The correct option is (C) 23.7%.
If the coefficient of determination is 0.237, the percentage of the data about the regression line that is explained is 23.7%.
The coefficient of determination is used to explain how much variability in a dependent variable is due to the independent variable in a regression analysis.
It is usually represented as R², which is a fraction between 0 and 1, or expressed as a percentage between 0% and 100%.
It measures the proportion of variability in the dependent variable that can be attributed to the independent variable(s).
In this problem, the coefficient of determination is given as 0.237, which means that 23.7% of the variability in the dependent variable is explained by the independent variable(s) in the regression line. Therefore, the correct option is (C) 23.7%.
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Yoko received a $90 gift card for a coffee store. She used it in buying some coffee that cost $8.23 per pound. After buying the coffee, she had $40.62 left on her card. How many pounds of coffee did she buy?
Answer: 6
Step-by-step explanation:
Step 1: Set up equation:
90=40.62+8.23p
^ the 'p' represents pounds.
Step 2: Subtract 40.62 from both sides
49.38 = 8.23p
Step 3: Divide both sides by 8.23 (We do this to get the variable "p" by itself)
6=p
Trust Milky. Milky Smart
Pls help me I don’t know how to do it!
Answer:
864
Step-by-step explanation:
To find the area of a shape, you have to multiply length times width.
Answer:
72 ft. sq.
Step-by-step explanation:
The way to look at it it to multiply length times width ( l x w )
there are 2 obvious numbers to get the answer, which is 8 and 9 so 8 x 9 = 72
Given that a^b=x, evaluate the following: a^26 PLS HELP NOW
Answer:
Step-by-step explanation:
Find the log of both the equation and the expression:
b log a = log x this can be solved for log a in terms of log x:
log x
log a = ------------
b
"Name" the second expression "y." Then y = a^26 and log y = 26 log a.
log x
Then log y = 26--------------
b
and the expression a^26 is found by taking the antilog of both sides of log y:
log y log x
10 = y = 10^(26----------)
b
Please double check to ensure that you have copied this problem down correctly.
ANSWER ASAP! DUE TODAY
5. Find the measure of arc AC.
Find mAC
A) 52
B) 72
C) 92
D) 112
Answer:
A. 52°
Step-by-step explanation:
The measure of an inscribed angle is 1/2 the measure of the intercepted arc.
m∠B = 1/2 (m of arc AC)
26 = 1/2 (m of arc AC)
52 = (m of arc AC)
PLEASE HELP ASAP A glass stand to display a doll is in the shape of a right triangular pyramid. The area of the base is 40.5 square inches with a height of 15 inches. What is the volume of the glass stand
Answer: [tex]202.5\ in.^3[/tex]
Step-by-step explanation:
Given
The area of the base is [tex]A=40.5\ in.^2[/tex]
height of the triangular prism is [tex]h=15\ in.[/tex]
The volume of a triangular prism is [tex]V=\dfrac{1}{3}Ah[/tex]
The volume of given Prism is
[tex]V=\dfrac{1}{3}\times 40.5\times 15=202.5\ in.^3[/tex]
The volume of the triangular prism is [tex]202.5\ in.^3[/tex]
On a map, 1 inch equals 5.2 miles. Two house are 3.5 inches apart on the map. What is the actual distance between the house?
Answer:
18.2 miles
Step-by-step explanation:
simple multiplication:
We know that 1 inch of distance on the map is equal to 5.2 miles.
Using this knowledge we can multiply the amount of inches by the miles per inch.
5.2*3.5=18.2 miles apart
Hope this helps!!
A point P(x, y) moves along the graph of the equation y = x3 + x2 + 6. The x-values are changing at the rate of 2 units per second. How fast are the y-values changing (in units per second) at the point Q(1, 8)?
The y-values are changing at a rate of 24 units per second at the point Q(1, 8).
