The 90% confidence interval for the ratio of variances is [0.05, 12.36].
Lets calculate the sample variances for each polymer.
For Epoxy:
Sample Variance (s₁²) = (1.74² + 2.15² + 2.02² + 1.95²) / (4 - 1)
= 0.135
For MMA Prepolymer:
Sample Variance (s₂²) = (1.78² + 1.57² + 1.72² + 1.67²) / (4 - 1)
=0.056
F statistic is F = (s₁²) / (s₂²)
F=0.135/0.056
=2.41
Determine the degrees of freedom for each sample.
For Epoxy: df₁ = 4 - 1 = 3
For MMA Prepolymer: df₂ = 4 - 1 = 3
Now find the critical F-value corresponding to a 90% confidence level with df₁ and df₂ degrees of freedom.
Using statistical tables or a calculator, the critical F-value for a 90% confidence level with df₁ = 3 and df₂ = 3 is approximately 5.14.
Calculate the lower and upper bounds of the confidence interval.
Lower Bound = (1 / F) × (s₁² / s₂²)
= (1 / 5.14) × (s₁² / s₂²)
= 0.050
Upper Bound = 5.14 × (s₁² / s₂²)
= F × (s₁² / s₂²)
=12.36
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Budget planners for a certain community have determined that $3,000,000 wel be required to provide a povernment service rester. The total property value in the communty 120,000,000 wat tax rate is required to meet the budgetary demands?
The tax rate required to meet the budgetary demands is 2.5%.
According to the given information;
Total property value in the community = $120,000,000
Total amount required to provide a government service = $3,000,000
Now, to find the tax rate required to meet the budgetary demands we will use the formula;
Tax Rate = (Total amount required to provide a government service / Total property value in the community) × 100
Substitute the given values in the above formula;
Tax Rate = ($3,000,000 / $120,000,000) × 100= 2.5%
Thus, the tax rate required to meet the budgetary demands for a community is 2.5 percent.
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Jordan runs to the end of his street and back home every day. The total distance of a trip to the end of the street and back home is 7/8 mile.
How many miles has Jordan run after 6 days?
Jordan has run 21/4 miles after 6 days.
Jordan runs to the end of his street and back home every day and the total distance of a trip to the end of the street and back home is 7/8 mile.
Since Jordan runs to the end of the street and back home every day, the distance he runs in one day is given by;
2 × (distance to the end of the street)
= 7/8 mile (distance to the end of the street)
= 7/16 mile
The distance Jordan runs in 6 days is;
6 × (distance to the end of the street and back home)
= 6 × 7/8 miles
= 42/8 miles
= 21/4 miles
Therefore, Jordan has run 21/4 miles after 6 days.
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Tammy read for 1/3 hour today. She read for 1/6 hour yesterday. How many hours did Tammy read in all?
Estimate the flow rate at t=9s.
Time (s) 0,1,5,8,11,15
Volume cm3 0,2,13.08,24.23,36.04,153.28
The estimated flow rate is approximately 3.94 cm3/s.
To estimate the flow rate at t=9s, we can use the formula:
flow rate = change in volume / change in time.
Using the data given, we can calculate the change in volume and change in time for the interval between t = 8s and t = 11s.
Change in volume = 36.04 - 24.23 = 11.81 cm³
Change in time = 11 - 8 = 3s
Now, we can plug these values into the formula to find the flow rate:
flow rate = change in volume / change in time = 11.81 cm3 / 3s ≈ 3.94 cm3/s
Therefore, the estimated flow rate at t=9s is approximately 3.94 cm3/s.
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The partial sum - 2 + ( − 6) + ( − 18) + ...... + (-486) =__________
The partial sum of the given sequence, -2 + (-6) + (-18) + ... + (-486), can be found using the formula Sₙ = a(1 - rₙ)/(1 - r), where a is the first term, r is the common ratio, and n is the number of terms. The value of the partial sum is 728.
To find the partial sum of the given sequence, we can use the formula for the sum of a geometric series, which is Sₙ = a(1 - rₙ)/(1 - r). In this case, the first term a is -2, and the common ratio r is -3. We need to determine the number of terms, n.
