help please.............​

(a) To prove that there exists a neighborhood W of the origin such that f(x) ≤ y for all x ∈ W, given that f is bounded above on a neighborhood V of the origin, we can use the linearity of f.

Since f is a linear function, it satisfies the following properties:

f(0) = 0

f(rx) = rf(x) for any scalar r and vector x

f(x + y) = f(x) + f(y) for any vectors x and y

Given that f is bounded above on V, there exists a positive number M such that f(x) ≤ M for all x ∈ V. Now, let's consider the neighborhood W defined as follows:

W = {x ∈ X | ||x|| < M}

We claim that for any x ∈ W, f(x) ≤ y.

Let x ∈ W. Since x is in the neighborhood W, we have ||x|| < M. By linearity, we can express x as x = rx' for some scalar r and vector x' with ||x'|| = 1.

Now, consider f(x):

f(x) = f(rx') = rf(x')

Since ||x'|| = 1, we have ||rx'|| = |r| ||x'|| = |r|.

Therefore, ||rx'|| < M implies |r| < M.

Using the fact that f is bounded above on V, we have f(x') ≤ M.

Combining these results, we get:

|f(x)| = |rf(x')| = |r| |f(x')| ≤ M

Since this inequality holds for any x ∈ W and |r| < M, we have shown that f(x) ≤ y for all x ∈ W, where W is a neighborhood of the origin.

(b) To prove that f is continuous, we can show that f is bounded above on any compact set in X. Let K be a compact set in X.

Since K is compact, it is also closed and bounded. By the linearity of f, we have:

f(K) = {f(x) | x ∈ K}

Since K is bounded, there exists a positive number M such that ||x|| ≤ M for all x ∈ K. By the linearity of f, we have:

f(K) = {f(x) | x ∈ K} ⊆ {f(x) | ||x|| ≤ M}

Thus, f(K) is bounded above by M.

By the previous result in part (a), if f is bounded above on a neighborhood of the origin, then it is bounded above on any neighborhood of the origin. Therefore, f is bounded above on the neighborhood V of the origin.

Since K is compact, it can be covered by finitely many neighborhoods of the origin, say V1, V2, ..., Vk. Thus, f is bounded above on each Vi, i = 1, 2, ..., k.

Now, consider the open cover {V1, V2, ..., Vk} of K. By compactness, there exists a finite subcover {V1, V2, ..., Vm}. Therefore, f is bounded above on K.

Since f is bounded above on any compact set K, it follows that f is continuous.

(c) The previous part (b) already proves that if f is bounded above on any compact set, it is continuous. Therefore, if f is bounded above on a set 2 with int(2) ≠ Ø (i.e., the interior of 2 is not empty), then f is continuous.

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Step-by-step explanation:

-.5f -5 < -1

-.5f < 4

f > -8

hope this helps

The required differential equation of the tractrix is dy/dx = (2y - y²)/s. This is obtained by substituting s = x/cos x into the differential equation 2y dy/dx - y² = 0 and simplifying. The differential equation dy/dx = (2y - y^2)/s is solved as shown below:

Solving the given differential equation In the given figure, let (x, y) be the coordinates of the point P. The weight is at (0, s) and the length of the rope is s. At a point P(x, y) on the tractrix, the tangent to the curve is parallel to the x-axis.The slope of the tangent is given by the differential coefficient dy/dx. We can determine dy/dx in terms of x and y by differentiating y^2 + x^2 = s^2 using implicit differentiation to get 2y dy/dx + 2x = 0.Differentiating again, we have d²y/dx² + y = 0.This differential equation is a second-order linear differential equation, with characteristic equation r² + 1 = 0. This yields r = ±i. Therefore, the general solution is y = c1cos x + c2 sin x, where c1 and c2 are constants. To find c1 and c2, we use the given initial condition that y = s when x = 0. This gives c1 = s and c2 = 0. Therefore, the solution to the differential equation is: y = s cos x. We can now eliminate x from the equation x² + y² = s² by substituting y = s cos x to obtain x² + s² cos² x = s². Solving for s, we have:s = x/cos x. Substituting this expression into the differential equation 2y dy/dx - y² = 0 yields the following equation:dy/dx = (2y - y²)/s This is the required differential equation of the tractrix.

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Answer:

Step-by-step explanation:

Answer:

I don't see an "x" or an equation-

The week se will give a 12-minute speech is week 22 (option A)

The table is a linear table. This is because the variables change by a fixed amount.

