Given that f, g ∈ L’[a, b], we need to prove the Cauchy-Schwarz inequality, |(f, g)| = ||$|| . ||$||.
The Cauchy-Schwarz inequality for inner product in L’[a, b] states that for all f, g ∈ L’[a, b],|(f, g)| ≤ ||$|| . ||$||Proof: Consider a function Q(t) = (f + tg, f + tg) for any real number t. Then, by using the rules of inner product, we can expand this expression and obtain a quadratic polynomial in t.$$Q(t) = (f + tg, f + tg) = (f, f) + t(f, g) + t(g, f) + t^2(g, g)$$$$ = (f, f) + 2t(f, g) + t^2(g, g)$$. Now, Q(t) > 0 because Q(t) is a sum of squares. So, Q(t) is a quadratic polynomial that can have at most one real root since Q(t) > 0 for all t ∈ R.
To find the discriminant of Q(t), we need to solve the equation Q(t) = 0.$$(f, f) + 2t(f, g) + t^2(g, g) = 0$$.
The discriminant of Q(t) is:$$D = (f, g)^2 - (f, f)(g, g)$$
Since Q(t) > 0 for all t ∈ R, the discriminant D ≤ 0.$$D = (f, g)^2 - (f, f)(g, g) ≤ 0$$$$\Right arrow (f, g)^2 ≤ (f, f)(g, g)$$$$\Right arrow |(f, g)| ≤ ||$|| . ||$||$$
Thus, |(f, g)| = ||$|| . ||$||, which proves the Cauchy-Schwarz inequality. Therefore, the given statement is true.
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ok so Im getting banned and these cant go to waist so have fun :D
Answer:
Thx u
Step-by-step explanation:
that's not good oof
:3
:)
:0
:>
What’s the answer???
Answer:
Question 7 = $2
Question 8 = $16
Which of the following expressions are equivalent to 10 – 12? Choose all answers that apply:
А. 2.5 - 6
B.2(5 - 6)
C.None of the above
Answer:
B: 2(5−6)
Step-by-step explanation:
Plz answer quickly will you brainlist
Answer:
Positive association is correct
Step-by-step explanation: The dots form are going up, forming a positive line. If they were to go down they would be negative. So in this case positive association is correct
A cylinder has a volume of 792 m and a radius of 6 m. Find its height.
Answer:
7 m
Step-by-step explanation:
you divide the volume by the radius I'm pretty sure. that's what I did and I got 7
Find the solution of the initial value problem Y" – 2y – 3y = 15tet, y(0) = 2, y'(0) = 0. = = - 2 NOTE: Enter an exact answer. y(t) =
The solution of the initial value problem is:[tex]Y(t) = e^(3t) + e^(-t) - 3tet[/tex]
To solve the initial value problem Y" – 2y – 3y = 15tet, y(0) = 2, y'(0) = 0, we can use the method of undetermined coefficients.
First, we find the general solution of the homogeneous equation Y" – 2y – 3y = 0.
The characteristic equation is:
[tex]r^2 - 2r - 3 = 0[/tex]
Factoring the quadratic equation, we have:
(r - 3)(r + 1) = 0
This gives us two distinct roots: r = 3 and r = -1.
Therefore, the general solution of the homogeneous equation is:
[tex]Yh(t) = C1e^(3t) + C2e^(-t)[/tex]
To find a particular solution Yp(t) for the non-homogeneous equation, we assume a solution of the form Yp(t) = Atet, where A is a constant to be determined.
Taking the first and second derivatives of Yp(t), we have:
[tex]Yp'(t) = Ate^t + Aet[/tex]
[tex]Yp"(t) = Ate^t + 2Aet[/tex]
Substituting these derivatives into the non-homogeneous equation, we get:
[tex](Ate^t + 2Aet) - 2(Atet) - 3(Atet) = 15tet[/tex]
Simplifying the equation, we have:
[tex]Ate^t + 2Aet - 2Ate^t - 3Ate^t = 15tet[/tex]
Combining like terms, we get:
[tex](-4A + 2A - 3A)te^t = 15tet[/tex]
Simplifying further, we have:
[tex]-5Ate^t = 15tet[/tex]
Cancelling out the common terms, we get:
-5A = 15
Solving for A, we find:
A = -3
Now, we have the particular solution Yp(t) = -3tet.