To find how fast the y-values are changing at the point Q(1, 8), we need to calculate the derivative of the given equation with respect to x. The derivative of y = x^3 + x^2 + 6 is dy/dx = 3x^2 + 2x.
At the point Q(1, 8), we substitute x = 1 into the derivative to find the rate of change.
dy/dx = 3(1)^2 + 2(1) = 3 + 2 = 5.
Therefore, the y-values are changing at a rate of 5 units per second at the point Q(1, 8). However, the rate at which the x-values are changing is given as 2 units per second.
Thus, to determine how fast the y-values are changing with respect to time, we multiply the rate of change in y by the rate of change in x: 5 units/second * 2 units/second = 10 units/second. Hence, the y-values are changing at a rate of 10 units per second at the point Q(1, 8).
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log(x)-log(x+6)=log(k)
Find x if k=1/2
Answer:
x = 6
Step-by-step explanation:
Using the rules of logarithms
logx - logy = log ([tex]\frac{x}{y}[/tex] )
logx = logy , then x = y
Given
log(x) - log(x + 6)
= log([tex]\frac{x}{x+6}[/tex] )
Then
log )[tex]\frac{x}{x+6}[/tex] ) = log([tex]\frac{1}{2}[/tex] ) , thus
[tex]\frac{x}{x+6}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )
2x = x + 6 ( subtract x from both sides )
x = 6
there is a line that includes the point 7,2 and has a slope of -1/3
Answer:
y= -1/3x + 13/3
Step-by-step explanation:
we can find the equation of the line thanks to this formula
y-y1 = m(x-x1) where y1 and x1 indicate the coordinates of the point, while m indicates the slope
y-2 = -1/3(x-7)
y = -1/3x+7/3+2
y= -1/3x + (7+6)/3
y = -1/3x+13/3
WHat do u do with the brainly credits or whatever
Answer:
You basically grow rep
Step-by-step explanation:
Brainly points you need for asking questions and people are more likely to trust you with their questions if you have alot!
Answer:
i actually dont know
Step-by-step explanation:
The aggregate claim amount is defined as S X₁, where Xį are independent and i=1 identically distributed claim amounts and N is the number of claims. (a) Suppose N follows a negative-binomial distribution with E(N) = k(1 − p)/p and variance Var(N) = k(1 − p)/p². Show that: k(1 − p) (i) E(S) = -m₁ and р k(1 − p) k(1-P)² m², (ii) Var (S) = -M2 + p² - (iii) Skew k(1 − p)m² 3k(1 − p)²m₁m2 p² + + 2k(1 − p)³m³ p³ P
The required results are:
(i) E(S) = -m₁
(ii) [tex]Var(S) = k(1 − p)(m₂ - m₁²)/p + [k(1-p)/p]² (m₁² + m₂ - m₁²)[/tex]
(iii) Skew = 3(1 − 2p)/sqrt{k(1 − p)}.
The aggregate claim amount is defined as S X₁, where Xį are independent and i=1 identically distributed claim amounts and N is the number of claims.
And also suppose N follows a negative-binomial distribution with
E(N) = k(1 − p)/p
and variance
Var(N) = k(1 − p)/p².
Now we have to show that:
k(1 − p)
(i) [tex]E(S) = -m₁ and р k(1 − p) k(1-P)² m²,[/tex]
(ii) Var (S) = -M2 + p² -
(iii) [tex]Skew k(1 − p)m² 3k(1 − p)²m₁m2 p² + + 2k(1 − p)³m³ p³ P(i) E(S) = -m₁[/tex]
Where S = X1 + X2 + ...... + XN.
Also, N and Xi's are independent.