By examining the sequence, we can see that each term is obtained by multiplying the previous term by -3. This indicates that the common ratio is -3, as each term is multiplied by -3 to obtain the next term.
To find the number of terms, we can determine the value of n using the formula rₙ = a * r^(n-1). In this case, we have -486 = -2 * (-3)^(n-1).
By solving this equation, we find n = 6.
Substituting the values into the formula for the partial sum, we have:
S₆ = -2(1 - (-3)^6)/(1 - (-3)),
= -2(1 - 729)/(1 + 3),
= -2(-728)/4,
= 728.
Therefore, the partial sum of the given sequence is 728.
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Andrew must cut a rope 9 1/7 yards long into 8 equal
pieces. How long will each piece of rope be?
Answer:
each rope should be 1 1/7 yard long
Step-by-step explanation:
9 1/7 ÷8
1 1/7
Answer:
8/7 yd per piece
Step-by-step explanation:
To answer this, divide the total rope length 9 1/7 yd by 8 pieces:
64 yd
------------------ = 8/7 yd per piece
7(8 pieces)
Check: does 8 times (8/7 yd/piece) come out to 64/7 yd, or 9 1/7 yd? YES
Could someone plz help and show work? Thanks
Answer:
3 cm
Step-by-step explanation:
if you use a level of significance in a two-tail hypothesis test, what decision will you make if zstat -1.58?
In a two-tail hypothesis test, if the calculated test statistic (z-statistic) is -1.58 and the level of significance is used, the decision will depend on comparing the z-statistic to the critical values of the standard normal distribution corresponding to the desired level of significance.
Explanation: In a two-tail hypothesis test, the null hypothesis assumes that there is no significant difference between the sample and population parameters. The alternative hypothesis, on the other hand, suggests a significant difference. The level of significance, denoted as α, determines the critical values that divide the rejection and non-rejection regions.
If the calculated test statistic, in this case -1.58, falls within the rejection region, which is determined by the critical values, we reject the null hypothesis. If the test statistic falls outside the rejection region, we fail to reject the null hypothesis.
To make a decision, we compare the z-statistic to the critical values corresponding to the level of significance. If the z-statistic of -1.58 falls outside the critical values, it means it is not extreme enough to reject the null hypothesis, and we fail to reject it. However, if the z-statistic falls within the critical values, we reject the null hypothesis in favor of the alternative hypothesis.
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Show that the function f(x) f(x) = x3, x < 0 1 x2 sin, x > 0 x is differentiable.
To show that the function f(x) = x³ for x < 0 and f(x) = x²sin(x) for x > 0 is differentiable, we need to demonstrate that the function has a derivative at every point in its domain.
Let's consider the function f(x) separately for x < 0 and x > 0.
For x < 0
In this case, f(x) = x³. The power rule tells us that the derivative of xⁿ with respect to x is nxⁿ⁻¹. Applying this rule, we find that the derivative of f(x) = x³ is f'(x) = 3x².
For x > 0
In this case, f(x) = x²sin(x). The product rule is used when we have a function that is the product of two other functions. The derivative of f(x) can be calculated as follows
f'(x) = (x²)' sin(x) + x² (sin(x))'
To find the derivative of x² sin(x), we use the product rule again
(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
Let f(x) = x² and g(x) = sin(x). We have
f'(x) = 2x
g'(x) = cos(x)
Substituting these values back into the product rule equation
f'(x) = (x²)' sin(x) + x² (sin(x))'
= (2x) sin(x) + x^2 cos(x)
Therefore, the derivative of f(x) = x²sin(x) is f'(x) = (2x) sin(x) + x²cos(x).
Now, we have found the derivatives of f(x) for both x < 0 and x > 0. To show that f(x) is differentiable, we need to verify that the derivatives from both cases match at x = 0.
As x approaches 0 from the left side (x < 0), we have
lim(x → 0⁻) f'(x) = lim(x → 0⁻) 3x² = 0
As x approaches 0 from the right side (x > 0), we have
lim(x → 0⁺) f'(x) = lim(x → 0⁺) (2x) sin(x) + x²cos(x) = 0
Since the limits of the derivatives from both cases are equal at x = 0, we can conclude that f(x) = x³ for x < 0 and f(x) = x²sin(x) for x > 0 is differentiable at every point in its domain.