The general solution of the non-homogeneous equation is the sum of the general solution of the homogeneous equation and the particular solution:
Y(t) = Yh(t) + Yp(t)
[tex]Y(t) = C1e^(3t) + C2e^(-t) - 3tet[/tex]
Using the initial conditions y(0) = 2 and y'(0) = 0, we can solve for the values of C1 and C2.
When t = 0:
[tex]Y(0) = C1e^(3(0)) + C2e^(-0) - 3(0)e^(0)[/tex]
2 = C1 + C2
Taking the derivative of Y(t) with respect to t and evaluating it at t = 0:
[tex]Y'(t) = 3C1e^(3t) - C2e^(-t) - 3te^(3t)Y'(0) = 3C1e^(3(0)) - C2e^(-0) - 3(0)e^(3(0))[/tex]
0 = 3C1 - C2
Solving these equations simultaneously, we find C1 = 1 and C2 = 1.
Therefore, the solution of the initial value problem is:
[tex]Y(t) = e^(3t) + e^(-t) - 3tet[/tex]
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A function is defined by f(x) = x²+2, ≥0. A region R is enclosed by y = f(x), the y-axis line y = 4.
Find the exact volume generated when the region R is rotated through 27 radians about the y-axis.
Given that a function is defined by `f(x) = x² + 2`, and the region R is enclosed by `y = f(x)`, the `y-axis` line `y = 4`. We need to find the exact volume generated when the region R is rotated through `27 radians` about the `y-axis`.
Explanation: The formula for finding the volume generated by rotating the region R about the `y-axis` is given by: `V = ∫ [from a to b] 2πxf(x) dx`. Here, the value of `a` is `0` because it's given that `f(x) = x² + 2`, and `f(x)` is greater than or equal to `0`. Also, the line `y = 4` intersects `f(x)` at `x = 2`. So, the value of `b` is `2`.
Therefore, the volume generated is given by: V = ∫ [from 0 to 2] 2πx (x² + 2) dx`=`2π ∫ [from 0 to 2] (x³ + 2x) dx`=`2π [(x⁴/4) + x²] {from 0 to 2}`=`2π [(2⁴/4) + 2²] - 0`=`2π [4 + 4]`=`16π` cubic units.
So, the exact volume generated when the region R is rotated through `27 radians` about the `y-axis` is `16π` cubic units.
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A study was conducted to determine if husbands and wives like the same TV shows. Married couples ranked 15th TV shows, and the Spearman rank correlation coefficient was found to be rs = 0.47.
a. Specify the competing hypotheses to determine if there is a positive correlation between the two rankings.
Null Hypothesis (H0): There is no positive correlation between the rankings of TV shows by husbands and wives in married couples.
Alternative Hypothesis (HA): There is a positive correlation between the rankings of TV shows by husbands and wives in married couples.
In simpler terms, the null hypothesis suggests that there is no relationship or association between the rankings of TV shows by husbands and wives.
The alternative hypothesis, on the other hand, proposes that there is a positive correlation between the rankings, indicating that husbands and wives tend to have similar preferences when it comes to TV shows.
The competing hypotheses in this study aim to determine whether there is evidence to support the idea that husbands and wives tend to like the same TV shows.
The null hypothesis assumes that there is no correlation between the rankings, meaning that the preferences of husbands and wives are independent of each other.
The alternative hypothesis, in contrast, suggests that there is a positive correlation, indicating a tendency for spouses to have similar preferences for TV shows.
To test these hypotheses, the researchers used the Spearman rank correlation coefficient (rs) to quantify the strength and direction of the relationship between the rankings.
The Spearman rank correlation is a statistical measure that assesses the monotonic relationship between two ranked variables, in this case, the rankings of TV shows by husbands and wives. A value of rs = 0.47 indicates a moderate positive correlation between the rankings.