So, E(S) = E(E(S|N))
From the law of total expectation,
E(S|N) = E(X1 + X2 +......+XN|N)
Using linearity of expectation
,E(X1 + X2 +......+XN|N) = E(X1|N) + E(X2|N) +......+ E(XN|N) = NE(X1)Now, E(S) = E(E(S|N)) = E(NE(X1))= NE(X1) = m₁
Therefore,
E(S) = -m₁.(ii) Var (S) = -M2 + p² -Where Var(S|N) = Var(X1 + X2 +......+XN|N) = Var(X1|N) + Var(X2|N) +......+ Var(XN|N)Also, E(S|N) = NE(X1)Thus, E(S) = E(E(S|N)) = E(NE(X1)) = NE(X1) = m₁ And Var(S) = E(Var(S|N)) + Var(E(S|N))
Using independence,
[tex]Var(S|N) = Var(X1 + X2 +......+XN|N) = Var(X1|N) + Var(X2|N) +......+ Var(XN|N)Var(S|N) = N Var(X1)Using independence, E(S|N) = NE(X1)Var(E(S|N)) = Var(NE(X1)) = N²Var(X1)Hence, Var(S) = E(Var(S|N)) + Var(E(S|N))= E(N Var(X1)) + Var(NE(X1))= (k(1 − p)/p) Var(X1) + N²Var(X1)= k(1 − p)Var(X1)/p + N²Var(X1)[/tex]
Also, [tex]Var(X1) = E(X1²) - (E(X1))² = m₂ - m₁²[/tex]
Therefore,
Var(S) = k(1 − p)(m₂ - m₁²)/p + N²(m₂ - m₁²)
And N follows a negative-binomial distribution with
E(N) = k(1 − p)/p and variance
[tex]Var(N) = k(1 − p)/p². Thus, E(N²) = Var(N) + (E(N))²E(N²) = k(1-p)/p² + [k(1-p)/p]²[/tex]
Therefore,
[tex]Var(S) = k(1 − p)(m₂ - m₁²)/p + [k(1-p)/p]² (m₁² + m₂ - m₁²)(iii) Skew = 3(1 − 2p)/sqrt{k(1 − p)}[/tex]
Therefore,
[tex]k(1 − p)m² 3k(1 − p)²m₁m2 p² + + 2k(1 − p)³m³ p³ P = 3(1 − 2p)/sqrt{k(1 − p)}.[/tex]To know more about binomial distribution, visit:
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(Time complexity, 20pt) For the following languages, sketch (high-level, no need to mention states) a 1-tape TM program that solves the problem and include a time analysis of your program showing how much time it takes, e.g. O(n),O(nlog n), O(n^2), O(n^3), etc.; n is the length of the input.
a) [10pt] L = {u#v: U, v E {0,1}* and u is a substring of v}
b) [CSC 389, 10pt] L = {0^2^k : for k >= 0}. E.g. 0,00,0000,00000000 € L. Note that n = 2^k.
a) Language L = {u#v: u, v ∈ {0,1}* and u is a substring of v}
b) Language L = {0^2^k : for k ≥ 0}
Time Analysis: The time complexity of this program is O(n), where n is the length of the input. This is because we only need to scan through the input string once, counting the consecutive '0' symbols, and check if the count is a power of 2.
a) Language L = {u#v: u, v ∈ {0,1}* and u is a substring of v}
To sketch a 1-tape Turing machine program for this language, we can follow these steps:
1. Start at the leftmost symbol of the input.
2. Scan to the right until we encounter the '#' symbol, marking all the scanned symbols.
3. Once we encounter the '#' symbol, start scanning to the right again, comparing each scanned symbol with the marked symbols.
4. If a match is found, continue scanning until the end of the input, ensuring that the remaining symbols match the unmarked symbols.
