To determine the intervals of increasing and decreasing for the function f(x) = sin(x)cos(x) + 9 on the interval (0, 2π), we need to analyze the sign of the derivative.
First, let's find the derivative of f(x) using the product rule:
f'(x) = (cos(x)cos(x) - sin(x)sin(x)) + 0
= cos^2(x) - sin^2(x)
Now, we can analyze the sign of the derivative to determine the intervals of increasing and decreasing:
For f'(x) = cos^2(x) - sin^2(x):
When cos^2(x) > sin^2(x), the derivative is positive, indicating an increasing interval.
When cos^2(x) < sin^2(x), the derivative is negative, indicating a decreasing interval.
When cos^2(x) = sin^2(x), the derivative is zero, indicating potential extrema.
To find the critical points where the derivative is zero, we solve the equation cos^2(x) - sin^2(x) = 0:
cos^2(x) = sin^2(x)
Taking the square root of both sides:
cos(x) = ±sin(x)
The critical points occur at x = π/4, 3π/4, 5π/4, and 7π/4.
Now, we can summarize the information as follows:
(a) The function is increasing on the intervals (0, π/4), (5π/4, 2π).
The function is decreasing on the intervals (π/4, 3π/4), (7π/4, 2π).
(b) To identify the relative extrema, we can apply the First Derivative Test by checking the sign of the derivative in the intervals around the critical points.
For x = π/4:
To the left of π/4, the derivative is positive (+), indicating a relative minimum.
To the right of π/4, the derivative is negative (-), indicating a relative maximum.
For x = 3π/4:
To the left of 3π/4, the derivative is negative (-), indicating a relative maximum.
To the right of 3π/4, the derivative is positive (+), indicating a relative minimum.
For x = 5π/4:
To the left of 5π/4, the derivative is positive (+), indicating a relative minimum.
To the right of 5π/4, the derivative is negative (-), indicating a relative maximum.
For x = 7π/4:
To the left of 7π/4, the derivative is negative (-), indicating a relative maximum.
To the right of 7π/4, the derivative is positive (+), indicating a relative minimum.
So, the relative extrema are:
Relative maxima: (x, y) = (π/4, f(π/4)), (7π/4, f(7π/4))
Relative minima: (x, y) = (3π/4, f(3π/4)), (5π/4, f(5π/4))
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Consider the region given by x <= 1, x+y >= 8, -2x+y-z <=2 x,y,z >= 0
(a) Formulate the problem using slack variables s₁ ≥ 0, s2 ≥ 0 and s3 ≥ 0. Write down the basic solution obtained by choosing the slack variables as pivots. Enter in the form (x, y, z, s1, s2, s3). ____
l(b) Is this a feasible solution? O True O False (c) Write down the basic solution obtained by choosing z, s1, s2 as pivots. Enter in the form (x, y, z, s1, s2, s3) _____ (d) Is this a feasible solution? O True O False (e) Write down the basic solution obtained by choosing x, y, z as pivots. Enter in the form (x, y, z, s1, s2, s3) _____
(f) Is this a feasible solution? O True O False
The basic solution obtained by choosing the slack variables as pivots is (0, 8, 2, 1, 0, 0). This is a feasible solution because all the variables are non-negative.
The region given by x <= 1, x+y >= 8, -2x+y-z <=2 x,y,z >= 0 can be formulated as the following linear programming problem:
maximize z
subject to
x <= 1
x+y >= 8
-2x+y-z <=2
x,y,z >= 0
The slack variables s₁ ≥ 0, s2 ≥ 0 and s3 ≥ 0 can be used to convert the inequalities to equalities. This gives the following system of equations:
x + s₁ = 1
x + y + s₂ = 8
-2x + y - z + s₃ = 2
The basic solution obtained by choosing the slack variables as pivots is the solution to this system of equations with s₁, s₂, and s₃ set to zero. This gives the solution (0, 8, 2, 1, 0, 0).
This solution is feasible because all the variables are non-negative.
The basic solution obtained by choosing z, s1, s2 as pivots is the solution to this system of equations with x, y, and s₃ set to zero. This gives the solution (0, 0, 0, 0, 1, 8).
This solution is not feasible because z is non-positive.
The basic solution obtained by choosing x, y, z as pivots is the solution to this system of equations with s₁, s₂, and s₃ set to zero. This gives the solution (1, 0, 0, 0, 0, 0).
This solution is feasible because all the variables are non-negative
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Let A be an m×n matrix. Prove that every vector x in R^n can be written in the form x=p+u, where p is in Row A and u is in Nul A. Also, show that if the equation Ax=b is consistent, then there is a unique p in Row A such that Ap=b Because Row A is a subspace of R^n, the ____ written as x= x^ +z wh The orthogonal comp That is, (Row A)
In linear algebra, it can be proven that for any matrix A, every vector x in R^n can be expressed as the sum of a vector p in the row space (Row A) of A and a vector u in the null space (Nul A) of A.
To prove that every vector x in R^n can be written in the form x=p+u, where p is in Row A and u is in Nul A, we need to consider the fundamental theorem of linear algebra. According to the theorem, the row space of A is orthogonal to the null space of A. Therefore, any vector x can be decomposed into two components: one in the row space (p) and the other in the null space (u).
If the equation Ax=b is consistent, it means that there exists a vector x that satisfies the equation. In this case, we can express b as Ap, where p is a vector in the row space of A. Since the row space is a subspace of R^n, there is a unique vector p that satisfies Ap=b.
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Simplify the expressions, give answers with positive exponents: a) [(x^6y^-3)/(27y^3/5)^-1/3 (3 marks) b) [a^3/2 / b^-1/2 (a-²/b³) (3 marks)
a) the expression [(x^6y^-3)/(27y^3/5)^-1/3], we can apply the rules of exponents:
Step 1: Simplify the denominator of the fraction inside the parentheses.