To evaluate the hypotheses, statistical tests can be conducted. The significance level (alpha) is typically set in advance (e.g., 0.05) to determine the threshold for accepting or rejecting the null hypothesis.
If the p-value associated with the test is less than the chosen significance level, the null hypothesis is rejected in favor of the alternative hypothesis, suggesting that there is evidence of a positive correlation between the rankings of TV shows by husbands and wives.
Conversely, if the p-value is greater than the significance level, there is insufficient evidence to reject the null hypothesis, and the data does not provide support for a positive correlation.
It is important to note that the interpretation of the Spearman rank correlation coefficient and the conclusions drawn from the study should consider other factors, such as the sample size, sampling method, and the specific characteristics of the TV shows ranked by the couples.
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Find the value of x and the value of y.
A. x= 2 squared root of 3, y= 4 squared root of 3
B. x= 3,y= 6 squared root of 3
C. x= 6 squared root of 3, y=12
D. 2 squared root of 3, y= 6
Answer:
Option A
Step-by-step explanation:
To find all missing sides of a right triangle we use the sine, cosine or tangent ratio as below,
sinθ = [tex]\frac{\text{Opposite side}}{\text{Hypotenuse}}[/tex]
cosθ = [tex]\frac{\text{Adjacent side}}{\text{Hypotenuse}}[/tex]
tanθ = [tex]\frac{\text{Opposite side}}{\text{Adjacent side}}[/tex]
Now we take an angle measuring 60°
sin(60°) = [tex]\frac{6}{y}[/tex]
[tex]\frac{\sqrt{3} }{2}=\frac{6}{y}[/tex]
y = 4√3
tan(60°) = [tex]\frac{6}{x}[/tex]
√3 = [tex]\frac{6}{x}[/tex]
x = 2√3
Therefore, Option A will be the correct option.
The 50th percentile of the numbers: 13. 10, 12, 10, 11 is
A. 125. B. 11 C. 10 D. 11.5
Answer:
B. 11
Step-by-step explanation:
The 50th percentile represents the halfway point of a data set and therefore, it is simply another name for the median.
We can use the following steps to find the median:
Step 1: Arrange the numbers in ascending numerical order:
10, 10, 11, 12, 13.
Step 2: Find the middle of the numbers:
Since there are 5 numbers, the median will have two numbers to the left and right of it. 11 satisfies this requirement so it is the median and thus the 50th percentile of the numbers.
A cylinder has a base diameter of 20 inches and a height of 11 inches. What is its
volume in cubic inches, to the nearest tenths place?
Step-by-step explanation:
Volume=base area * height
=πr^2h
22/7 * 10^2 *11
=3457.1cm3
Answer:
The volume of a cylinder with a diameter of 10 inches and height of 20 inches is 1,570.8 cubic inches.
Please help with the square roots
Step-by-step explanation:
-√81=-√(9)²=-9
A number inside a Square root cannot ne negative. So, the second one is not a real no.
Answer:
[tex] - \sqrt{81} = - 9 \\ \sqrt{ - 25 = - 5} [/tex]
Step-by-step explanation:
-81/9 9+9+9+9+9+9+9+9+9(81)
-9
-25/5 (5+5+5+5+5(25)
-5
Studies have shown that a high percentage of analytical models actually used in the business world are simply wrong.
What's a good strategy - which I've repeatedly emphasized in this class - to avoid depending on wrong answers?
(Limit your answer to 10 words of less.)
A good strategy to avoid depending on wrong answers is to conduct rigorous testing and validation.
In the business world, many analytical models are found to be incorrect, as studies have shown. To avoid relying on flawed answers, it is crucial to implement a strategy that emphasizes rigorous testing and validation. This involves thoroughly evaluating the model's performance by comparing its outputs with known or expected outcomes. By subjecting the model to various scenarios and testing its predictions against real-world data, discrepancies can be identified and corrected.
Regularly testing and validating analytical models helps to uncover potential flaws and inaccuracies. This iterative process allows for adjustments and improvements to be made, ensuring that the model provides reliable and accurate results. By implementing a robust testing and validation strategy, businesses can minimize the risks associated with using incorrect analytical models and make informed decisions based on reliable insights.
