/Kb

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Answer:

answer is 2.5

Step-by-step explanation:

Answer: 54

Step-by-step explanation: bc ? you just divid 54 by 3 and 6 then you divid 77 by 4 and 9 then divide them both tada

Answer:

sodas cost $5 and a large popcorn costs $8

create formulas out of the situations making the variable p=popcorn and s=soda

so,

3p + 6s = 54

and

4p + 9s = 77

then choose one variable to solve for

3p = 54 - 6s

divide both sides by 3

p = 18 - 2s

then plug your answer for p into the other equation:

4(18 - 2s) + 9s = 77

finally, solve for s

72 - 8s + 9s = 77

s = 5

if s=5 then

3p + 6(5) = 54

p = 8

Answer:

[tex]y = x^2 + 6x + 8[/tex]

Step-by-step explanation:

⭐What is the equation of a parabola in standard form?

⭐Whenever you factorise a quadratic (parabola), you are writing the parabola in terms of its x-intercepts.

⭐ Therefore, the expressions of the x-intercepts are the factors of your quadratic.

⭐What are factors of a number?

We have to reverse-engineer this problem to find the quadratic by:

1. Re-write the quadratic in terms of its factors

If one of the x-intercepts is -2, then the expression for said x-intercept is: [tex](x+2)[/tex]

If one of the x-intercepts is -4, then the expression for said x-intercept is:

[tex](x+4)[/tex]

2. Multiply the factors

[tex]y = (x+2)(x+4)[/tex]

[tex]y = x^2 + 4x + 2x + 8[/tex]

3. Simplify the products

[tex]y = x^2 + 6x + 8[/tex]

The weight of harsh on the moon if he weighs 54 pounds on earth is 9 pounds

Weight varies directly with gravity

Let

Weight of a person = w

Gravity = g

So,

w = k × g

Where,

k = constant of proportionality

If w = 180 pounds and g = 30 pounds

w = k × g

180 = k × 30

180 = 30k

divide both sides by 30

k = 180/30

k = 6

If w = 54 pounds g = ?

w = k × g

54 = 6 × g

54 = 6g

divide both sides by 6

g = 54/6

g = 9 pounds

Therefore, harsh weighs 9 pounds on the moon.

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The expression 10% of 40 is 4 and 10% more than 40 is 44

From the question, we have the following parameters that can be used in our computation:

10% of 40

Express "of" as products

So, we have

10% of 40 = 10% * 40

Evaluate

10% of 40 = 4

In this case, we have:

10% more than 40

This means that

10% more than 40 = 40 * (1 + 10%)

Evaluate

10% more than 40 = 44

Hence. 10% more than 40 is 44

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The difference between the model and the data will get smaller.

A sample is a percentage of the total population in statistics. You can use the data from a sample to make inferences about a population as a whole.

Given here, she 50 people and finds that only 75% and additional 100 people hence changing the sample size

As our sample size increases, the confidence in our estimate increases, our uncertainty decreases and we have greater precision. Thus with the increase in sample size, the data would move closer to the estimated probability.

Hence The difference between the model and the data will get smaller.

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Answer:

The difference between the data and the model will get smaller.

Step-by-step explanation:

Plato/Edmentum

The difference between Tom and Alfie is  1 kg

Let's call the weight of Tom T, the weight of Harry H, the weight of Freddie F, and the weight of Alfie A. We are given that T = H + 5, H = F + 3, and F = A - 2.

Substituting the second and third equations into the first equation gives us:

T = (A - 2) + 3 + 5

T = A - 2 + 3 + 5

T = A + 1

So the difference between Tom and Alfie is T - A = A + 1 - A = 1 kg.

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After integration, the required function f is (2x³ - sin (x) + Cx + D).

What is the integration of 'xⁿ' and 'sin (x)'?