5. If a mismatch is found at any point, reject the input.
6. If the end of the input is reached and all symbols match, accept the input.
Time Analysis: The time complexity of this program is O(n^2), where n is the length of the input. This is because, in the worst case, we need to compare each symbol in the input string with the marked symbols, and we may need to scan through the input string twice.
b) Language L = {0^2^k : for k ≥ 0}
To sketch a 1-tape Turing machine program for this language, we can follow these steps:
1. Start at the leftmost symbol of the input.
2. Scan to the right, counting the number of consecutive '0' symbols until a different symbol is encountered.
3. If the count of '0' symbols is not a power of 2, reject the input.
4. If a different symbol is encountered, check if it is the end of the input. If it is, accept the input.
5. If a different symbol is encountered and it is not the end of the input, reject the input.
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Solve for x
HELP ME PLEASEEEE
Answer:
x = 9
Step-by-step explanation:
5x + 7 = 52
5x = 52-7
5x = 45
x= 9
5potatos + 56 potatos
Answer:
61
Step-by-step explanation:
Porportion equation and solution: a punch recipe calls for 3 liters of ginger ale and 1.5 liters of tropical fruit juice how many quarts of punch will the recipe make (1 Liter = 1.6 quarts)
Answer:
The recipe will make 7.2 quarts of punch.
Step-by-step explanation:
A punch recipe calls for 3 liters of ginger ale and 1.5 liters of tropical fruit juice.
Then the recipe will make 4.5 liters of punch, obtained by the sum of ginger ale and tropical fruit juice.
The ration is the comparison of two quantities and is measured from the division of two values, then: [tex]\frac{a}{b}[/tex].
The proportion is the equality between two or more ratios. That is, [tex]\frac{a}{b}=\frac{c}{d}[/tex] equals a proportion.
In this case, being 1 L= 1.6 quarts, you have:
[tex]\frac{1.6 quarts}{1 L}=\frac{x}{4.5 L}[/tex]
Solving:
[tex]x=4.5 L*\frac{1.6 quarts}{1 L}[/tex]
x= 7.2 quarts
So, the recipe will make 7.2 quarts of punch.
Test the claim about the population mean, h, at the given level of significance using the given sample statistics. Claim: h = 30; « = 0.08; 3 = 3.31. Sample statistics: x = 28.9, n = 65 Calculate the standardized test statistic. The standardized test statistic is -2.68 (Round to two decimal places as needed.) Determine the critical value
The standardized test statistic is -2.68 and the critical value is 1.44.
Given that, Claim: h = 30 Sample statistics: x = 28.9, n = 65 α = 0.08 σ = 3.31
We need to calculate the standardized test statistic and critical value.
Test statistic: The standardized test statistic can be calculated as follows:
Z = (x - μ) / (σ / √n)
Substitute the given values in the above formula to get
Z = (28.9 - 30) / (3.31 / √65) = -2.68
Critical value: Since the level of significance α is given, we can calculate the critical value using the z-score table for standard normal distribution.
Since α is given, the right-tailed test is conducted.
The critical value is obtained by subtracting the α-level from 1 and then find the z-score value.
Let's find the z-score value corresponding to the area 0.92 in the z-score table.
We get the corresponding z-score as 1.44, which is the critical value.
Hence, the standardized test statistic is -2.68 (Round to two decimal places as needed.) and the critical value is 1.44.
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Write down the vectors along the lines representing those pipes, find the cross product between them from which to create the unit vector n, define a vector that spans two points on each line, and finally determine the minimum distance between the lines. (Take the origin to be at the lower corner of the first pipe.) Similarly, you may also develop the symmetric equations for each line and substitute directly into your formula.
Vectors along the lines:
To find vectors along the lines, we need two points on each line. Let's denote the points on Line A as A₁ and A₂, and the points on Line B as B₁ and B₂.
Vector along Line A: v₁ = A₂ - A₁
Vector along Line B: v₂ = B₂ - B₁.
Define the vectors along the lines:
Let's say we have two lines represented by the following equations:
Line 1: r₁(t) = a₁ + t * u₁
Line 2: r₂(t) = a₂ + t * u₂
Here, a₁ and a₂ are points on the lines, and u₁ and u₂ are the direction vectors of the lines.