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What is the solution to this system of equations?
X+ 2y = 4
2x-2y = 5
0 (3.-52
0 (3.3
O no solution
infinitely many solutions
Answer:
3, 1/2
Step-by-step explanation:
x + 2y = 4
2x - 2y = 5
_______________ +
2x + x + 2y + (-2y) = 4 + 5
3x + 0 = 9
3x = 9
x = 9/3
x = 3
if you want to find the value of y, you just have to choose one of the equation. I will choose x + 2y = 4, even if you choose 2x - 2y = 5 the result remains same
x + 2y = 4
3 + 2y = 4
2y = 4 - 3
y = 1/2
x, y = 3, 1/2
#CMIIWi'm sorry, i'm not good at english ^^
A researcher was interested in seeing if cats or dogs are more playful with their owners overall. The null hypothesis of this study is
a. dogs will play with their owners more than cats
b. cats will play with their owners more than dogs
c. cats and dogs play with their owners at the same rate
d. more information is needed
The null hypothesis of this study is the statement that there is no significant difference between the playfulness of cats and dogs with their owners. In other words, the researcher assumes that both cats and dogs will play with their owners at the same rate. This is option c.
To test this hypothesis, the researcher would need to collect data on the playfulness of both cats and dogs with their owners. This could involve observing the animals during playtime or asking owners to self-report how often their pets play with them. The data would then be analyzed using statistical tests to determine if there is a significant difference in the average rates of playfulness between cats and dogs.
It is important to note that the null hypothesis does not necessarily reflect the researcher's personal beliefs or assumptions about the topic. Instead, it serves as a baseline assumption that can be tested through empirical research. If the data collected suggests that cats and dogs do not play with their owners at the same rate, then the null hypothesis would be rejected, and the researcher would need to explore alternative explanations for the observed differences.
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15 points!!! PLEASE HELP!!!
Find the solution to the following equations below and identify either one solution, no solution, or infinite solutions. Be able to explain your choice.
3(x+4)=3x+11
-2(x+3)=-2x-6
4(x-1) = 1/2(x-8)
3x-7=4+6 +4x
Answer:
3(x+4)=3x+11
3x + 12 = 3x + 11
3x + 1 = 3x
No solution
-2(x+3)=-2x-6
-2x - 6 = -2x - 6
x = x
Infinite solutions
4(x-1) = 1/2(x-8)
4x - 4 = 1/2x - 4
8x - 8 = x - 8
7x = 0
x = 0
One solution
3x-7=4+6 +4x
3x - 7 = 10 + 4x
x = -17
One solution
Elena cashed a check for $$4350 at Quick Cash. The fee to cash
a check is 12% of the amount of the check. How much did Elena
pay to cash her check?
Answer:
522
Step-by-step explanation:
What is the total weight of 3 bags if their individual weights are 2/5, 7/10 and 3/5 pound? Give your answer as a mixed number in siplest form
Answer:
1 7/10
Step-by-step explanation:
Given that:
Weight of bag 1 = 2/5 pounds
Weight of bag 2 = 7/10 pounds
Weight of bag 3 = 3/5 pounds
Total weight of the three bags :
2/5 + 7/10 + 3/5
Take the lcm of 5 and 10
Lcm of 5 and 10 = 10
(4 + 7 + 6) / 10
17 /10
= 1 7/10
Find the perimeter of the figure
Answer: 132ft
Step-by-step explanation:
9x4=36+36=72
30-9=21
21+21+9+9=60
60+72=132ft
Branliest!!!! 100 points! solve for x in the image below:
also here's the equation: 3x+9= 90 degrees
Answer:
Hi! The answer to your question is x = 27
Step-by-step explanation:
☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆
☁Brainliest is greatly appreciated!☁
Hope this helps!!
- Brooklynn Deka
Answer:
x=27
Step-by-step explanation:
3x+9=90
subtract 9 from both sides
3x=90-9
3x=81
divide by 3
x= 27
checking
3x27 +9= 90
81+9=90
I hope this makes sense
Consider the rectangular prism.What is the surface area of the rectangular prism?