[tex]\int {x^{n} } \, dx = \frac{x^{n+1} }{n+1} + C\\\\\\\int {sinx} \, dx = -cosx + C[/tex]

Given, f''(x) = 12x + sin x

Therefore,

[tex]\int {f''(x)} \, dx \\\\=\int {f'(x)} \, dx \\\\\\= \int{12x + sin x} \, dx + C\\\\= 6x^{2} - cosx + C\\[/tex]

Again, f'(x) = 6x - cos (x) + C

Therefore,

[tex]\int {f'(x)} \, dx\\ \\=\int {f(x)} \, dx \\\\= \int {6x^{2} - cosx + C } \, dx \\\\= 2x^{3} - sinx + Cx + D[/tex]

Therefore, the required function is (2x³ - sin (x) + Cx + D).

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Note that at any time during business hours, about 7 shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store. This is solved using Little Law.

Little's Law asserts that the long-term average number of individuals in a stable system, L, is equal to the long-term average effective arrival rate,λ, multiplied by the average duration a customer spends in the system, W.

To compute,

First, let's make sure that all the variables use the same time unit.

Thus, if there are 84 shoppers who are making a purchase per house, then there will be: 84/60 minutes

= 1.4

This means that  λ (rate) = 1.4 and the average time (W) is 5 minutes

Using Little's Law which states that the queuing formula is:

L = λW

Note that:

λ = 1.4 (computed)

W = 5 mintues (given)

Hence, the Average number of people in queue per time is:

L = 1.4 x 5

L = 7 shoppers.

Thus, it is correct to state that at any time during work hours, about 7 shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store.

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Full Question:

If shoppers enter a store at an average rate of r shoppers per minute and each stays in the store for an average time of T minutes, the average number of shoppers in the store, N, at any one time is given by the formula N=rT. This relationship is known as Little's law.

The owner of the Good Deals Store estimates that during business hours, an average of 3 shoppers per minute enter the store and that each of them stays an average of 15 minutes. The store owner uses Little's law to estimate that there are 45 shoppers in the store at any time.

Little's law can be applied to any part of the store, such as a particular department or the checkout lines. The store owner determines that, during business hours, approximately 84 shoppers per hour make a purchase and each of these shoppers spend an average of 5 minutes in the checkout line. At any time during business hours, about how many shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store?

The joint relative frequency for those students who don't like to run, but however enjoy cycling, is C. 32%.

The joint frequency for students who like cycling, but do not like running, can be found by the formula :
= Number of students who enjoy cycling but don't enjoy running / Number of students in total

Number of students who enjoy cycling but don't enjoy running = 64

Number of students in total = 200

The joint frequency is :

= 64 / 200 x 100%

= 0.32 x 100 %

= 32 %

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The {-4, 0, 1} represents members of the domain of the graphed function.

What is the domain of the function?

A function is a mathematical object that accepts input, appears to apply a rule to it, and returns the result.

A function can be thought of as a machine that requires in a number, performs some operation(s), and then outputs the result.

The domain of a function is the collection of all its inputs. Its codomain is the set of possible outputs.

The range refers to the outputs which are actually used.

Domain: {-4, 0, 1}.

Simply list the domain as -4 < x < 2, which would imply ALL values between -4 and 2 inclusive.    

Yes, this is a function. No x-values repeat, and it passes the Diagonal Line Test for functions.

Hence, the {-4, 0, 1} represents members of the domain of the graphed function.

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The polynomial function whose graph passes through (7,11) (11,-12) and (0,4) will be y = -0.614x² + 5.295x + 4.

A function is an assertion, concept, or principle that establishes an association between two variables. Functions may be found throughout mathematics and are essential for the development of significant links.