Find the cross product:
To find the unit vector n, we can take the cross product of the direction vectors:
n = u₁ × u₂
Note: × represents the cross product operator.
Define a vector spanning two points on each line:
Let's choose two points on each line, which we'll call b₁ and b₂. The vector spanning these points on Line 1 is given by:
v₁ = b₁ - a₁
Similarly, the vector spanning the points on Line 2 is given by:
v₂ = b₂ - a₂
Determine the minimum distance:
The minimum distance (d) between the lines can be found using the formula:
d = |(v₁ × u₂) · n| / |n|
In this formula, · represents the dot product operator, and | | represents the magnitude or length of the vector.
Alternatively, if you prefer using symmetric equations, you can substitute the equations of the lines directly into the formula to find the minimum distance.
Remember to substitute the appropriate values for a₁, a₂, u₁, u₂, b₁, and b₂ based on the specific problem you are solving.
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Plsssss helps 50 points
Answer:
2000.2 cm³
Step-by-step explanation:
Finding the volume of a cylinder is quite simple if you under what pi (3.14) is. You can watch khan academy. It has some useful videos on solving for volume. It also explains very well. Hope this helps!
QUESTION 45
What is the value of x?
3x
Answer:
1/67
Step-by-step explanation:
Complete question
If X+2/3x is equal to 45. find the value of x
Given the equation
x+2/3x = 45
Cross multiply
x+2 = 3x(45)
x+2 = 135x
x - 135x = -2
-134x = -2
x = 2/134
x = 1/67
Hence the value of x is 1/67
HELP ASAP!!! find the lines of symmetry. Select all that apply
Answer:
B. and D.
Step-by-step explanation:
If you folded that shape along the lines o and m, it would match up.
There are two lines of symmetry m and o. Options B and D are correct.
From the given figure we have to determine the axes of symmetries.
A line is a straight curve connecting two points or more showing the shortest distance between the initial and final points.
Here,
As of the given figure,
There are 2 axes of symmetry first line m and the other is line o.
Line l and line n are not symmetrical lines because the reflection of one portion will not be superimposed with the other.
Thus, there are two lines of symmetry m and o. Options B and D are correct.
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In a class of 30 students, 6 play an instrument and 17 play a sport. There are 9
students who do not play an instrument or a sport. What is the probability that a
student chosen randomly from the class plays neither a sport nor an instrument?
Answer:
9/30
Step-by-step explanation:
Plz help me we are almost done
Answer:
1) D
2) B
Step-by-step explanation:
Hope this Helps!
:)
HELPP I WILL GIVE YIU BRAINLIEST!! i think the answer is 42x + 12 but not entirely sure!!!
Answer:
yes your answer is correct
Step-by-step explanation:
Each of the following sentences has three errors in grammar, capitalization, usage, or spelling. Write a correct version for each sentence. Avoid adding new phrases, starting new sentences, or rewriting in your own words.
1) Among the oddly named towns in the United States are: What Cheer, Iowa, Peculiar, Missouri, and Cheesecake, New Jersey.
2) After our supervisor and her returned from their meeting at 2:00 p.m., we were able to sort the customers names more quickly.
3) 6 of the 18 workers in my department were fired, as a result we had to work harder to achieve our goals.
4) Jeffreys presentation to a nonprofit group netted him only three hundred dollars, which is a tenth of his usual honorarium but he believes in pro bono work.
5) Of all the discoveries and inventions in human history the 4 greatest are said to be these speech, fire, agriculture, and the wheel.
6) Our latest press release written by our corporate communication department announces the opening of three Asian offices.
7) In his justification report dated September first, Justin argued that expansion to twelve branch offices could boost annual revenue to 22 million dollars.
8) The practicality and advisability of opening 12 branch offices are what will be discussed in the consultants feasibility report.
9) 3 categories of Apps (social media, productivity, and entertainment—are most popular for installing on smartphones.