124 in
208 in
240 in
248 in
Answer: 240 in 2
Step-by-step explanation:
just did it
A random variable has a normal distribution with mean 7.5 and standard deviation 2.5. For which of the following is the probability equal to 0?
a) Less than 5.0
b) Between 4.0 and 10.0
c) Greater than 9.0
d) Between 6.0 and 8.0
The probability is equal to 0 for the option "a) Less than 5.0."Explanation:Given, the mean of the normal distribution, μ = 7.5 and the standard deviation of the normal distribution, σ = 2.5 The formula to find the Z-score, z is given by;z = (x - μ)/σwhere x is the value of the random variable under consideration.
a) To find the probability of the random variable being less than 5, we find the Z-score;z = (5 - 7.5)/2.5 = -1 Therefore, P(X < 5) = P(Z < -1)Using the standard normal table, the probability corresponding to the Z-score -1 is 0.1587.Therefore, the probability of the random variable being less than 5 is 0.1587.
b) To find the probability of the random variable being between 4.0 and 10.0, we find the Z-score corresponding to each value and calculate the difference in their probability. The probability required will be the absolute value of this difference. z 1 = (4 - 7.5)/2.5 = -1.4 z 2 = (10 - 7.5)/2.5 = 1 Therefore, P(4 < X < 10) = P(-1.4 < Z < 1) = P(Z < 1) - P(Z < -1.4) = 0.8413 - 0.0808 = 0.7605
c ).To find the probability of the random variable being greater than 9.0, we find the Z-score;z = (9 - 7.5)/2.5 = 0.6 Therefore, P(X > 9) = P(Z > 0.6)Using the standard normal table, the probability corresponding to the Z-score 0.6 is 0.2743.Therefore, the probability of the random variable being greater than 9.0 is 0.2743
d) To find the probability of the random variable being between 6.0 and 8.0, we find the Z-score corresponding to each value and calculate the difference in their probability. The probability required will be the absolute value of this difference. z 1 = (6 - 7.5)/2.5 = -0.6z2 = (8 - 7.5)/2.5 = 0.2Therefore, P(6 < X < 8) = P(-0.6 < Z < 0.2) = P(Z < 0.2) - P(Z < -0.6) = 0.5793 - 0.2743 = 0.305Therefore, the probability is equal to 0 for the option "a) Less than 5.0."
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A random variable has a normal distribution with mean 7.5 and standard deviation 2.5. The following is the probability equal to 0.
Option a) Less than 5.0 is correct.
Note that the normal distribution is symmetrical. So, the probabilities of events occurring are equal on either side of the mean.
The probability is zero when the range of events is beyond the limits of the standard normal distribution, which is from -3 to +3. Now let's standardize the values below:
a. less than 5.0: The formula to standardize is
[tex]z = (x - \mu) / \sigma[/tex]
z = (5 - 7.5) / 2.5
z = -1
Thus, the area of the left side of the standard normal distribution is zero, indicating that the probability of less than 5 is zero. Therefore, option a) is correct.
Other options are: b. Between 4.0 and 10.0: The probability that the values fall between 4 and 10 is 0.974 c. Greater than 9.0: The probability that the values are greater than 9 is 0.080 d. Between 6.0 and 8.0: The probability that the values fall between 6 and 8 is 0.329.
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Evaluate the requested derivatives: a) g(x) = 3x^3 -8x^2 -2x + 35 Find g'(2). b) k(x) = 1 /x^5 Find k"(1) c) n(x) = (-4x + 2)(3x^2 - 5x + 2) Find n'(0)
a) The derivative of g(x) at x=2 is g'(2) = 2.
b) The second derivative of k(x) at x=1 is k"(1) = -30.
c) The derivative of n(x) at x=0 is n'(0) = -18.
a) To find g'(x), we need to take the derivative of g(x) with respect to x. Let's differentiate each term separately:
g(x) = 3x³ - 8x² - 2x + 35
The derivative of 3x³ is obtained by applying the power rule, which states that if we have a term of the form [tex]ax^n[/tex], the derivative is given by [tex]nax^{(n-1)[/tex]. In this case, the derivative of 3x³ is 3 * 3x², which simplifies to 9x².