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Maxie spent 15 hours doing her homework last week this week she spent 18 hours doing her homework she says that she spent 120% more time doing homework this week is she correct
Answer: She's wrong.
Step-by-step explanation:
Numbers of hours used in solving homework last week = 15
Numbers of hours used in solving homework this week = 18
Percentage increase = (18 - 15) / 15 × 100
= 3/15 × 100
= 1/5 × 100
= 20%
Since Maxie said that she spent 120% more time doing homework this week, she's wrong. She only spent 20% more.
You are given the line y=-2x-5, and it then shifted up 2 units. Write your equation of the new line.
Answer:
i think the answer is 119943147893471987 yes its as easy as 1+1
Step-by-step explanation:
Answer: y=-2x-3
Step-by-step explanation:
What would the command "-index(F5:F277,randbetween(1.273))" do when entered in the spreadsheet with survey responses that we use for Project 2 (and will use for Project 3)? Return the most frequent answer to "Pineapple on pizza?" Return a random number between 1 and 273. Average the values in Column F. Change an answer in Column F at random. Pick a response at random from the responses to the question "Pineapple on pizza?" Return the greatest response to a random question
The command "-index (F5:F277, rand between (1.273))" when entered in the spreadsheet with survey responses that we use for Project 2 (and will use for Project 3) would pick a response at random from the responses to the question.
So, the correct option is: Pick a response at random from the responses to the question "Pineapple on pizza."
The INDEX function is an Excel worksheet function that finds the value or reference to a value within an array. It returns a reference to the location of the value, rather than the value itself. The INDEX function in Excel is a lookup and reference function.
The INDEX function allows you to search a spreadsheet and find the value contained in a given cell. The INDEX function takes two arguments, the array and the index number. The array is the range of cells that you want to search, while the index number is the position of the value you want to return.
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If 10 is the area of a circle what is the radius?
Answer: 1.785
Step-by-step explanation:
Answer:
Step-by-step explanation:
To determine the radius of a circle given its area, we can use the formula:
Area = π * radius^2
Given that the area is 10, we can set up the equation as follows:
10 = π * radius^2
To solve for the radius, we need to isolate it on one side of the equation. Dividing both sides by π, we get:
10 / π = radius^2
To find the radius, we can take the square root of both sides of the equation:
radius = √(10 / π)
Using a calculator to approximate the value of π as 3.14159, we can calculate:
radius ≈ √(10 / 3.14159)
radius ≈ √(3.1831)
radius ≈ 1.7849
Therefore, the radius of the circle is approximately 1.7849 when the area is 10.
hope it helps!!
Bridget has captured many purple-footed bog frogs. She weighs each one
and then counts the number of yellow spots on its back. This trend line is a
fit for these data.
$
Number of spots
NA DONN
1 2 3 4 5 6 7 8 9 10 11 12
Weight (g)
A. strong
B. parabolic
c. negative
D. weak
Answer:
A. Strong
Step-by-step explanation:
I took the test
Have a great day! ;)
This trend line is a strong fit for these data. Then the correct option is A.
What is the linear system?A linear system is one in which the parameter in the equation has a degree of one. It might have one, two, or even more variables.
Bridget has captured many purple-footed bog frogs.
She weighs each one and then counts the number of yellow spots on its back.
This trend line is a strong fit for these data.
Because all the points are closer to the line.
Then the correct option is A.
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U is the set of integers. G is the set of negative integers. What is the complement of set G in universive U?
Pls help fast
Answer:
Wouldn't the answer be C) Positive Integers?
Step-by-step explanation: Because the complement of a set include all of the elements not included in the indicated set. I hope I'm making some sense. :)
1.Explain why we need unit root test for stationary and the meaning of a spurious regression.
2.What are autocorrelation and Durbin-Watson test? And how are they related?
3. Explain the concept of VAR and VEC model and how they differ?
4.Explain the characteristic of an ARDL and its application
5.Explain the concept of cointegration and show how to perform the test for cointegration
6.Briefly explain these following tasks of heteroskedasticity: (1) the meaning of heteroskedasticity; (2) how to detect heteroskedasticity; (3) heteroskedasticity consequences for the OLS Estimation
1. Unit root tests are used to determine whether a time series variable is stationary or contains a unit root. Stationarity is a property of a time series where its statistical properties (such as mean, variance, and autocovariance) remain constant over time.