Rate of change = change in length of speech / change in week

(180 - 150) / (4 - 3)

= 30 / 1 = 30 seconds  

The next step is to convert minutes to seconds

1 minute = 60 seconds

60 x 12 = 720 seconds

Length of speech in week 2 = 150 - 30 = 120 seconds

Length of speech in week 1 = 120 - 30 = 90

Week the speech would have a length of 720 seconds = 90 +[ 30 x (week number - 1)]

720 = 90 + [30 x (x -1)

720 = 90 + 30x - 30

720 = 60 + 30x

720 - 60 = 30x

660 = 30x

x = 660 / 30

x = 22

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Answer:

I got 38 and 11

Step-by-step explanation:

Answer: -32

Step-by-step explanation: -14 - 6 - 12 = -32

Answer:

772 hamburgers sold on monday

Step-by-step explanation:

add the problem

Answer: 12.5

Step-by-step explanation:

So, basically there is 5 minutes and the number of times occuring is 5 so its 5x1 which is 5 and then there is 10 minutes and the number of times occuring  is twice so its 2x10 which is 20. Next, is 15 minutes and the times occuring  is 2 so, 2x15 which is 30. And finally theres  20 minutes and the amount of times occuring  is once so 20x1 which is 20. Then you add them all up which is 5+20+30+20=75. And the formula for mean/average is The sum of all number/ The amount of numbers there are so 75/6 is 12.5.

Answer:

B

Step-by-step explanation:

9A:

The problem says 35% of the 3560 applications are from boys who lived in other states.

This can be expressed as:

3560*35% = 3560*0.35 = 1246 applications.

9B:

The problem says applications to the university (3560 applications) represented 40% of all applications.

This can be expressed as:

3560 = 40% * A, where A = number of applications received in all.

To find how many applications received in all, just solve for A.

A = 3560 / 0.4 = 8900 applications.

The key to these types of problems is that "of" signals multiplication. For example, 40% of all applications is 40% * all applications.

Answer:

Step-by-step explanation:

Answer:

its 3, -6,-1... got 100%

Step-by-step explanation:

Answer:

bit.[tex]^{}[/tex]ly/3a8Nt8n

Step-by-step explanation:

Answer:

v^ 3 = 1000

so v = cube root of 1000

    v = 10

Answer:

24 days

Step-by-step explanation:

Given

[tex]Pages = 528[/tex]

[tex]Rate = 22[/tex]

Required

The number of days

The number of days (d) is calculated as:

[tex]d = \frac{Pages}{Rate}[/tex]

[tex]d = \frac{528}{22}[/tex]

[tex]d = 24[/tex]

Answer:

4 horas para compatir com su familia

Answer:

Option B

Step-by-step explanation:

→ Find 2 coordinates through which the line passes through

( -2 , -5 ) and ( 0 , -1 )

→ Work out the gradient

[tex]\frac{-1--5}{0--2} =2[/tex]

→ Write into y = mx + c

y = 2x + c

→ Substitute in the coordinates ( 0 , -1 )

-1 = 0 + c so c = -1

→ Write equation

y = 2x - 1

→ Expand out all the equations and simply to see which gives y = 2x - 1

Option B

Answer:

16ft by 16 ft: NO

10ft by 15ft: YES

15ft by 20ft: NO

12ft by 16ft: YES

Step-by-step explanation:

First, we know that the scale used is:

1 in = (1 + 1/2) ft.

This means that each inch on the drawing is equivalent to (1 + 1/2) ft.

We know that Michael uses a paper that has the measures:

width = (8 + 1/2) in

length = 11in

Then the maximum dimensions that can be represented with this paper are:

WIDTH = (8 + 1/2)*(1 + 1/2) ft. = (8 + 8/2 + 1/2 + 1/4) ft

            = (8 + 4 + 2/4 + 1/4) ft

            = (12 + 3/4) ft

LENGTH = 11*(1 + 1/2) ft = (11 + 11/2)ft = (11 + 10/2 + 1/2)ft

               = (11 + 5 + 1/2)ft = (16 + 1/2) ft.

Now let's analyze the options, we can only draw the rooms in the paper if the measures are equal or smaller than the ones we found above:

Where the measures are written as: "width by length".

a) 16ft by 16 ft.

width = 16ft

length = 16ft

We can not draw this, because the maximum width that we can draw is (12 + 3/4) ft, which is smaller than 16ft.

b) 10 ft by 15 ft

width = 10ft

length = 15ft

Both are smaller than the maximum measures we found, then yes, we can draw this room.

c) 15 ft by 20 ft

width = 15ft

length = 20ft

Both are larger than the maximum measures, so no, we can not draw this.

d) 12ft by 16ft

width = 12ft < (12 + 3/4) ft = maximum width

lenth = 16ft < (16 + 1/2) ft = maximum length.

Both measures are smaller than the maximum ones, then we can draw this one

Based on the information given, it should be noted that the critical value is +2.33.

We are given that the sample size is 42, the sample proportion is 0.247, and the significance level is 0.05. We want to test the claim that the population proportion is 0.325.