The derivative of -8x² is obtained in a similar manner, resulting in -16x.
The derivative of -2x is -2.
Since 35 is a constant term, its derivative is zero.
Now, let's combine these derivatives to find g'(x):
g'(x) = 9x² - 16x - 2
To find g'(2), we substitute x = 2 into the derivative:
g'(2) = 9(2)² - 16(2) - 2
= 9(4) - 32 - 2
= 36 - 32 - 2
= 2
Therefore, g'(2) = 2.
b) To find k"(x), we need to take the second derivative of k(x) with respect to x. Let's differentiate each term:
k(x) = 1 / [tex]x^5[/tex]
The derivative of 1/[tex]x^5[/tex] can be found using the power rule and the chain rule. The power rule states that the derivative of [tex]x^n[/tex] is n[tex]x^{(n-1)[/tex], and the chain rule applies when we have a function within another function. In this case, the derivative of 1/[tex]x^5[/tex] is -5/[tex]x^6[/tex].
Taking the derivative of -5/[tex]x^6[/tex], we apply the power rule again, resulting in 30/[tex]x^7[/tex].
Now, let's find k"(x) by differentiating -5/[tex]x^6[/tex] again:
k"(x) = -30/[tex]x^7[/tex]
To find k"(1), we substitute x = 1 into the second derivative:
k"(1) = -30/([tex]1^7[/tex])
= -30/1
= -30
Therefore, k"(1) = -30.
c) To find n'(x), we need to take the derivative of n(x) with respect to x. We can apply the product rule to differentiate the two factors of n(x):
n(x) = (-4x + 2)(3x² - 5x + 2)
Using the product rule, the derivative of n(x) is given by:
n'(x) = (-4x + 2)(d/dx)(3x² - 5x + 2) + (3x² - 5x + 2)(d/dx)(-4x + 2)
To differentiate each term, we use the power rule:
(d/dx)(3x² - 5x + 2) = 6x - 5
(d/dx)(-4x + 2) = -4
Substituting these derivatives back into n'(x), we get:
n'(x) = (-4x + 2)(6x - 5) + (3x² - 5x + 2)(-4)
Now, let's find n'(0) by substituting x = 0 into the derivative:
n'(0) = (-4(0) + 2)(6(0) - 5) + (3(0)² - 5(0) + 2)(-4)
= (2)(0 - 5) + (0 - 0 + 2)(-4)
= (2)(-5) + (2)(-4)
= -10 - 8
= -18
Therefore, n'(0) = -18.
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The county recreation department cleared 3/4 of a mile for a trail in Washington Park. There will be a small sign every 1/12 mile along the trail. How many signs are needed?
Answer:
its 9 signs
Step-by-step explanation:
in order to find number of signs you gonna divide total distance by distance of a small sign
3/4 ÷ 1/12
= 3/4 × 12/1 = 3 × 3
therefore, the answer is 9 sign
PUT THEM IN ORDER PLEASE
Answer:
3,2,4, then 1
Find the value of x. 2 3 6
What is the length of the legs of the triangle if the hypotenuse of an isosceles right triangle is (sqrt)23 feet?
4 feet
8 feet
16 feet
16.2 feet
please help me ..........
Answer:
It’s either A OR B
Step-by-step explanation:
What is the equation in standard form of
the line that passes through the points
(3, 5) and (-7, 2)?