(27y^3/5)^-1/3 = (27^(-1/3) * y^(3/5))^(-1/3) [using the property (ab)^c = a^c * b^c]
= 27^(-1/3 * -1/3) * y^(3/5 * -1/3) [using the property (a^b)^c = a^(b*c)]
= 27^(1/9) * y^(-1/15) [multiplying the exponents]
= (3^3)^(1/9) * y^(-1/15) [expressing 27 as 3^3]
= 3^(3/9) * y^(-1/15) [raising a power to a power]
Step 2: Simplify the numerator and rewrite it with positive exponents.
x^6y^-3 = x^6 * (1/y^3) [since y^-3 is equivalent to 1/y^3]
Now rewrite the expression:
[(x^6y^-3)/(27y^3/5)^-1/3] = (x^6 * (1/y^3)) / (3^(3/9) * y^(-1/15))
further, combine the x and y terms:
= x^6 * (1/y^3) * (1 / (3^(1/3) * y^(-1/15))) [using the property a / b = a * (1/b)]
Now, simplify the denominator:
= x^6 * (1/y^3) * (1 / (3^(1/3) * 1/y^(1/15))) [since y^(-1/15) is equivalent to 1/y^(1/15)]
= x^6 * (1/y^3) * (y^(1/15) / 3^(1/3)) [simplifying the denominator]
Finally, simplifying the expression:
= x^6 * (y^(1/15) / (3^(1/3) * y^3))
= x^6 * y^(1/15 - 3) * 3^(-1/3)
= x^6 * y^(-44/15) * 3^(-1/3)
b) the expression [a^3/2 / b^-1/2 (a^(-2)/b^3)], use the rules of exponents:
Step 1: Simplify the numerator and rewrite it with positive exponents.
a^3/2 = a^(3/2)
Step 2: Simplify the denominator and rewrite it with positive exponents.
b^-1/2 (a^(-2)/b^3) = (1/b^(1/2)) * (a^(-2)/b^3) [using the property b^-n = 1/b^n]
Now, rewrite the expression:
[a^3/2 / b^-1/2 (a^(-2)/b^3)] = a^(3/2) / ((1/b^(1/2)) * (a^(-2)/b^3))
simplify further by combining the a and b terms:
= a^(3/
2) / ((1/b^(1/2)) * (a^(-2)/b^3))
= a^(3/2) * (b^(1/2) / (a^(-2) * b^3))
Finally, simplifying the expression:
= a^(3/2 + 2) * b^(1/2 - 3)
= a^(7/2) * b^(-5/2)
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Factor completely if possible: a. 54q²-144q +42 b. p-1000 c. 6x²-19x-7 d. solve for t: 15t³+40t=70t²
a) The expression is 54q² - 144q + 42. To factor it completely, we follow these steps:
Step 1: the greatest common factor (GCF) of the coefficients.
The GCF of 54, 144, and 42 is 6.
Step 2: Divide each term by the GCF.
(6)(9q² - 24q + 7)
Step 3: Factor the quadratic expression.
The quadratic expression 9q² - 24q + 7 cannot be factored further using simple integer factors.
Therefore, the completely factored form of the expression 54q² - 144q + 42 is:
6(9q² - 24q + 7).
b) The given expression is p - 1000. Since there are no common factors other than 1, the expression is already completely factored.
c) The given expression is 6x² - 19x - 7. To factor it completely, we can use either factoring techniques or the quadratic formula. Let's use the quadratic formula in this case.
Using the quadratic formula, which is x = (-b ± √(b² - 4ac)) / (2a), we can find the roots of the quadratic equation 6x² - 19x - 7 = 0.
The coefficients are a = 6, b = -19, and c = -7.
Plugging these values into the quadratic formula, we can calculate the roots of the equation.
d) To solve the equation 15t³ + 40t = 70t², we can rearrange it as follows:
15t³ + 40t - 70t² = 0
Step 1: Factor out the common factor, which is t:
t(15t² + 40 - 70t) = 0
Step 2: Simplify the expression inside the parentheses:
t(15t² - 70t + 40) = 0
Step 3: Factor the quadratic expression:
t(3t - 4)(5t - 10) = 0
Now we have factored the equation completely. The solutions are t = 0, t = 4/3, and t = 2.
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checkers cab company is trying to decide which brand of tires to use for the coming year. based on current price and prior experience they have narrowed their choice to brands, beltex and roadmaster. a recent study examined the durability of these tires by using a machine with a metallic device that wore down the tires. the time it took (in hours) for the tire to blowout was recorded. the blowout times (hours) were as follows: Beltex: 3.82, 3.11, 4.21, 2.64, 4.16, 3.91, 2.44, 4.52, 2.84, 3.26, 3.74, 3.04, 2.56, 2.58, 3.15
Roadmaster: 4.16, 3.92, 3.94, 4.22, 4.15, 3.62, 4.11, 3.45, 3.65, 3.82, 4.55. 3.82, 3.85, 3.62, 4.88
a. construct a 90% confidence interval for the difference in the mean blowout times between the two brands of tires and interpret the results.
b. can we conclude that the average blowout times are in fact not the same? Use a significance level of .10.
To compare the mean blowout times, a 90% confidence interval will be constructed, and using a significance level of 0.10, we will determine whether we can conclude about the average blowout times.
a. To construct a 90% confidence interval for the difference in the mean blowout times between the two brands of tires, we will calculate the sample mean and standard deviation for each brand. Using the formula for the confidence interval of the difference between two means, we can determine the range within which we can be 90% confident that the true difference lies. The interpretation of this confidence interval is that it provides a range of values that likely contains the actual difference in the mean blowout times between the Beltex and Roadmaster tire brands, with 90% confidence.
b. To test whether the average blowout times are significantly different between the two brands, we can perform a hypothesis test. Using a significance level of 0.10, we can compare the means of the two samples. If the p-value is less than 0.10, we can conclude that the average blowout times are significantly different. However, if the p-value is greater than or equal to 0.10, we do not have sufficient evidence to conclude that the average blowout times are different.
By analyzing the results of the hypothesis test, we can determine whether we can reject the null hypothesis that the average blowout times are the same.