Unit root tests are important because many econometric models and statistical techniques assume stationarity. If a variable is non-stationary, it can lead to spurious regression.
Unit root tests help identify such cases by testing the null hypothesis of a unit root presence in the time series.
2. Autocorrelation refers to the correlation between the observations of a time series with their lagged values. It indicates the presence of a systematic relationship or dependence between the current observation and past observations.
The Durbin-Watson test is a statistical test used to detect autocorrelation in the residuals of a regression model.
The Durbin-Watson test statistic ranges from 0 to 4. A value close to 2 indicates no significant autocorrelation, while values significantly below 2 suggest positive autocorrelation, and values significantly above 2 suggest negative autocorrelation.
3. VAR models represent a system of equations where each variable is regressed on its own lagged values and the lagged values of all other variables in the system.
VAR models are widely used for forecasting, impulse response analysis, and studying dynamic relationships in macroeconomic and financial data.
VEC models, on the other hand, are a special case of VAR models designed to capture long-run equilibrium relationships among variables. VEC models incorporate error correction terms that help adjust for any deviations from the long-run equilibrium.
They are particularly useful when studying variables that exhibit cointegration, as they allow for the analysis of both short-run dynamics and long-run equilibrium relationships.
4. The Autoregressive Distributed Lag (ARDL) model is a regression model commonly used when dealing with time series data that may have a mix of stationary and non-stationary variables.
The ARDL model finds applications in macroeconomics, finance, and other fields where the relationship between variables may exhibit mixed order of integration.
5. Cointegration refers to the long-run equilibrium relationship between non-stationary time series variables. Cointegration implies that a linear combination of the variables is stationary, indicating a stable relationship.
6. Heteroskedasticity refers to the condition where the variance of the error term in a regression model is not constant across all levels of the independent variables. This violates the assumption of homoscedasticity, which assumes constant variance.
To detect heteroskedasticity, several methods can be used:
a) Graphical Analysis: Plotting the residuals against the predicted values or the independent variables to visually examine patterns of heteroskedasticity.
b) White's Test: A statistical test that regresses the squared residuals on the independent variables to test for heteroskedasticity.
Heteroskedasticity has consequences for Ordinary Least Squares (OLS) estimation:
a) OLS estimates of coefficients remain unbiased, but they are no longer efficient (standard errors are incorrect).
b) The t-tests and F-tests become invalid, leading to incorrect inference.
c) Confidence intervals and hypothesis tests may be distorted.
Correcting for heteroskedasticity can be done using robust standard errors or weighted least squares (WLS) estimation, which takes into account the heteroskedasticity structure of the error terms.
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Mrs. Habib has 46.25 feet of border for a bulletin board for her classroom. the board is 37.5 feet tall and 8.3 feet wide. how many feet of border will Mrs habib have left after she puts border around the board?
It’s not 22.15 I’ve tried.
Answer:
Mrs. Habib will have 22.25 feet of border left after she puts border around the board.
Step-by-step explanation:
You must find the perimeter of the board and subtract it from the amount of border she has to find how much she will have left after she uses it. The formula for perimeter is [tex]P=2(l+w)[/tex], where [tex]l=[/tex] the length of the board, and [tex]w=[/tex] the width of the board. You will add those together and multiply them by 2 because there are 4 sides to a rectangle. That means this equation will look like:
[tex]P=2(8.25+3.75)[/tex]
Now you can just solve for the perimeter.
[tex]P=2(12)[/tex]
[tex]P=24[/tex]
The perimeter is 24 feet. That means it will take 24 feet of border to cover her board. In order to find out how much she'll have left over, just subtract 24 from the total amount of border she has.
[tex]46.25-24=22.25[/tex]
Therefore Mrs. Habib will have 22.25 feet of border left over after she covers the bulletin board.