Assume the polynomial function is quadratic. Then the equation is given as,

y = ax² + bx + c

At (0, 4), we have

4 = a(0)² + b(0) + c

c = 4

Then the equation is written as,

y = ax² + bx + 4

At (7, 11), we have

y = ax² + bx + 4

11 = 49a + 7b + 4             ...1

At (11, -12), we have

- 12 = 121a + 11b + 4        ...2

Equations 1 and 2 are solved by a calculator. Then we have

a = - 0.614 and b = 5.295

The polynomial capability whose diagram goes through (7,11) (11,- 12) and (0,4) will be y = - 0.614x² + 5.295x + 4.

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Total  no. of cups of baking soda used for the experiment in improper fraction is [tex]\frac{91}{8}[/tex].

Initial no. of cups of baking soda used for an experiment = [tex]1\frac{3}{4}[/tex]

Converting mixed fraction to improper fraction,

[tex]1\frac{3}{4}=\frac{7}{4}[/tex]

No. of times the experiment performed by students = [tex]6\frac{1}{2}[/tex]

Converting mixed fraction to improper fraction,

[tex]6\frac{1}{2} = \frac{13}{2}[/tex]

Total  no. of cups of baking soda  =(Initial no. of cups)* (No. of times)

                                                        [tex]=\frac{7}{4} *\frac{13}{2} \\\\=\frac{91}{8} \\\\[/tex]

                                                        = 11.4 (decimal form)

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The Divergence Test is inconclusive because lim ak = ∞

The Divergence Test is used to determine whether a given infinite series converges or diverges.

In order to use the Divergence Test, we must examine the limit of the terms of the series as n approaches infinity.

In the given series, the limit of the terms as n approaches infinity is not defined. Therefore, the Divergence Test is inconclusive.

The Divergence Test is inconclusive because lim ak = ∞

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Answer: 7

Step-by-step explanation:

x=12-5

x=7

Answer:

x=7

Step-by-step explanation:

12-5=7 or 7+5=12

Given the set of data, we can calculate that the mean is 9.5, the variance is 17.58 and the standard deviation is 4.19

Mean, variance, and standard deviation

The concepts of mean, variance and standard deviation are very basic but very important in statistics.

Mean is the average of all data in a sample group, which is obtained by adding up all the data values, then dividing by the number of samples.

[tex]mean = \frac{sum all of the data}{size of data}[/tex]

Variance is a value that describes the variation of data, by measuring how far each piece of data is spread from the average of a data set.

[tex]variance = \frac{sum (x_{i} - mean)^{2} }{n}[/tex]

Standard Deviation is a measure of the spread of observations in a data set relative to their mean. it measures how many observations in a data set differ from the mean and is the square root of the variance.

σ = [tex]\sqrt{variance}[/tex]

Now we can calculate the mean of the given data as follows:

[tex]mean = \frac{sum all of the data}{size of data}[/tex]

         = (3 + 6 + 8 + 12 + 13 + 15) / 6

         = 57 / 6

         = 9.5

Then we can calculate the variance as follows:

[tex]variance = \frac{sum (x_{i} - mean)^{2} }{n}[/tex]

              = [tex]\frac{((3-9,5)^{2} + (6-9,5)^{2} + (8-9,5)^{2} + (12-9,5)^{2} + (13-9,5)^{2} + (15-9,5)^{2} )}{6}[/tex]

              = (42,25 + 12,25 + 2,25 + 6,25 + 12,25 + 30,25) / 6

              = 105.5 / 6

              = 17.58

Standard deviation can be calculated from the variance:

σ = [tex]\sqrt{variance}[/tex]

  = [tex]\sqrt{17.58}[/tex]

  = 4.19

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Answer:

Step-by-step explanation:

A.) To show that triangle ABC is similar to triangle DEC, we need to prove that the ratios of the sides of the two triangles are equal.

First, we can write the ratios of the sides of triangle ABC as follows:

AC/AB = 20/16 = 5/4

BC/AB = 24/16 = 3/2

Now, we can write the ratios of the sides of triangle DEC as follows:

CE/CD = 18/15 = 6/5

AC/CD = 20/15 = 4/3

Since the ratios of the sides of the two triangles are equal, it follows that triangle ABC is similar to triangle DEC.