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A 99% confidence interval for the percentage of people in a town who have a college degree is constructed based on a simple random sample of people in the town, and it goes from 65% to 71%. Which of the following statements are correct conclusions based on this result? Check all that apply.
Group of answer choices
A. There is a 99% chance that a confidence interval constructed this way from a simple random sample contains the true percentage of people with a college degree in the town.
B. Between 65% and 71% of people in the town have a college degree.
C. All the values between 65% and 71% are reasonable estimates of the percentage of people with college degree in the town.
D. The sample percentage was 68.
E. There's a 99% chance the percentage of people with a college degree in the town is somewhere between 65% and 71%.
Option (E) there's a 99% chance the percentage of people with a college degree in the town is somewhere between 65% and 71%.
Is the confidence interval a guarantee of the exact percentage of people with a college degree in the town?We can analyze the statements and determine which ones are correct conclusions based on the provided 99% confidence interval for the percentage of people in a town who have a college degree.
A. There is a 99% chance that a confidence interval constructed this way from a simple random sample contains the true percentage of people with a college degree in the town.
This statement is incorrect.
The interpretation of a confidence interval is not based on the probability of containing the true parameter value.
Instead, it means that if we were to repeat the sampling and construct confidence intervals in the same way, 99% of those intervals would contain the true parameter value.
B. Between 65% and 71% of people in the town have a college degree.
This statement is incorrect.
The confidence interval provides a range of values that is likely to contain the true percentage of people with a college degree in the town, but it does not guarantee that the true value lies within that range.
It only provides a level of confidence in the estimation.
C. All the values between 65% and 71% are reasonable estimates of the percentage of people with a college degree in the town.
This statement is incorrect.
The confidence interval provides a range of reasonable estimates, but individual values within that range cannot be considered as precise estimates.
The true percentage could be any value within the interval, not just the endpoints.
D. The sample percentage was 68.
This statement is not directly supported by the given information.
The sample percentage is not explicitly provided, and we cannot determine its value based solely on the confidence interval.
E. There's a 99% chance the percentage of people with a college degree in the town is somewhere between 65% and 71%.
This statement is correct. The confidence interval indicates that, with 99% confidence, the true percentage of people with a college degree in the town is likely to fall between 65% and 71%. It provides a range within which the true value is expected to lie.
Therefore, the correct conclusions based on this result are:
Statement E: There's a 99% chance the percentage of people with a college degree in the town is somewhere between 65% and 71%.
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Find the distance between the pair of points. Also, find the midpoint of the line segment joining them. (2-3).(2.0) Find the distance. d= (Simplify your answer. Type an exact answer, using it as needed.) Find the midpoint. (Simplify your answer. Type an ordered pair. Type an exact answer, using as needed.)
The distance between the points (2, -3) and (2, 0) is 3 units. The midpoint of the line segment joining these points is (2, -1.5).
To find the distance between two points in a coordinate plane, we can use the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Given the points (2, -3) and (2, 0), we can substitute the coordinates into the distance formula:
d = √((2 - 2)^2 + (0 - (-3))^2)
= √(0 + 9)
= √9
= 3
Therefore, the distance between the points (2, -3) and (2, 0) is 3 units.
To find the midpoint of the line segment joining the two points, we can average the x-coordinates and the y-coordinates separately:
Midpoint (x, y) = ((x1 + x2)/2, (y1 + y2)/2)
Substituting the coordinates:
Midpoint (x, y) = ((2 + 2)/2, (-3 + 0)/2)
= (4/2, -3/2)
= (2, -1.5)
Therefore, the midpoint of the line segment joining the points (2, -3) and (2, 0) is (2, -1.5).
The distance between the points (2, -3) and (2, 0) is 3 units, and the midpoint of the line segment joining these points is (2, -1.5).
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Assume that sales in year 2002 was 168,909 and sales in year 2003 was 380,554. State the percentage increase of sales in DECIMAL FORM to the nearest two decimal places. (Example, if sales increased 70%, write your answer as .70 and do not write the % sign in your answer.
the percentage increase in sales from 2002 to 2003 is approximately 125.34%. In decimal form, this is 1.2534 (rounded to two decimal places).To calculate the percentage increase in sales from 2002 to 2003, we use the following formula:
Percentage Increase = ((New Value - Old Value) / Old Value) * 100
Given that sales in 2002 were 168,909 and sales in 2003 were 380,554, we can substitute these values into the formula:
Percentage Increase = ((380,554 - 168,909) / 168,909) * 100
Calculating this expression, we get:
Percentage Increase = (211,645 / 168,909) * 100 ≈ 125.34
Therefore, the percentage increase in sales from 2002 to 2003 is approximately 125.34%. In decimal form, this is 1.2534 (rounded to two decimal places).
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If you wish to estimate a population mean with a sampling distribution error SE=0.21 using a 95% confidence interval and you know from prior sampling that σ2 is approximately equal to 3.3, how many observations would have to be included in your sample?
The number of observations that would have to be included in your sample is _____
To estimate the required sample size, we can use the formula for the margin of error in a confidence interval:
Margin of error = Z * (σ / sqrt(n))
Where:
Z is the Z-score corresponding to the desired confidence level (for a 95% confidence interval, Z ≈ 1.96)
σ is the standard deviation of the population
n is the sample size
Given that the sampling distribution error (SE) is 0.21, we can equate the margin of error to the SE:
0.21 = 1.96 * (sqrt(3.3) / sqrt(n))
Simplifying the equation:
sqrt(n) = 1.96 * sqrt(3.3) / 0.21
Squaring both sides:
n = (1.96 * sqrt(3.3) / 0.21)^2
Calculating the value:
n ≈ 61.34
Since we cannot have a fraction of an observation, we need to round up to the nearest whole number. Therefore, the minimum number of observations required in the sample is 62 in order to achieve a sampling distribution error of approximately 0.21 and obtain a 95% confidence interval.