HELLPP WORTH 20 POINTS ✨
Answer:
it's radius is 9 and diameter is 18
Answer:
diameter=18cm
radius =diameter/2=18/2=9cm
According to a recent survey, the probability that the driver in a fatal vehicle accident is female (ovont F) is 0.2907 The probability that the driver is 24 years old or less (event A) is 0.1849. The probability that the driver is female and is 24 years old or less is 0.0542.
a. Find the probability of FUA
b. Find the probability of F'UA
The probability of F'UA is 0.9458.
According to the given data; the probability of ovont F is 0.2907, the probability of event A is 0.1849 and the probability of the driver is female and is 24 years old or less is 0.0542.
Here are the required probabilities;
a. The probability of FUA:F: Female U: 24 years old or less A: Fatal vehicle accident We can find the probability of FUA using the formula; P(FUA) = P(F ∩ U ∩ A)
We know that the probability of the driver in a fatal vehicle accident is female is 0.2907P(F) = 0.2907 Also, we know that the probability that the driver is 24 years old or less is 0.1849.P(U) = 0.1849
We also know that the probability that the driver is female and is 24 years old or less is 0.0542.P(F ∩ U) = 0.0542Now we can use the formula; P(FUA) = P(F ∩ U ∩ A)= P(F) x P(U) x P(A|FU)= 0.2907 × 0.1849 × (0.0542 / 0.2907)= 0.0542
So, the probability of FUA is 0.0542.
b. The probability of F'UA: It can be calculated by using the complement of FUA.P(F'UA) = 1 - P(FUA)= 1 - 0.0542= 0.9458
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Answer:
Step-by-step explanation:
Given data: The probability that the driver in a fatal vehicle accident is female (event F) is 0.2907. The probability that the driver is 24 years old or less (event A) is 0.1849. The probability that the driver is female and is 24 years old or less is 0.0542.
a) The probability of FUA is 0.4214.
b) The probability of F'UA is 0.5786.
a) The probability of FUA can be calculated as follows:
P(FUA) = P(F) + P(A) - P(F ∩ A) [By Addition Law], Where P(F) = 0.2907, P(A) = 0.1849, P(F ∩ A) = 0.0542.
By putting these values in the above equation we get:
P(FUA) = P(F) + P(A) - P(F ∩ A)
= 0.2907 + 0.1849 - 0.0542
= 0.4214
Therefore, the probability of FUA is 0.4214.
b) The probability of F'UA can be calculated as follows:
P(F'UA) = P(F' ∩ A') [By Complement Law], Where
P(F' ∩ A') = 1 - P(FUA)
= 1 - 0.4214
= 0.5786
Therefore, the probability of F'UA is 0.5786.
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Let f, g and h be the functions from the set of integers to the set of integers defined by f(x) = 2x +3, g(x) = 3x + 2 and h(x) = x3 +1.
(a) Find (fºg)(x) (b) Find (gof)(x) (c) l'ind (f)(x) (d) Find (h+h)(x) (e) Find h-1(x)
Let f, g and h be the functions from the set of integers to the set of integers defined by f(x) = 2x +3, g(x) = 3x + 2 and h(x) = x3 +1.
(a) To find (f º g)(x), we substitute g(x) into f(x) as follows:
(f º g)(x) = f(g(x)) = f(3x + 2) = 2(3x + 2) + 3 = 6x + 4 + 3 = 6x + 9.
(b) To find (g º f)(x), we substitute f(x) into g(x) as follows:
(g º f)(x) = g(f(x)) = g(2x + 3) = 3(2x + 3) + 2 = 6x + 9 + 2 = 6x + 11.
(c) To find the inverse of f(x), denoted as l'ind (f)(x), we solve for x in terms of f(x):
x = (f(x) - 3) / 2.
Rearranging the equation, we get f^(-1)(x) = 1/2x - 3/2.
(d) To find (h + h)(x), we add h(x) to itself:
(h + h)(x) = h(x) + h(x) = ([tex]x^3[/tex] + 1) + (x^3 + 1) = 2[tex]x^3[/tex] + 2.
(e) To find the inverse of h(x), denoted as h^(-1)(x), we solve for x in terms of h(x):
x = (h(x) - 1)^(1/3).
Rearranging the equation, we get h^(-1)(x) = (x - 1)^(1/3).
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2) A 95% confidence interval estimate for a population mean u is (23, 45). Which of the following is a true statement?