B.) To find the length of DC, we can use the fact that triangle ABC is similar to triangle DEC. Since the ratios of the sides of the two triangles are equal, we can set up a proportion to solve for DC.

First, we can write the ratio of the sides of triangle ABC as follows:

AC/AB = DC/CE

Then, we can substitute the known values for AC, AB, and CE:

20/16 = DC/18

Then, we can cross-multiply to solve for DC:

DC = (20/16) * 18

= (5/4) * 18

= 45/4

= 11.25 cm

Therefore, the length of DC is approximately 11.25 cm.

Answer:

Step-by-step explanation:

To show that triangle ABC is analogous to triangle DEC, we need to prove that the rates of the sides of the two triangles are equal.

First, we can write the rates of the sides of triangle ABC as follows

AC/ AB = 20/16 = 5/4

BC/ AB = 24/16 = 3/2

Now, we can write the rates of the sides of triangle DEC as follows

CE/ CD = 18/15 = 6/5

AC/ CD = 20/15 = 4/3

Since the rates of the sides of the two triangles are equal, it follows that triangle ABC is analogous to triangleDEC.

B.) To find the length of DC, we can use the fact that triangle ABC is analogous to triangle DEC. Since the rates of the sides of the two triangles are equal, we can set up a proportion to break for DC.

First, we can write the rate of the sides of triangle ABC as follows

AC/ AB = DC/ CE

also, we can substitute the known values for AC, AB, and CE

20/16 = DC/ 18

also, we cancross-multiply to break for DC

DC = (20/16) * 18

= (5/4) * 18

= 45/4

= 11.25 cm

Thus, the length of DC is roughly11.25 cm.

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Answer:

15.5

Step-by-step explanation:

Pepe= 2/3 x 12 = 8

Paula= 1/4 x 30 = 7.5

8+7.5=15.5

Answer:

$32.70 ÷ 5

Each case is $6.54.

Step-by-step explanation:

$3270 for 5 cases, meaning you would split $32.70 into 5.

$32.70 ÷ 5 phone cases = $6.54

This also means each case is $6.54. To prove this, multiply by 5.

A fraction means a part of something or a number of parts of something. The number on the bottom shows how many parts something has been divided into ½ means 1 part of something that has been divided into 2 parts. We call this a half. If you add two halves together you get one.

Answer: A fraction is a numerical quantity that is not a whole number

Step-by-step explanation:

The simple formula or rule that generates the terms of an integer sequence that begins with the given list is given below.

Sequence is the order in which elements, such as numbers, letters, or symbols, follow one another in a specific order. It is a fundamental concept in mathematics and computer science, and is used in many other areas including music, language, and art. In mathematics, it is often used to describe patterns or to represent numerical values. In computer science, it is used to represent data structures, such as lists and arrays.
a. The sequence can be expressed as 3x2^n, where n is the position of the number in the sequence. The next three terms are 127, 162, and 201.

b. The sequence can be expressed as 4n+3, where n is the position of the number in the sequence. The next three terms are 47, 51, and 55.

c. The sequence can be expressed as 2^n+n-1, where n is the position of the number in the sequence. The next three terms are 1111, 10000, and 10011.

d. The sequence can be expressed as the number of times the previous value is repeated, starting with 1. The next three terms are 5, 5, and 8.

e. The sequence can be expressed as 4x3^n, where n is the position of the number in the sequence. The next three terms are 5906, 17718, and 53154.

f. The sequence can be expressed as (n+1)!+(n-1)!+2, where n is the position of the number in the sequence. The next three terms are 27027025, 94594500, and 34459425.

g. The sequence can be expressed as the number of times the previous number is repeated, starting with 1. The next three terms are 0, 0, and 1.

h. The sequence can be expressed as 4^n, where n is the position of the number in the sequence. The next three terms are 4294967296, 17179869184, and 68719476736.