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S Problem 4 You run a meal kit delivery startup and you need to determine the typical time required for a customer to prepare the meal kit into a meal. The research department takes a random sample of 70 customers and calculates the sample average time spent for the meal preparation ž to be 97 minutes and the sample standard deviation s to be 14 minutes. a. Calculate the 99% confidence interval for u, the population mean time for meal preparation. The z- critical value for this interval is Za = 20.01 = 20.005 = 2.575. b. Interpret the confidence interval in words. c. The vice president of research asserts that the mean time for meal preparation is 95 minutes. Do you agree with this? Explain how you know. d. Suppose your company needs a narrower confidence interval, so it can decide whether the recipe testing department needs more staff. Calculate the sample size necessary to estimate the population mean within 1 minute. 2 2
The 99% confidence interval for the population mean meal preparation time, based on a sample of 70 customers, is estimated to be between 92.26 and 101.74 minutes.
What is the 99% confidence interval for the population mean meal preparation time, based on a sample of 70 customers?The 99% confidence interval is a statistical measure that helps estimate the true population mean time for meal preparation. In this case, the sample average time spent on meal preparation is 97 minutes, with a sample standard deviation of 14 minutes. Using a z-critical value of 2.575 for a 99% confidence level, the margin of error is calculated to be approximately 4.74 minutes.
The confidence interval of 92.26 to 101.74 minutes suggests that, based on the sample data, we can be 99% confident that the population mean time for meal preparation falls within this range. This interval provides valuable information for decision-making, such as recipe testing department staffing, as it indicates the likely time customers will spend on meal preparation.
In regards to the Vice President's assertion that the mean time for meal preparation is 95 minutes, the data does not support this claim. The lower bound of the confidence interval is 92.26 minutes, which is lower than the asserted value of 95 minutes. Therefore, it is unlikely that the mean time is 95 minutes based on the available data.
To obtain a narrower confidence interval, which provides a more precise estimate of the population mean, a larger sample size is required. Calculating the necessary sample size involves considering the desired margin of error. If the company aims to estimate the population mean within 1 minute, a sample size of approximately 757 customers would be needed.
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If A₁ = [ Jand -1 4 and 42 3].is B- [ 3 52 7 -3 2] 58 in span(A1, A2)? Explain. (6 points)
Given matrices: A₁ = [1 -1 4; 4 2 3]B = [3; 5; 7; -3; 2]We have to check whether the matrix B lies in span(A1, A2) or not. Now, we need to find A₂ such that the matrix B lies in span(A1, A2) i.e.
it can be represented as a/ of A₁ and A₂.We can find A₂ as follows:Let A₂ = [a b c; d e f]We want B to be a linear combination of A₁ and A₂i.e. there exist constants x and y such that:B = xA₁ + yA₂= x[1 -1 4; 4 2 3] + y[a b c; d e f]Now, the above equation can be written in the form:[1 -1 4; 4 2 3 | 3; 5; 7] [a b c; d e f | -3; 2]
This can be written in the form of an augmented matrix as:[1 -1 4 3; 4 2 3 5] [a b c -3; d e f 2]Now, we perform row operations to put the matrix in echelon form:[1 -1 4 3; 0 6 -13 -7] [a b c -3; 0 -2 5 5]Now, we perform back-substitution to find the values of a, b, c, d, e and f:Since the above matrix is not in echelon form, we cannot perform back-substitution, thus, we can say that the matrix B does not lie in span(A1, A2).Hence, the matrix B = [3; 5; 7; -3; 2] does not lie in span(A₁, A₂).
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Divide and express the result in standard form. 3) 2-4i /5-7i
To divide the complex numbers (2 - 4i) by (5 - 7i), we can use the concept of multiplying by the conjugate of the denominator to simplify the expression.
The conjugate of the denominator (5 - 7i) is (5 + 7i). To simplify the division, we multiply both the numerator and denominator by the conjugate:
[(2 - 4i) * (5 + 7i)] / [(5 - 7i) * (5 + 7i)]
Expanding the numerator and denominator, we get:
[(10 + 14i - 20i - 28i^2)] / [(25 - 49i^2)]
Simplifying further, we have:
[(10 - 6i - 28i^2)] / [(25 + 49)]
Since i^2 is equal to -1, we can substitute -1 for i^2:
[(10 - 6i - 28(-1))] / [25 + 49]
Simplifying the expression, we get:
[(10 - 6i + 28)] / [74]
Combining like terms, we have:
(38 - 6i) / 74
Now, we can simplify the expression by dividing both the real and imaginary parts by 2:
38/74 - (6/74)i
In standard form, the result is:
19/37 - (3/37)i
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Find the product using the rectangle method. Determine the individual terms that should appear in the rectangles, and then give the final product. (x+2)(x+6) Complete the table below. (Simplify your answers.) The final product is (Simplify your answer.)
The product of (x+2)(x+6) using the rectangle method is [tex]x^2 + 8x + 12.[/tex]
What is the result of multiplying (x+2) and (x+6) using the rectangle method, and what is the final simplified product?To find the product (x+2)(x+6) using the rectangle method, we can create a rectangle and divide it into smaller sections based on the terms in the expression.
The rectangle can be divided into four sections:
The top-left section will have x as its term.The top-right section will have 2 as its term.The bottom-left section will have x as its term.The bottom-right section will have 6 as its term.Now, let's complete the table below to visualize the product:
| | x | 2 |
|-----|-----|-----|
| x | [tex]x^2[/tex] | 2x |
| 6 | 6x | 12 |
By multiplying the corresponding terms in each section and adding them together, we get the final product:
(x+2)(x+6) = [tex]x^2 + 2x + 6x + 12[/tex]
Simplifying this expression, we have:
(x+2)(x+6) = [tex]x^2 + 8x + 12[/tex]
Therefore, the final product is [tex]x^2 + 8x + 12.[/tex]
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Please provide proving
Show that in a line there exists an infinite number of
points.
This result aligns with our intuition and the geometric properties of lines, which extend infinitely in both directions.
To show that there exists an infinite number of points on a line, we can use a proof by contradiction. Let's assume that there are only a finite number of points on the line.