(A) There is 0.95 probability that μ is between 23 and 45.
(B) If 95% confidence intervals are calculated from all possible samples of the given size, μ will be in 95% of these intervals.
(C) If 95% confidence intervals are calculated from all possible samples of the given size, 95% of them will be
(23, 45).
(D) We are 95% confidence that the interval from (23, 45) contains the sample mean x
(E) The margin of error of this confidence interval is 22.
The correct statement for the 95% confidence interval is given by
option (B) If 95% confidence intervals are calculated from all possible samples of the given size, μ will be in 95% of these intervals.
Confidence interval = 95%
Population mean μ
A confidence interval is an estimate of a population parameter the population mean μ based on sample data.
The interpretation of a 95% confidence interval is that ,
Sample from the population and construct 95% confidence intervals,
Approximately 95% of these intervals would contain the true population parameter.
Therefore, statement (B) accurately reflects the concept of confidence intervals.
It states that if we calculate 95% confidence intervals from all possible samples of the given size,
The true population mean μ will be within 95% of these intervals.
This aligns with the interpretation of a confidence interval as a measure of the precision or reliability of our estimate.
The other statements which are not accurate,
(A) There is no probability associated with a specific confidence interval.
Confidence intervals provide a range of plausible values, but they do not represent probabilities of the parameter being within that range.
(C) Calculating confidence intervals from all possible samples will not guarantee that 95% of them will be (23, 45).
The specific values of the confidence intervals will vary across samples.
(D) Confidence intervals provide a range in which we are confident the true parameter lies.
But it does not imply that the sample mean x falls within that range with 95% certainty.
(E) The margin of error is the half-width of the confidence interval, which represents the maximum amount of error we expect in our estimate.
Here, the margin of error would be (45 - 23) / 2 = 11, not 22.
Therefore , for the confidence interval 95% option B is correct.
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Help me please ahhhh...Simplify the expression below.
2.5x. 4
Answer:
Maybe multiply 2 and 4 then multiply 5x of product of 2 and 4
Step-by-step explanation:
Use the following function rule to find f(6).
f(x)=6x+11
Answer:
X=-11/6
Steps
f(x)=6x+11
simplify,
0=6x+11
-6x+11
Divide both sides,
Answer is x = -¹¹/6
Daniel has 280 baseball cards. 15% of there are rare collector's items. How
many baseball cards does Daniel possess that are rare? *
Answer:
the answer is 42. hope this helped
First turn 15% into a decimal.
You get .15
Then multiply .15 by 280
You get 42
Find the center and radius of the circle: x^2 + y^2 + 4x + 14y +52 = 0
The center of the circle is point: C=(−2,−7).
Please help! I know its a lot and I'm sorry but I REALLY NEED HELP!!! I just don't understand this and I don't want to fail my brain just is not smart with math. Even if you answer just ONE question it would mean the WORLD TO ME thanks!
Answer:16
Step-by-step explanation:
What is the range of the absolute value function shown in the graph?
A. 3 ≤ y < ∞
B. -∞ < y ≤ 3
C. -6 ≤ y < ∞
D. -∞ < y < ∞
Answer:
C. -6 ≤ y < ∞
C is correct
Step-by-step explanation:
edmentum
What is the length of the diameter?
is there any instructions above the circle?
Estimate the derivative using forward finite divided difference applying both truncated and more accurate formula using 0.5 and step sizes of ht=0.25 and tu=0.125 4x12x2 + x3 -1 #x) = 5 + 3sinu = 2x1 + x2 + x3 = 4 2xy + 2x2 + x3 = 3
The more accurate forward finite divided difference estimates for the derivatives are
f₁'(x₁) = 0
f₂'(x₂) = 0
f₃'(x₃) = 0
To make it easier to work with, let's rearrange the equations in terms of the variables:
4x₁ + 2x₂ + x₃ = 1
2x₁ + x₂ + x₃ = 4
2x₁ + 2x₂ + x₃ = 3
The truncated formula for estimating the derivative using the forward finite divided difference is given by:
f'(x) ≈ (f(x + ht) - f(x)) / ht
Here, f(x) represents the function we want to differentiate, and ht is the step size.