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Complete questions as follows-
For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence.
a. 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, . . .
b. 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, . . .
c. 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, . . .
d. 1, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, . . .
e. 0, 2, 8, 26, 80,242,728,2186,6560,19682, . . .
f. 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, . . .
g. 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, . . .
h. 2, 4, 16, 256, 65536, 4294967296, . . .

f cannot have a jump discontinuity at [tex]$x_0 \in(a, b)$[/tex] and  [tex]$$ \lim _{x \uparrow x_0} f(x) < \lim _{x \mid x_0} f(x) .$$[/tex]

f cannot have a removable discontinuity at [tex]$$x_0 \in(a, b) $$[/tex] and [tex]\alpha=\lim _{x \rightarrow x_0} f(x) < f\left(x_0\right)[/tex]

Let f be a function defined on (a, b) satisfies intermediate value property.

Claim: f ca not have removable on jump discontinuity.

Suppose f has a jump discontinuity at [tex]$x_0 \in(a, b)$[/tex]

We take [tex]$\theta$[/tex] such that

[tex]$$\lim _{x \rightarrow x_0} f(x) < \theta < \lim _{x \downarrow x_0} f(x) \text { and } \theta \neq f\left(x_0\right)$$[/tex]

Now there exist [tex]$\delta > 0$[/tex] such that [tex]$f(x) < \theta$[/tex] for all [tex]$x \in\left[x_0-\delta, x_0\right)$[/tex] and [tex]$f(x) > \theta$[/tex] for all [tex]$x \in\left(x_0, x_0+\delta\right]$[/tex]

Now [tex]$f\left(x_0-\delta\right)[/tex][tex]< \theta < f\left(x_0+\delta\right)$[/tex] for all [tex]$x \in\left[x_0-\delta, x_0+\delta\right] \backslash\left\{x_2\right\}$[/tex] and [tex]$f\left(x_0\right) \neq \theta$[/tex].

Therefore the point [tex]$\theta$[/tex] has no preimage under f

that is, there does not exists [tex]$y \in\left[x_0-\delta, x_0+\delta\right][/tex] for which

[tex]$$f(y)=\theta[/tex] because [tex]\left\{\begin{array}{l}y=x_0 \Rightarrow f(y) \neq \theta \\y > x_0 \Rightarrow f(y) > \theta \\y < x_0 \Rightarrow f(y) < \theta\end{array}\right.$$[/tex]

Therefore f does not satisfies intermediate value property on [tex]$\left[x_0-\delta, x_0+\delta\right]$[/tex],

Hence f does not satisfies IVP on (a, b) which is not possible because we assume f satisfies IVP on (a, b),

Therefore f can not have a jump discontinuity.

Suppose f has a removable point of discontinuity at [tex]$x_0 \in(a, b)$[/tex],

Let [tex]$\alpha=\lim _{\alpha \rightarrow x_0} f(x)$[/tex],

Let [tex]\alpha < f\left(x_0\right)$[/tex] so [tex]$f\left(x_0\right)-\alpha > 0$[/tex].

Now [tex]$\lim _{x \rightarrow x_0} f(x)=\alpha$[/tex] then [tex]\exists$ \delta > 0$[/tex] such that

[tex]$$\begin{aligned}& |f(x)-\alpha| < \frac{f\left(x_0\right)-\alpha}{2} \text { for all } x \in\left\{x_0-\delta, x_0-\alpha\right]-\left\{x_0\right\} \\& \Rightarrow \quad f(x) < \alpha+\frac{f\left(x_0\right)-\alpha}{2} \text { for all } x \in\left[x_0-\delta, x_0+\delta\right]-\left\{x_0\right\}\end{aligned}$$[/tex]

So [tex]$f(x) < \frac{f\left(x_0\right)+\alpha}{2}$[/tex] for all [tex]$x \in\left[x_0-\delta, x_0\right]-\left\{x_0\right\}$[/tex]

Now [tex]$f\left(x_0\right) > \alpha$[/tex].