Suppose we have a line segment AB, where A and B are two distinct points on the line. Since we assumed that there are only a finite number of points on the line, let's label these points as A1, A2, A3, ..., An, where n is the total number of points.
Now, consider the midpoint of the line segment AB. We can label this midpoint as M. Since M is the midpoint, it divides the line segment AB into two equal parts, AM and MB.
If we look closely at the points on the line segment AB, we notice that M is not one of the labeled points, A1, A2, A3, ..., An, because M is the exact midpoint and not one of the endpoints. Therefore, we have discovered a new point, M, on the line segment AB that is not included in our initial assumption of a finite number of points.
Now, let's continue this process by dividing each of the line segments AM and MB into halves and identifying the midpoints. We can label these new midpoints as M1, M2, M3, ..., Mn, where each Mi is the midpoint of the line segment connecting two consecutive points from the previous step.
By repeating this process infinitely, we generate an infinite number of midpoints on the line segment AB. Each midpoint is distinct from the others because it is not one of the labeled points from the previous step. Therefore, we have found an infinite number of points on the line segment AB.
Since this argument applies to any line segment AB on the line, we can conclude that there exists an infinite number of points on the line itself.
In summary, by assuming a finite number of points on a line and showing the existence of an infinite number of points on any line segment, we have proved that there exists an infinite number of points on a line. This result aligns with our intuition and the geometric properties of lines, which extend infinitely in both directions.
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How to prove that a straight line is an infinite set of points?
which of the following python lines returns subset data for only the variables ""survived"" and ""age"" from a dataframe called ""titanic""?
The correct Python line to return subset data for the variables "survived" and "age" from a dataframe called "titanic" is subset = titanic[['survived', 'age']].
To extract a subset of data containing only the variables "survived" and "age" from the dataframe "titanic" in Python, you can use double brackets to specify the columns of interest. The line subset = titanic[['survived', 'age']] achieves this.
Here's a breakdown of the line:
titanic[['survived', 'age']] is used to select the columns 'survived' and 'age' from the dataframe 'titanic'. The double brackets create a new dataframe with only the specified columns.
The resulting subset dataframe is then assigned to the variable 'subset' using the assignment operator '='.
You can use 'subset' to perform further operations or analyze the data containing only the 'survived' and 'age' variables.
By executing this line of code, you will obtain a new dataframe named 'subset' that contains only the columns 'survived' and 'age' from the original 'titanic' dataframe.
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Use the following information to complete parts (a) through (e) below A researcher wanted to determine the effectiveness of a new cream in the treatment of warts. She identified 151 individuals who had two warts. She applied cream A on B on the second wart. Test whether the proportion of successes with cream A is different from cream B at the α 0.10 level of significance Treatment A Success Failu 10 53 reatment BIsucces 63 Success Failure25 one wart and cream (a) What type of test should be used? Choose the correct answer below OA. A hypothesis test regarding the difference of two means using a matched-pairs design. OB. A hypothesis test regarding two population standard deviations. OC. A hypothesis test regarding the difference between two population proportions from dependent samples. OD. A hypothesis test regarding the difference between two population proportions from independent samples.
In this scenario, the researcher is interested in comparing the proportion of successes between two treatments (cream A and cream B) for the treatment of warts.
The data consists of two independent samples with dichotomous outcomes (success/failure), so a hypothesis test regarding the difference between two population proportions from independent samples should be used, which is option OD.
To perform the hypothesis test, we can use the z-test for two proportions. First, we calculate the sample proportions for each treatment: pA = 10/151 = 0.0662 for cream A, and pB = 63/151 = 0.4172 for cream B. We then calculate the pooled sample proportion: p = (10 + 63)/(151 + 151) = 0.2417. Next, we calculate the standard error of the difference between the sample proportions:
SE = sqrt(p(1 - p)(1/nA + 1/nB)) = sqrt(0.2417(1 - 0.2417)(1/151 + 1/151)) = 0.0648
Finally, we calculate the test statistic:
z = (pA - pB) / SE = (-0.351) / 0.0648 = -5.42
Using a standard normal distribution table, we find that the probability of obtaining a z-score as extreme or more extreme than -5.42 is less than 0.0001. Therefore, at the α = 0.10 level of significance, we reject the null hypothesis that the proportion of successes for cream A is equal to the proportion of successes for cream B. We conclude that there is sufficient evidence to suggest that the new cream A is less effective than cream B in treating warts.
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2. Simulate an M/M/1 queue with 1 = 1 and u = 1.25, this time recording customer time in the system (from arrival to departure) as the performance measure for 4000 customers. Make replications and use a mean plot to determine an appropriate number of customer to delete when starting the system empty, with 4 customers initially in the system, and with 8 customers initially in the system. How does this warmup period change with these different initial conditions? What does this suggest about how to initialize simulations?
The warmup period changes with different initial conditions in an M/M/1 queue simulation. Deleting a specific number of customers when starting the system empty, with 4 customers initially, and with 8 customers initially affects the warmup period.
What impact does different initial conditions have on the warmup period in an M/M/1 queue simulation?The warmup period in an M/M/1 queue simulation refers to the time it takes for the system to stabilize and reach a steady-state behavior after initialization. By conducting simulations with different initial conditions, specifically starting the system empty, with 4 customers initially, and with 8 customers initially, the warmup period can be analyzed.
When starting the system empty, there are no customers present, and the warmup period is generally longer compared to scenarios with initial customers. This is because the system needs time to receive incoming customers and build up a queue.
With 4 customers initially in the system, there is already some workload present. This reduces the warmup period compared to an empty system since there are customers in the queue and the system can start processing them immediately.
Similarly, when starting with 8 customers initially, the warmup period further decreases as there are even more customers in the system from the beginning. This allows for faster processing and a shorter time required to stabilize.
These findings suggest that initializing simulations with some initial customers can help reduce the warmup period. Having customers in the system from the start allows for more accurate representations of real-world scenarios and avoids extended periods of transient behavior.