Let's calculate the derivatives using the truncated formula for the given equations:
For x₁:
f₁'(x₁) ≈ (f₁(x₁ + ht) - f₁(x₁)) / ht
= (4(x₁ + ht) + 2x₂ + x₃ - 4x₁ - 2x₂ - x₃) / ht
= (4x₁ + 4ht + 2x₂ + x₃ - 4x₁ - 2x₂ - x₃) / ht
= (4ht) / ht
= 4
Similarly, we can calculate the derivatives for x₂ and x₃.
For x₂:
f₂'(x₂) ≈ (f₂(x₂ + ht) - f₂(x₂)) / ht
= (2x₁ + (x₂ + ht) + x₃ - 2x₁ - x₂ - x₃) / ht
= (x₂ + ht - x₂) / ht
= ht / ht
= 1
For x₃:
f₃'(x₃) ≈ (f₃(x₃ + ht) - f₃(x₃)) / ht
= (2x₁ + 2x₂ + (x₃ + ht) - 2x₁ - 2x₂ - x₃) / ht
= (x₃ + ht - x₃) / ht
= ht / ht
= 1
So, the truncated forward finite divided difference estimates for the derivatives are:
f₁'(x₁) = 4
f₂'(x₂) = 1
f₃'(x₃) = 1
The more accurate formula for estimating the derivative using the forward finite divided difference is given by:
f'(x) ≈ (-3f(x) + 4f(x + ht) - f(x + 2ht)) / (2ht)
Let's calculate the derivatives using the more accurate formula for the given equations:
For x₁:
f₁'(x₁) ≈ (-3f₁(x₁) + 4f₁(x₁ + ht) - f₁(x₁ + 2ht)) / (2ht)
= (-3(4x₁ + 2x₂ + x₃) + 4(4(x₁ + ht) + 2x₂ + x₃) - (4(x₁ + 2ht) + 2x₂ + x₃)) / (2ht)
= (-12x₁ - 6x₂ - 3x₃ + 16x₁ + 8ht + 4x₂ + 2x₃ - 4x₁ - 8ht - 2x₂ - x₃) / (2ht)
= (-12x₁ + 16x₁ - 4x₁ + 8ht - 8ht) / (2ht)
= 0
Similarly, we can calculate the derivatives for x₂ and x₃.
For x₂:
f₂'(x₂) ≈ (-3f₂(x₂) + 4f₂(x₂ + ht) - f₂(x₂ + 2ht)) / (2ht)
= (-3(2x₁ + x₂ + x₃) + 4(2x₁ + (x₂ + ht) + x₃) - (2x₁ + (x₂ + 2ht) + x₃)) / (2ht)
= (-6x₁ - 3x₂ - 3x₃ + 8x₁ + 4x₂ + 4ht + 4x₃ - 2x₁ - x₂ - x₃) / (2ht)
= (-6x₁ + 8x₁ - 2x₁ - 3x₂ + 4x₂ - x₂ - 3x₃ + 4x₃ - x₃ + 4ht) / (2ht)
= 0
For x₃:
f₃'(x₃) ≈ (-3f₃(x₃) + 4f₃(x₃ + ht) - f₃(x₃ + 2ht)) / (2ht)
= (-3(2x₁ + 2x₂ + x₃) + 4(2x₁ + 2x₂ + (x₃ + ht)) - (2x₁ + 2x₂ + (x₃ + 2ht))) / (2ht)
= (-6x₁ - 6x₂ - 3x₃ + 8x₁ + 8x₂ + 4x₃ + 4ht - 2x₁ - 2x₂ - x₃) / (2ht)
= (-6x₁ + 8x₁ - 2x₁ - 6x₂ + 8x₂ - 2x₂ - 3x₃ + 4x₃ - x₃ + 4ht) / (2ht)
= 0
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Fill in the blanks below in order to justify whether or not the mapping shown
represents a function.
Set A
Set B
4.
5
9
→ 2
-1
-3
Answer:
Step-by-step explanation:
How to write numbers in standard form:
Write the first number 8.
Add a decimal point after it: 8.
Now count the number of digits after 8. There are 13 digits.
So, in standard form: 81 900 000 000 000 is 8.19 × 10¹³