And  [tex]$f(x) < \frac{f\left(x_0\right)+\alpha}{2} < f\left(x_0\right)$[/tex] for all [tex]$x \in\left[\left(x_0 \delta, x_0\right)\right.$[/tex]

Let [tex]$\mu=\frac{f\left(x_0\right)+\alpha}{2}$[/tex].

Then there does not exist [tex]$e \in\left[x_0-\delta, c\right]$[/tex] such that [tex]$f(c)=\mu$[/tex]

Because for [tex]$e=x_0 \quad f(e) > \mu$[/tex]

                for [tex]$c < x_0 \quad f(c) < \mu$[/tex].

Therefore f does not satisfy IVP on [tex]$\left[x_0-\delta_1 x_0\right]$[/tex] which contradict our hypothesis,

therefore [tex]$\alpha \geqslant f\left(x_0\right)$[/tex]

Let [tex]$\alpha > f\left(x_0\right)$[/tex]. so [tex]$\alpha-f\left(x_0\right) > 0$[/tex]

[tex]$\lim _{x \rightarrow x_0} f(x)=\alpha$[/tex]

Then [tex]\exists $ \varepsilon > 0$[/tex] such that

[tex]$|f(x)-\alpha| < \frac{\alpha-f\left(x_0\right)}{2}$[/tex] for all [tex]$\left.x \in\left[x_0-\varepsilon_0 x_0+\varepsilon\right]\right\}\left\{x_i\right\}$[/tex]

[tex]$\Rightarrow f(x) > \alpha-\frac{\alpha-f\left(x_0\right)}{2}$[/tex] for all [tex]$x \in\left[x_0-\varepsilon_1, x_0+\varepsilon\right] \backslash\left\{x_0\right\}$[/tex]

[tex]$\Rightarrow f(x) > \frac{\alpha+f\left(x_0\right)}{2}$[/tex] for all [tex]$x \in\left[x_0-\varepsilon_1, x_0\right)$[/tex]

Now [tex]$f\left(x_0\right) < \alpha$[/tex]

The [tex]$f(x) > \frac{f\left(x_0\right)+\alpha}{2} > f\left(x_0\right)$[/tex].

So [tex]$f\left(x_0\right) < \frac{f\left(x_0\right)+\alpha}{2} < f(x)$[/tex] for all [tex]$x \in\left[x_0 \varepsilon, \varepsilon_0\right)$[/tex]

Let [tex]$\eta=\frac{f\left(x_e\right)+\alpha}{2}$[/tex]

Then there does not exist [tex]$d \in\left[x_0-\varepsilon, x_0\right]$[/tex] such that [tex]$f(d)=\xi$[/tex].

Because if [tex]$d=x_0, f(d)=f\left(x_0\right) < \eta$[/tex] if [tex]$d E\left[x_0-\varepsilon, x_0\right)$[/tex]

Then [tex]$f(d) > \eta$[/tex]

Therefore f does not satisfies IVP on [tex]$\left[x_0-\varepsilon, x_0\right]$[/tex] which contradict olio hypothesis.

Therefore [tex]$\alpha \leq f\left(x_0\right)$[/tex] (b) From (a) and (b) it follows [tex]$\alpha=f\left(x_0\right)=\lim _{x \rightarrow x_0} f(x)$[/tex]. Therefore f can not have a removable discontinuous

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The distance the duck fly in all is 99 miles

From the question, we have the following parameters that can be used in our computation:

The distance covered in all can be calculated as

Distance = The sum of the product of speed and time

Substitute the known values in the above equation, so, we have the following representation

Distance = 18 * 3 + 15 * 2

Evaluate the products

This gives

Distance = 54 + 45

Evaluate the sum

Distance = 99 miles

Hence, the distance is 99 miles

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