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Ut = 3 Uxx 0 < x < 4, t≥0 with boundary/initial conditions: u(0, t) = 0, and_u(x,0) = {{ 3, 0≤x≤2 u(4, t) = 0, 0, 2
The general solution of the given partial differential equation in the prompt is: u(x,t) = tg(x,t) - 3t + 0 for 0 ≤ x ≤ 2, and t≥0
To solve the partial differential equation (PDE) given in the prompt, we will use the method of characteristics. The method of characteristics is a technique that allows us to find the general solution of a PDE by studying the behavior of a particular solution as it moves along a characteristic curve.
In this case, the characteristic curve is the path in the x-y plane that is given by the equation U(x,t) = t. The initial condition u(0,t) = 0 and u(x,0) = {3, 0 ≤ x ≤ 2} imply that the solution must satisfy the condition U(x,0) = 0 for all x.
The characteristic equation is U(x,t) = t, which means that the general solution of the PDE can be written as:
u(x,t) = tg(x,t) + h(x,t)
where g(x,t) and h(x,t) are two unknown functions. We can use the boundary/initial conditions to determine the values of g(x,t) and h(x,t).
The boundary condition u(4,t) = 0 and the initial condition u(x,0) = {3, 0 ≤ x ≤ 2} give us the following system of equations:
g(4,t) = 0
3 = h(4,t)
0 = h(x,0) for 0 ≤ x ≤ 2
We can use the first two equations to eliminate h(x,t) and solve for g(x,t). Substituting the second equation into the first, we get:
3 = h(4,t)
Substituting the initial condition into the second equation, we get:
0 = h(x,0) for 0 ≤ x ≤ 2
We can eliminate h(x,0) by substituting the first equation into the second:
3 = h(4,t) + h(x,0) for 0 ≤ x ≤ 2
Substituting the initial condition for h(x,0), we get:
3 = h(4,t) + 3 for 0 ≤ x ≤ 2
Simplifying this equation, we get:
h(4,t) = -3
Substituting this value into the first equation, we get:
g(4,t) = -3t
Finally, we can use the initial condition to find the value of g(0,t):
g(0,t) = 0
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identify the curve by finding a cartesian equation for the curve. r = 5cos ϴ
The curve can be identified as a circle.
What shape does the given equation represent?
The given polar equation, r = 5cos(θ), represents a circle in the Cartesian coordinate system. In polar coordinates, r represents the distance from the origin (0,0), and θ represents the angle measured counterclockwise from the positive x-axis. By substituting the given equation into the standard conversion equations r = √(x^2 + y^2) and x = rcos(θ), we can derive the corresponding Cartesian equation. Simplifying the equation, we get x^2 + y^2 = 25cos^2(θ). Since cos^2(θ) ranges from 0 to 1, the equation simplifies to x^2 + y^2 = 25, which is the equation of a circle with radius 5 and centered at the origin (0,0).
To delve deeper into the topic of polar equations and their conversion to Cartesian equations, it would be beneficial to study trigonometry and coordinate systems. Understanding the relationship between polar and Cartesian coordinates helps in visualizing and analyzing various curves, shapes, and functions. Exploring the concepts of sine, cosine, and the unit circle can provide a solid foundation for comprehending polar coordinates and their transformations. Additionally, learning about conic sections, specifically circles, can enhance your knowledge of equations representing curves in the Cartesian coordinate plane.
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Agroup of College Algebra students were trying to determine whether f(x) = 4x and g(x)= are inverse functions. The students correctly found that g(f(x)) = a) Give an equation for (fog)(x) = f(g(x)). Simplify as much as possible. f(g(x)) =_ b) Use your answer to part a) and the students' conclusion that f(g(x)) = to decide if f(x) and g(x) are inverse functions. Of and g ARE inverse functions Of and g ARE NOT inverse functions Explain:
a) To find the equation for (fog)(x) = f(g(x)), we substitute g(x) into f(x):
(fog)(x) = f(g(x)) = f(4x)
Now, we substitute the expression 4x into f(x):
(fog)(x) = f(g(x)) = f(4x) = 4(4x) = 16x
Therefore, (fog)(x) = 16x.
b) Based on the students' conclusion that g(f(x)) = , we need to compare it with the equation for (fog)(x) obtained in part a), which is (fog)(x) = 16x.
If g(f(x)) = and (fog)(x) = 16x, then f(x) and g(x) are inverse functions.
Explanation: Two functions, f(x) and g(x), are considered inverse functions if their composition results in the identity function. In this case, g(f(x)) = , which matches the equation for (fog)(x) = 16x. Therefore, the functions f(x) = 4x and g(x) = are indeed inverse functions.
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Region R is the base of a solid. For the solid, each cross section perpendicular to the x-axis is a square. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid.
To find the volume of the solid, we can use integrals and consider each cross section perpendicular to the x-axis as a square.
Let's assume the length of each side of the square cross section at a given x-coordinate is denoted by s(x). We can express the volume of each small square slice as dV = s(x)^2 dx, where dx represents an infinitesimally small length along the x-axis.
To find the total volume, we need to integrate this expression over the entire region R. The integral will have the form:
V = ∫[a,b] s(x)^2 dx,
where [a, b] represents the interval over which region R extends along the x-axis.
By integrating the square of the side length of each cross section over the entire interval, we obtain the volume of the solid.
Note: The specific function s(x) that determines the side length of each square cross section would need to be provided to evaluate the integral and find the actual volume.
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find x.y.z using Cramer's Law 4x-y+z=-5
2x+24+3z=10
5x-24+6z=1
x ≈ -0.6875, y ≈ 4.0375, and z ≈ 0.35. To solve the system of equations using Cramer's Law, we need to find the determinants of the coefficient matrix and the matrices obtained by replacing each column of the coefficient matrix with the column on the right side of the equations.