Answer:
3x-10y=-41
Step-by-step explanation:
"standard form of the line" is ax+by=c, where a, b, and c are free coefficients
first, we need to find the slope (m) of the line
that is calculated with the formula (y2-y1)/(x2-x1)
we have the points (3,5) and (-7,2)
label the points:
x1=3
y1=5
x2=-7
y2=2
substitute into the equation
m=(2-5)/(-7-3)
m=-3/-10
m=3/10
the slope is 3/10
before we put a line into standard form, we need to put it into another form first-- like slope-intercept form (y=mx+b, where m is the slope and b is the y intercept)
we already know the slope
here's our line so far:
y=3/10x+b
we need to find b; since the line will pass through the points (3,5) and (-7,2) we can use either one of them to find b
let's use (3,5) as an example. Substitute into the equation
5=3/10(3)+b
5=9/10+b
41/10=b
b is 41/10
this is the equation:
y=3/10x+41/10
now we can find the equation in standard form. Subtract 3/10x from both sides
-3/10x+y=41/10
a (the number in front of x cannot be negative OR less than one. First, let's multiply both sides by -1)
3/10x-y=-41/10
multiply both sides by 10 to clear the fraction
3x-10y=-41
^^ is the equation
hope this helps!
For the following nonlinear system, 73 2 = y + 3x2 + 3x 91(x, y) = 7 y2 + 2y – X – 2 92(x, y) = 2 y = use the initial approximation (po, qo) = (-0.3, -1.3), and compute the next three approximations to the fixed point using (a) Jacobi iteration (b) Seidel iteration.
To compute the next three approximations to the fixed point using Jacobi iteration and Seidel iteration, we will use the initial approximation (p0, q0) = (-0.3, -1.3) for the given nonlinear system.
(a) Jacobi Iteration:
In Jacobi iteration, we update the variables simultaneously using the previous values.
(b) Seidel Iteration:
In Seidel iteration, we update the variables using the most recently computed values.
To compute the next three approximations to the fixed point using Jacobi iteration and Seidel iteration, we will use the initial approximation (p0, q0) = (-0.3, -1.3) for the given nonlinear system.
(a) Jacobi Iteration:
In Jacobi iteration, we update the variables simultaneously using the previous values. The iteration formula for this system is as follows:
p(n+1) = (y(n) + 3x(n)^2 + 3x(n) - 73)/3
q(n+1) = (7y(n)^2 + 2y(n) - x(n) - 2)/92
Using the initial approximation (-0.3, -1.3), we can compute the next three approximations as follows:
Iteration 1:
p(1) = (1 + 3(-0.3)^2 + 3(-0.3) - 73)/3 ≈ -8.300
q(1) = (7(-1.3)^2 + 2(-1.3) - (-0.3) - 2)/92 ≈ -1.317
Iteration 2:
p(2) = (1 + 3(-8.300)^2 + 3(-8.300) - 73)/3 ≈ -209.034
q(2) = (7(-1.317)^2 + 2(-1.317) - (-8.300) - 2)/92 ≈ -2.924
Iteration 3:
p(3) = (1 + 3(-209.034)^2 + 3(-209.034) - 73)/3 ≈ -14314.328
q(3) = (7(-2.924)^2 + 2(-2.924) - (-209.034) - 2)/92 ≈ -6.344
(b) Seidel Iteration:
In Seidel iteration, we update the variables using the most recently computed values. The iteration formula for this system is as follows:
p(n+1) = (y(n) + 3x(n)^2 + 3x(n) - 73)/3
q(n+1) = (7y(n+1)^2 + 2y(n+1) - x(n) - 2)/92
Using the initial approximation (-0.3, -1.3), we can compute the next three approximations as follows:
Iteration 1:
p(1) = (1 + 3(-0.3)^2 + 3(-0.3) - 73)/3 ≈ -8.300
q(1) = (7(-1.3)^2 + 2(-1.3) - (-0.3) - 2)/92 ≈ -1.315
Iteration 2:
p(2) = (1 + 3(-1.315)^2 + 3(-1.315) - 73)/3 ≈ -8.264
q(2) = (7(-8.264)^2 + 2(-8.264) - (-1.315) - 2)/92 ≈ -3.471
Iteration 3:
p(3) = (1 + 3(-3.471)^2 + 3(-3.471) - 73)/3 ≈ -1.252
q(3) = (7(-1.252)^2 + 2(-1.252) - (-3.471) - 2)/92 ≈ -1.100
These are the next three approximations to the fixed point using Jacobi iteration and Seidel iteration with the given initial approximation.
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