The critical value is the z-score that separates the rejection region from the non-rejection region. The rejection region is the area under the standard normal curve where we would reject the null hypothesis. The non-rejection region is the area under the standard normal curve where we would fail to reject the null hypothesis.

The z-table shows that the critical value for a two-tailed test with a significance level of 0.05 is +2.33. This means that if the z-score is greater than or equal to +2.33, we would reject the null hypothesis.

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Answer:

A

Step-by-step explanation:

A secant is a line that goes through a circle at two points.

Answer:

A

Step-by-step explanation:

a straight line that cuts a curve in two or more parts.

Answer: It would be "290 in.2"

Step-by-step explanation:

The curvature of the function y = x³ at x = 1 is 6

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

y = x³

To start with, we need to differentiate the function

So, we have

y' = 3x²

Next, we differentiate the function for the second time to calculate the curvature of the function

So, we have

y'' = 6x

The value of x is 1

So, we have

y'' = 6(1)

Evaluate

y'' = 6

Hence, the curvature of the function y = x³ at x = 1 is 6

See attachment for the graph

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This question is incomplete, the complete question is;

Alicia would like to know if there is a difference in the average price between two brands of shoes. She selected and analyzed a random sample of 40 different types of Brand A shoes and 33 different types of Brand B shoes. Alicia observes that the boxplot of the sample of Brand A shoe prices shows two outliers. Alicia wants to construct a confidence interval to estimate the difference in population means.

Is the sampling distribution of the difference in sample means approximately normal?

a) Yes, because Alicia selected a random sample

b) Yes, because for each brand it is reasonable to assume that the population size is greater than ten times its sample size.

c) Yes, because the size of each sample is at least 30

d) No, because the distribution of Brand A shoes has outliers

e) No, because the shape of the population distribution is unknown.

Answer:

the sampling distribution of  the difference in sample means is approximately normal because the size of each sample is al least 30

Option (c) Yes, because the size of each sample is at least 30 ) is the correct answer.

Step-by-step explanation:

Given the data in the question;

sample size of brand A shoes [tex]n_A[/tex] = 40

sample size of brand B shoes [tex]n_B[/tex] = 33

As we can notice,

[tex]n_A[/tex] = 40 > 30

[tex]n_B[/tex] = 33 > 30

Hence, we consider both samples to be large.

Therefore, the sampling distribution of  the difference in sample means is approximately normal because the size of each sample is al least 30.

Option (c) Yes, because the size of each sample is at least 30 ) is the correct answer.

The largest two-digit integer a = 99 satisfies the congruence 3x ≡ 4

To obtain the largest two-digit integer that satisfies the congruence 3x ≡ 4 (mod 7), we need to find an integer value for x that satisfies the congruence equation.

First, we can rewrite the congruence as an equation:

3x = 4 + 7k, where k is an integer.

Next, we can iterate through two-digit integers in descending order to find the largest value of a that satisfies the equation.

Starting with a = 99, we substitute it into the equation:

3(99) = 4 + 7k

297 = 4 + 7k

By trying different values of k, we can see that k = 42 satisfies the equation:

297 = 4 + 7(42)

Therefore, x = 99 is a solution to the congruence equation 3x ≡ 4 (mod 7).

However, we need to find the largest two-digit integer, so we continue the iteration.

Next, we try a = 98:

3(98) = 4 + 7k

294 = 4 + 7k

By trying different values of k, we can see that k = 42 also satisfies the equation:

294 = 4 + 7(42)

Therefore, x = 98 is another solution to the congruence equation 3x ≡ 4 (mod 7).

Since we have found the largest two-digit integer a = 99 that satisfies the congruence, we can conclude that a = 99 is the answer to the problem.

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Answer:

37

Step-by-step explanation:

Answer:

The answer is 16.99

Answer:

It's 17!

Step-by-step explanation:

Answer:

A- In one simulated SRS of 100 students, the average GPA was  = 3.45.

Step-by-step explanation:

Just took the assignment myself, but here's a bit of insight to help anyone later on!

The x-bar means sample statistic. Since it was one sample taken out of the dot plot, 3.45 is a sample statistic.

Answer:

b. y=19x

Step-by-step explanation:

Answer:

Her results changed because the power increased due to the fact that she reduced variability in her data as a result of using a sample that had lesser variability.

Step-by-step explanation:

She is trying to find if there is difference between people who consumed 1 versus 2 beverages.

Now initially she surveyed 40 people between ages 18 to 38 years and of huge weights and found out there was no difference. However, she decided to run the survey again on only women aged 18 to 22 years with normal body weight.

Since she used a lesser sample size and surveyed only women, it means there will be lesser variability in the result as the sample is smaller and streamlined to only women.

Thus, her result changed because the power increased due to the fact that she reduced variability in her data as a result of using a sample that had lesser variability.