The given system of equations can be written in matrix form as:
| 4 -1 1 | | x | | -5 |
| 2 2 3 | * | y | = | 10 |
| 5 -2 6 | | z | | 1 |
Let's calculate the determinants using the following notation:
D = determinant of the coefficient matrix
Dx = determinant obtained by replacing the x-column with the column on the right side
Dy = determinant obtained by replacing the y-column with the column on the right side
Dz = determinant obtained by replacing the z-column with the column on the right side
Coefficient matrix:
| 4 -1 1 |
| 2 2 3 |
| 5 -2 6 |
Column matrix:
| -5 |
| 10 |
| 1 |
Calculating D:
D = | 4 -1 1 |
| 2 2 3 | = 4(2(6) - (-2)(3)) - (-1)(2(6) - 5(-2)) + 1(2(-2) - 5(2))
| 5 -2 6 |
= 4(12 + 6) - (-1)(12 + 10) + 1(-4 - 10)
= 4(18) + 22 - 14
= 72 + 22 - 14
= 80
Calculating Dx:
Dx = | -5 -1 1 |
| 10 2 3 | = -5(2(6) - (-2)(3)) - (-1)(10(6) - 1(3)) + 1(10(-2) - 1(2))
| 1 -2 6 |
= -5(12 + 6) - (-1)(60 - 3) + 1(-20 - 2)
= -5(18) + 57 + (-22)
= -90 + 57 - 22
= -55
Calculating Dy:
Dy = | 4 -5 1 |
| 2 10 3 | = 4(10(6) - (-5)(3)) - (-5)(2(6) - 5(3)) + 1(2(10) - 4(3))
| 5 1 6 |
= 4(60 + 15) - (-5)(12 - 15) + 1(20 - 12)
= 4(75) + 5(3) + 1(8)
= 300 + 15 + 8
= 323
Calculating Dz:
Dz = | 4 -1 -5 |
| 2 2 10 | = 4(2(1) - 2(-5)) - (-1)(2(10) - 5(2)) + (-5)(2(2) - 5(2))
| 5 -2 1 |
= 4(2 + 10) - (-1)(20 - 10) + (-5)(4 - 10)
= 4(12) + 10 + (-30)
= 48 + 10 - 30
= 28
Finally, we can find x, y, and z using Cramer's Law:
x = Dx / D = -55 / 80 = -0.6875
y = Dy / D = 323 / 80 = 4.0375
z = Dz / D = 28 / 80 = 0.35
Therefore, x ≈ -0.6875, y ≈ 4.0375, and z ≈ 0.35.
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Incorrect Question 1 What is a common procedure for data collection? collecting data O preparing data data analysis O conducting a survey 0/1 pts
A common procedure for data collection involves conducting a survey.
Conducting a survey is a widely used and effective method for data collection. Surveys involve gathering information from a sample of individuals or organizations through a structured set of questions.
This approach allows researchers to collect data on a variety of topics, such as opinions, preferences, behaviors, or demographic information. Surveys can be conducted through various means, including face-to-face interviews, phone interviews, online surveys, or paper-based questionnaires.
The process of conducting a survey typically involves several steps. First, researchers need to define their research objectives and identify the target population they want to survey. They then design a survey instrument, which includes formulating relevant questions and response options.
Next, the survey is administered to the selected sample, either by directly interacting with participants or by distributing the survey through various channels. Once the data is collected, it needs to be carefully organized and prepared for analysis. This may involve cleaning the data, coding responses, and ensuring data accuracy. Finally, researchers can conduct data analysis to draw meaningful conclusions and insights from the collected data.
In summary, conducting a survey is a common procedure for data collection. It involves designing and administering a structured set of questions to a sample of individuals or organizations to gather information on a particular topic.
The collected data is then prepared and analyzed to extract valuable insights and draw conclusions.
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find the arclength of the following curve c(t) = (√2t, 1 2 t2, ln t ) , 1 ≤ t
The arc length of the curve c(t) = (√2t, 1/2 t^2, ln(t)), where 1 ≤ t, is given by (1/3)b^3 + b - 1/b + 4/3.
To find the arc length of the curve defined by c(t) = (√2t, 1/2 t^2, ln(t)), where 1 ≤ t, we can use the arc length formula:
L = ∫[a,b] √[dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2 dt
Let's calculate the derivatives first:
dx/dt = (√2)'t = √2
dy/dt = (1/2 t^2)' = t
dz/dt = (ln(t))' = 1/t
Now we can substitute these derivatives into the arc length formula:
L = ∫[1,b] √(√2)^2 + t^2 + (1/t)^2 dt
L = ∫[1,b] 2 + t^2 + 1/t^2 dt
L = ∫[1,b] (2t^2 + t^4 + 1) / t^2 dt
Now, we can simplify the integrand:
L = ∫[1,b] (t^2 + 1 + 1/t^2) dt
L = ∫[1,b] (t^2 + 1) dt + ∫[1,b] 1/t^2 dt
Integrating each term separately:
∫(t^2 + 1) dt = (1/3)t^3 + t + C1
∫1/t^2 dt = -1/t + C2
Now, we can evaluate the definite integral from t = 1 to t = b:
L = [(1/3)b^3 + b] - [(1/3)(1)^3 + 1] - [-1/1 + 1/b]
L = (1/3)b^3 + b - 4/3 + 1 + 1 - 1/b
Simplifying further:
L = (1/3)b^3 + b - 1/b + 4/3
Therefore, the arc length of the curve c(t) = (√2t, 1/2 t^2, ln(t)), where 1 ≤ t, is given by (1/3)b^3 + b - 1/b + 4/3.
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Draw out the CAST system. Which trigonometric ratios are
positive for angles in the first quadrant? Second quadrant? Third
quadrant? Fourth quadrant? What do the letters in each quadrant
stand for?
The CAST (Compound Angle Sum and Difference Trigonometry) system is a mnemonic device utilized to remember the sign of the trigonometric ratios in each of the four quadrants of the unit circle. Here is the CAST system for remembering the signs of the trigonometric ratios:Sine is positive in the first and second quadrants.