10) Because some organizations prefer single spaced reports be sure to check with your organization to learn it’s preference.
Below are the corrected versions of the sentences with grammar, capitalization, usage, and spelling errors addressed.
Oddly named towns in the United States include What Cheer, Iowa; Peculiar, Missouri; and Cheesecake, New Jersey.After our supervisor and she returned from their meeting at 2:00 p.m., we were able to sort the customers' names more quickly.Six of the eighteen workers in my department were fired. As a result, we had to work harder to achieve our goals.Jeffrey's presentation to a nonprofit group netted him only three hundred dollars, which is a tenth of his usual honorarium. But he believes in pro bono work.Of all the discoveries and inventions in human history, the four greatest are said to be speech, fire, agriculture, and the wheel.Our corporate communication department has written the latest press release, which announces the opening of three Asian offices.In his justification report dated September 1st, Justin argued that expanding to twelve branch offices could boost annual revenue to 22 million dollars.The feasibility report from the consultants will discuss the practicality and advisability of opening twelve branch offices.Three categories of apps (social media, productivity, and entertainment) are most popular for installing on smartphones.Because some organizations prefer single-spaced reports, be sure to check with your organization to learn its preference.The sentence lists the oddly named towns correctly, but the capitalization of "Iowa" and "New Jersey" should be in lowercase, and colons are not needed in this context.The pronoun "her" should be replaced with "she" to be grammatically correct. "Customers names" should be "customers' names" to show possession.The numbers "6" and "18" should be written as words, and the sentence should be divided into two sentences for clarity."Jeffreys" should be corrected to "Jeffrey's" to indicate possession. "Pro bono work" should be in italics or quotation marks for correct formatting.The sentence lacks proper punctuation and capitalization. The four greatest discoveries and inventions should be separated by commas.The sentence is grammatically correct, but the use of "our" twice can be avoided for better flow.The date should be written as "September 1st" instead of "September first" for proper formatting.The subject-verb agreement is incorrect. "The practicality and advisability" should be followed by "is," not "are."The number "3" should be written as "Three." The parentheses should be placed correctly, and a comma is needed after "entertainment.""It's" should be corrected to "its" as it denotes possession, not a contraction. The sentence should be divided into two sentences for clarity.Learn more about spelling errors here:
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Which groups of numbers could 1/3 fall under. Natural,whole,integer,or rational
Answer:
The number 1/3 is a rational number. A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.
1/3 is not a natural number, whole number, or integer. Natural numbers are the set of positive integers (1, 2, 3, …). Whole numbers are the set of non-negative integers (0, 1, 2, 3, …). Integers are the set of whole numbers and their additive inverses (-3, -2, -1, 0, 1, 2, 3, …).
Step-by-step explanation:
d = 18 in. Fine the radius or diameter of each circle with the given dimensions.
Answer:
r= 9
Step-by-step explanation:
radius (r) = diameter (D)/2
= 18/2
= 9
Suppose that instead of the model inc; = Be + B, ed; + B2 sex, + &i (sex equals 1 for males and 0 for females) we posited inc; = 20 + a, ed, + a2 sexi + ei = 365 where sex* equals 0 for males and 1 for females. What would you expect concerning the relationship between the parameter estimates and the interpretation of these estimates from OLS regressions of these alternative formulations?
The two alternative formulations of the OLS regressions have differences in their parameter estimates and interpretations. The first model can be represented as: inci = Be + B1edi + B2sexi + εi, where sexi equals 1 for males and 0 for females. The second model is given by: inci = a0 + a1edi + a2sexi + ei, where sexi equals 0 for males and 1 for females.
The first model, the intercept term B0 represents the average income earned by men (when sexi equals zero) who have no education. The coefficient B1 represents the increase in income for each additional year of education. B2 represents the average difference in income between men and women when education is the same. Thus, B2 is the coefficient of the dummy variable for sex.In the second model, a0 represents the average income earned by women (when sexi equals one) who have no education. a1 represents the increase in income for each additional year of education, and a2 represents the average difference in income between men and women when education is the same. Thus, a2 is the coefficient of the dummy variable for sex.The interpretation of the coefficients is different between the two models. In the first model, B2 represents the average difference in income between men and women. In the second model, a2 represents the average difference in income between women and men. Thus, the coefficients of the dummy variable for sex have opposite signs and different interpretations. Also, the intercept terms in the two models are different, which reflects the difference in the average income earned by men and women in the two models. The parameter estimates from the OLS regressions for these alternative formulations will differ as well.
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whats 40/8 but not 5? if u get it right, u get branliest cuz u big brain
Answer:
it could also be [tex]\frac{20}{4}[/tex] or [tex]\frac{10}{2}[/tex]
Step-by-step explanation:
Simplify.
[tex]\frac{40}{8}[/tex] ÷ 2 = [tex]\frac{20}{4}[/tex]
[tex]\frac{20}{4}[/tex] ÷ 2 = [tex]\frac{10}{2}[/tex]
Rewrite in simplest terms: -8(9r – 1) – 9(-85 + 2)
The Answer is:
−72 r + 755
Answer:
-72r + 755
Step-by-step explanation:
-8(9r – 1) – 9(-85 + 2)
-72r + 8 - 9(-83)
-72r + 8 + 747
-72r + 755
Let f : R → R be a function. Define a sequence fn : R → R by 1 fn(x) = f ( x + = = (₁ + ₁/ ). n (a) Prove that if f is uniformly continuous, then fn → f uniformly. (b) Prove that uniform continuity of f is a necessary assumption to conclude that fn → ƒ uniformly. That is, give an example of a function ƒ : R → R such that f is continuous, but that fn does not converge to f uniformly.
(a) If f is uniformly continuous, then fn converges to f uniformly.
(b) Uniform continuity of f is necessary for the uniform convergence of fn to f; a counterexample is provided with a continuous f but non-uniform convergence of fn.
(a)If f is uniformly continuous, then we want to show that fn converges to f uniformly.
Given ε > 0, since f is uniformly continuous, there exists δ > 0 such that for any x, y in R, if |x - y| < δ, then |f(x) - f(y)| < ε.
Now, consider fn(x) = f(x + 1/n) for each n. We need to show that for any ε > 0, there exists N such that for all n ≥ N, |fn(x) - f(x)| < ε for all x in R.
Fix ε > 0. By the uniform continuity of f, there exists δ > 0 such that for any x, y in R, if |x - y| < δ, then |f(x) - f(y)| < ε.
Now, for any x in R, let N > 1/δ. Then for all n ≥ N and for any y = x + 1/n, we have |x - y| = |x - (x + 1/n)| = 1/n < δ.
By choosing y = x + 1/n, we have |fn(x) - f(x)| = |f(x + 1/n) - f(x)| < ε.
Since ε was chosen arbitrarily, this shows that fn converges to f uniformly.
(b) Counterexample: To show that uniform continuity of f is necessary for the uniform convergence of fn to f, we need to find an example where f is continuous but fn does not converge to f uniformly.
Consider the function f(x) = x on R. This function is continuous.
Now, let's examine the sequence fn(x) = f(x + 1/n). For any x in R, we have:
|fn(x) - f(x)| = |(x + 1/n) - x| = 1/n.
Notice that for any fixed x, as n approaches infinity, 1/n approaches 0. However, the convergence is not uniform since the choice of x affects the convergence rate.
For any ε > 0, no matter how large N is chosen, there will always exist x such that |fn(x) - f(x)| = 1/n ≥ ε, for n > 1/ε.
Therefore, fn does not converge uniformly to f(x) = x, despite f(x) being continuous.
This counterexample demonstrates that the assumption of uniform continuity of f is necessary for the uniform convergence of fn to f.
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