Cosine is positive in the first and fourth quadrants.Tangent is positive in the first and third quadrants.In the first quadrant, all the ratios are positive: sin(θ), cos(θ), and tan(θ).In the second quadrant, only sin(θ) is positive, but cos(θ) and tan(θ) are both negative.In the third quadrant, only tan(θ) is positive, but sin(θ) and cos(θ) are both negative.In the fourth quadrant, only cos(θ) is positive, but sin(θ) and tan(θ) are both negative.The letters A, S, T, and C stand for All, Sine, Tangent, and Cosine, respectively. The CAST acronym and its meaning may be shown in the following table:QuadrantLettersMeaning of LettersFirstAllSine, Cosine, and Tangent are all positive.SecondSineSine is positive.ThirdTangentTangent is positive.FourthCosineCosine is positive.The CAST mnemonic is helpful for remembering the sign of the trigonometric ratios in each quadrant of the unit circle.
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Solve the system of equations by graphing: Sy+22 ly 8 2.0 + 10 Enter your answer as a numbers. If the system is inconsistent (has no solutions), enter DNE (for "does not exist") into each box. If the system is dependent (infinite number of solutions), enter oo into each box (for infinity. These are double letter o's, no zero's.) Answer: (x, y) =
The solution to the system of equations is (x, y) = (2, -1).
To solve the system of equations, we can graph the equations and find the point of intersection.
The given system of equations is:
3x + 2y = 8
2x + 10y = 20
To graph the first equation, we can rearrange it in terms of y:
y = (8 - 3x) / 2
Similarly, for the second equation:
y = (20 - 2x) / 10
Now we can plot the graphs of these equations on a coordinate plane.
By observing the graphs, we can see that they intersect at the point (2, -1). Therefore, the solution to the system of equations is (x, y) = (2, -1).
The system of equations is solved by graphing, and the solution is (x, y) = (2, -1).
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Let the joint density of the continuous random variables X1 and X2 be k(1 – X2) f(X1, X2 if 0 < x1 < x2 < 1 elsewhere (a) Find the value of k that makes this a probability density function. Answer:k = 6 (b) Compute P(X1 < , X2 > ) Answer: 32 (c) Find the marginal density functions for X1 and X2. Answer: fi(x1) = 3(1 – x 1)?,0 < x1 1, 2(x2) = 6x2(1 – x2),0 X2 < 1 (d) Compute P(X2 < }|X1 < 3) Answer:32
In summary, the joint density of the continuous random variables X1 and X2 is given by k(1 – X2) f(X1, X2) if 0 < X1 < X2 < 1, and 0 elsewhere. To make this a probability density function, the value of k is determined to be 6.
Now, let's calculate the probability P(X1 < X2). We integrate the joint density function over the region where X1 is less than X2. The limits of integration for X1 are 0 to X2, and for X2, it is from X1 to 1. Evaluating the integral, we find that P(X1 < X2) = 32.
Next, we need to find the marginal density functions for X1 and X2. The marginal density function of X1, denoted as f1(x1), is obtained by integrating the joint density function over all possible values of X2. Similarly, the marginal density function of X2, denoted as f2(x2), is obtained by integrating the joint density function over all possible values of X1. After performing the integrations, we find that f1(x1) = 3(1 – x1) for 0 < X1 < 1, and f2(x2) = 6x2(1 – x2) for 0 < X2 < 1.
Finally, let's compute the probability P(X2 < X1|X1 < 3). We integrate the joint density function over the region where X2 is less than X1 and X1 is less than 3. The limits of integration for X2 are 0 to X1, and for X1, it is from 0 to 3. Evaluating the integral, we find that P(X2 < X1|X1 < 3) = 32.
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Find the values of mand nif Kmn has an Euler circuit
a. Kₘ,ₙ has an Euler circut when both mand nare even and m, n>0.
b. It can never have an Euler circuit. c. Kₘ,ₙ has an Euler circuit when mis odd, n is even, and m, n > 0.
d. Kₘ,ₙ has on Euler circuit when mis even, nis odd, and n>0 e. Kₘ,ₙ has an Euler circuit when both mandnare odd and m, n>0
The values of mand nif Kmn has an Euler circuit Kₘ,ₙ has an Euler circuit when both m and n are even and m, n > 0.
An Euler circuit is a closed walk in a graph that visits every edge exactly once and returns to the starting vertex. In the case of a complete graph Kₘ,ₙ, it has m + n vertices and (m * n) / 2 edges.
For Kₘ,ₙ to have an Euler circuit, the number of edges (m * n) / 2 must be even. This requires both m and n to be even. If either m or n is odd, then (m * n) / 2 will be a non-integer, and Kₘ,ₙ cannot have an Euler circuit.
Additionally, both m and n must be greater than 0, as a graph with no vertices cannot have an Euler circuit.
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Seven seniors B, D, F, J, K, L and M are looking for positions after they graduate. The university has posted open positions for an accountant (a), consultant (c), editor (e), programmer (p), reporter (r), secretary (s), and teacher (t). Each of the seven students has applied for some of these positions: B:c,e D: a, c, p, s,t F: c,r J:c, e,r K: a, e, p, s L: e,r M: p, r, s, t - Page 3 of 3 Construct a graph representation of the problem and determine whether or not all seven students can be hired for the position they like.
A graph representation of the problem is shown in the image attached. The graph shows the seven seniors and the open positions posted by the university.
Each node in the graph represents a senior or a position. An edge connects a senior and a position if the senior has applied for that position. For instance, B has applied for positions c and e, so there are edges between node B and nodes c and e.
Similarly, there are edges between each senior and the positions they have applied for. The graph is bipartite since there are two sets of nodes, seniors and positions, and all edges connect a senior to a position (and vice versa). It is not possible to hire all seven students for the position they like since there are only four reporter positions available, but three students (F, J, and L) have applied for the reporter position. Therefore, at least one of these students will not be able to get the position they like.
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21. Graph the quadratic function f(x) = 2(x + 4)²-1. Find and label the vertex and axis of symmetry. Vertex Axis of symmetry.
Answer:Pitch of the sound depends upon its frequency. As the pitch of the sound is directly proportional to frequency, Low-frequency sounds are said to have low pitch whereas sounds of high frequency are said to have the high pitch.
Step-by-step explanation:
