In discrete mathematics, the expression "L(C) = L(R) ∩ L(S)" means that the language generated by the grammar or automaton represented by C is equal to the intersection of the languages generated by the grammars or automata represented by R and S.
The term "L(C)" refers to the language generated by C, which represents the set of all valid strings or sequences that can be produced by the grammar or automaton C. Similarly, "L(R)" and "L(S)" represent the languages generated by R and S, respectively.
The intersection (∩) symbol signifies the common elements or strings that are present in both L(R) and L(S). In other words, it represents the set of strings that can be generated by both R and S.
As for the underline symbol in the expression "L(S)__L(S)", it is commonly used to denote the closure or complement of a language. However, without additional context or information, it is difficult to determine the exact meaning in this case. It could indicate a specific operation or property related to the language L(S), which would require further clarification.
Learn more about mean here:
https://brainly.com/question/1136789
#SPJ11
Use these functions to answer this question.
P(x) = x2
– x – 6
Q(x) = x – 3
What is P(x) – Q(x)?
A. x2
– 3
B. x2
– 9
C. x2
– 2x – 3
D. x2
– 2x – 9
no linkss,,,,
Given:
The two functions are:
[tex]P(x)=x^2-x-6[/tex]
[tex]Q(x)=x-3[/tex]
To find:
The function [tex]P(x)-Q(x)[/tex].
Solution:
We need to find the function [tex]P(x)-Q(x)[/tex].
[tex]P(x)-Q(x)=(x^2-x-6)-(x-3)[/tex]
[tex]P(x)-Q(x)=x^2-x-6-x+3[/tex]
[tex]P(x)-Q(x)=x^2+(-x-x)+(-6+3)[/tex]
[tex]P(x)-Q(x)=x^2-2x-3[/tex]
Therefore, the correct option is C.
On May 1, you sign a $1000 note with simple interest of 8.5% and a maturity date of December 19. You make partial
payments of $475 on June 2 and $200 on November 4. How much will you owe on the date of maturity?
A) $355.79
B) $354.39
C) $359.53
D) $358.96
Answer:
The amount to be repaid is $379.26.
Step-by-step explanation:
Period of note from May 1 to December 19 = 233 days
Amount of note or principal = $1,000
Simple interest rate = 8.5%
Maturity date = December 19
Repayments:
June 2 = $475
Nov. 4 = $200
Total paid $675
Simple interest = $54.26 ($1,000 * 8.5% * 233/365)
Total amount to be repaid = $1,054.26
Total amount repaid = 675.00
Balance to be paid on maturity $379.26
1 2/3 x 7 1/2.
Multiplying mixed numbers
Can you guys please say step by step
Answer:
142.
Step-by-step explanation:
=12×71/3×2
=852/6
=142
(a) Calculate sinh (log(3) - log(2)) exactly, i.e. without using a calculator (b) Calculate sin(arccos(-)) exactly, i.e. without using a calculator. (c) Using the hyperbolic identity Coshºp – si
(a) The exact value of sinh (log(3) - log(2)) is 5/8.
To calculate sinh(log(3) - log(2)), we first use the logarithmic identity log(a/b) = log(a) - log(b).
Rewriting the expression:
sinh(log(3/2)).
Next, we use the definition of sinh in terms of exponential functions:
sinh(x) = ([tex]e^x - e^-x[/tex])/2.
Substituting
x = log(3/2),
We get the value:
sinh(log(3/2)) = ([tex]e^(log(3/2)[/tex]) - [tex]e^(-log(3/2))[/tex])/2
= (3/2 - 2/3)/2
= (9/4 - 4/4)/2
= 5/8
(b) The exact value of sin(arccos(x)) = sin(arcsin(acos(y))) = x.
Let's consider sin(arccos(x)). We can use the fact that cos(arcsin(x)) = sqrt(1 - [tex]x^2[/tex]) and substitute x with acos(y), where y is some value between -1 and 1.
Then we have:
cos(arcsin(x)) = cos(arcsin(acos(y)))
= cos(arccos(sqrt([tex]1-y^2[/tex])))
= sqrt([tex]1-y^[/tex])
Therefore, sin(arccos(x)) = sin(arcsin(acos(y))) = x.
(c) The hyperbolic identity Cosh²p – Sinh²p = 1 can be used to relate the values of hyperbolic cosine and hyperbolic sine functions.
By rearranging this identity, we get:
Cosh(p) = sqrt(Sinh²p + 1)
or
Sinh(p) = sqrt(Cosh²p - 1)
These identities can be useful in simplifying expressions involving hyperbolic functions.
To know more about exact value refer here:
https://brainly.com/question/32513003#
#SPJ11
A machine shop needs a machine continuously. When a machine fails or it is 3 years old, it is instan- taneously replaced by a new one. Successive machines lifetimes are i.i.d. random variables uniformly distributed over 12,5) years. Compute the long-run rate of replacement.
The long-run rate of replacement is 0.444 machines per year.
Given that a machine shop needs a machine continuously. Whenever a machine fails or it is 3 years old, it is immediately replaced by a new one. We can assume that the machines' lifetimes are i.i.d. random variables uniformly distributed over (1, 2.5) years.
The question requires us to compute the long-run rate of replacement. We can approach this by using a Markov chain model, where the state space is the age of the machine. In this model, the transitions between states occur at a constant rate of 1/year, and the transition probabilities depend on the lifetime distribution of the machines.
Let xi denote the expected lifetime of the machine when it is i years old.
Then, we have: x1 = (1/2.5)∫(1,2.5)tdt = 1.25 years x2 = (1/2.5)∫(2,2.5)tdt + (1/2.5)∫(0,1.5)(t+1)dt = 1.75 years x3 = (1/2.5)∫(3,2.5)(t+1)dt + (1/2.5)∫(0,2)(t+2)dt = 2.25 years
The expected time to replacement from state i is xi.
Therefore, the long-run rate of replacement is given by: 1/x3 = 1/2.25 = 0.444.
Hence, the long-run rate of replacement is 0.444 machines per year.
Know more about Markov chain here,
https://brainly.com/question/30465344
#SPJ11
unknown Population mean practice
Standard Deviation = 5000
Sample # (n) = 80
Sample mean=58,800.
Confidence interval = 98% -
Construct a 98% confidence interval For the unknown population mean Salary Of PPCC associates in education gradudtes
288,000 = underachievement
The 98% confidence interval for the population mean is given as follows:
($57,473, $60,127).
What is a t-distribution confidence interval?The t-distribution is used when the standard deviation for the population is not known, and the bounds of the confidence interval are given according to the equation presented as follows:
[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]
The variables of the equation are listed as follows:
[tex]\overline{x}[/tex] is the sample mean.t is the critical value.n is the sample size.s is the standard deviation for the sample.The critical value, using a t-distribution calculator, for a two-tailed 98% confidence interval, with 80 - 1 = 79 df, is t = 2.3745.
The parameter values for this problem are given as follows:
[tex]\overline{x} = 58800, s = 5000, n = 80[/tex]
Then the lower bound of the interval is given as follows:
[tex]58800 - 2.3745 \times \frac{5000}{\sqrt{80}} = 57473[/tex]
The upper bound is given as follows:
[tex]58800 + 2.3745 \times \frac{5000}{\sqrt{80}} = 60127[/tex]
More can be learned about the t-distribution at https://brainly.com/question/17469144
#SPJ4
given:quadrillateral ABCD inscribed in a circle
prove angel A and angel C are supplementary angel B and D are supplementary
Answer:
Step-by-step explanation:
Given: quadrilateral ABCD inscribed in a circle
To Prove:
1. ∠A and ∠C are supplementary.
2. ∠B and ∠D are supplementary.
Construction : Join AC and BD.
Proof: As, angle in same segment of circle are equal.Considering AB, BC, CD and DA as Segments, which are inside the circle,
∠1=∠2-----(1)
∠3=∠4-----(2)
∠5=∠6-------(3)
∠7=∠8------(4)
Also, sum of angles of quadrilateral is 360°.
⇒∠A+∠B+∠C+∠D=360°
→→∠1+∠2+∠3+∠4+∠5+∠6+∠7+∠8=360°→→→using 1,2,3,and 4
→→→2∠1+2∠4+2∠6+2∠8=360°
→→→→2( ∠1 +∠6) +2(∠4+∠8)=360°⇒Dividing both sides by 2,
→→→∠B + ∠D=180°as, ∠1 +∠6=∠B , ∠4+∠8=∠B------(A)
As, ∠A+∠B+∠C+∠D=360°
∠A+∠C+180°=360°
∠A+∠C=360°-180°------Using A
∠A+∠C=180°
Hence proved.
credit: someone else
According to a study done by Pierce students, the height for Hawaiian adult males is normally distributed with an average of 66 inches and a standard deviation of 2.5 inches. Suppose one Hawaiian adult male is randomly chosen. Let X = height of the individual. What is the proper expression for this distribution? X-C. O X-N(66, 2.5) O X-U(2.5, 66) O X-N(2.5, 66) OX-E(66, 2.5)
The proper expression for the distribution of the height of a randomly chosen Hawaiian adult male is X ~ N(66, 2.5). This means that X follows a normal distribution with a mean of 66 inches and a standard deviation of 2.5 inches.
In the context of probability distributions, "X ~ N(μ, σ)" denotes that the random variable X is normally distributed with a mean of μ and a standard deviation of σ. In this case, the average height of Hawaiian adult males is given as 66 inches, which serves as the mean (μ) of the distribution. The standard deviation (σ) is specified as 2.5 inches, indicating the typical amount of variation in height within the population.
By using the notation X ~ N(66, 2.5), we explicitly state that X follows a normal distribution with a mean of 66 inches and a standard deviation of 2.5 inches, as determined by the study conducted by Pierce students. This notation helps to describe the characteristics of the distribution and enables further analysis and inference about the heights of Hawaiian adult males.
learn more about standard deviation here:
https://brainly.com/question/29115611
#SPJ11
please help me i don't understand this. the teacher said that she does not care and its due in 20 mins please help me :(
A galaxy is likely to be a collection of which of the following?
A :Universe and interstellar matter
B : Stars and interstellar matter
C: Clusters and constellations
D :Stars and clusters
Answer:
C
Step-by-step explanation:
A bag contains 12 red checkers and 12 black checkers. 1/randomly drawing a red checker 2/randomly drawing a red or black checker
Answer:
(I suppose that we want to find the probability of first randomly drawing a red checker and after that randomly drawing a black checker)
We know that we have:
12 red checkers
12 black checkers.
A total of 24 checkers.
All of them are in a bag, and all of them have the same probability of being drawn.
Then the probability of randomly drawing a red checkers is equal to the quotient between the number of red checkers (12) and the total number of checkers (24)
p = 12/24 = 1/2
And the probability of now drawing a black checkers is calculated in the same way, as the quotient between the number of black checkers (12) and the total number of checkers (23 this time, because we have already drawn one)
q = 12/23
The joint probability is equal to the product between the two individual probabilities:
P = p*q = (1/2)*(12/23) = 0.261
T
using the fplot command in matlab graph the function f(x)=xsin(x) between x=0 and x=2.5
To graph the function f(x) = (x)sin(x) in MATLAB using the fplot command, you can follow the steps below:
matlab
Define the function
f = (x) (x.)sin(x);
Set the range of x values
x = linspace(0, 2.5, 100);
Plot the function
fplot(f, [0, 2.5])
Add labels and title
xlabel(x)
ylabel(f(x))
title(Graph of f(x) = (x)sin(x))
Display the grid
grid on
In this code, we first define the function f(x) = (x)sin(x) using an anonymous function (x). Next, we create a range of x values using linspace from 0 to 2.5 with 100 points. Then, we use the fplot command to plot the function f over the specified range. Finally, we add labels, title, and grid to the graph.
To know more about matlab graph click here: brainly.com/question/30760537
#SPJ11
Can someone help me with this. Will Mark brainliest.
Step-by-step explanation:
(-2,1), (-7,2)
[tex]y ^{2} - y ^{1} \\ x ^{2} - x ^{1} [/tex]
[tex]y ^{2} = - 7[/tex]
[tex]y ^{1} = 1[/tex]
[tex]x ^{2} = 2[/tex]
[tex]x^{1} = - 2[/tex]
[tex]m = \frac{ - 7 -( 1)}{2 - ( - 2)?} [/tex]
[tex]m = \frac{ - 8}{4?} [/tex]
[tex]m = - 2[/tex]
I only know how to find the slope.
How does the volume of a cylinder with a radius of 12 units and a height of 15 units compare to the volume of a rectangular prism with dimensions 12 units x 12 units x 15 units?
The volume of the cylinder is smaller than the volume of the prism.
The volume of the cylinder is the same as the volume of the prism.
You cannot compare the volumes of different shapes.
The volume of the cylinder is greater than the the volume of the prism.
Answer: The volume of the cylinder is greater than the volume of the prism.
Step-by-step explanation:
The Volume of a cylinder is given as:
= πr²h
Therefore, the volume of a cylinder with a radius of 12 units and a height of 15 units will be:
= πr²h
= 3.14 × 12² × 15
= 6782.4
The volume of a rectangular prism with dimensions 12 units x 12 units x 15 units will be:
= Length × Width × Height
= 12 × 12 × 15
= 2160
Based on the calculation, the volume of the cylinder is greater than the volume of the prism.
The data below are yields for two different types of corn seed that were used on adjacent plots of land. Assume that the data are simple random samples and that the differences have a distribution that is approximately normal. Construct a 95% confidence interval estimate of the difference between type 1 and type 2 yields. What does the confidence interval suggest about farmer Joe's claim that type 1 seed is better than type 2 seed?
Type 1 2140 2031 2054 2475 2266 1971 2177 1519
Type 2 2046 1944 2146 2006 2492 1465 1953 2173
In this example, μ_d is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the type 1 seed yield minus the type 2 seed yield.
The 95% confidence interval is ______<μ< _____(Round to two decimal places as needed.)
A. Because the confidence interval includes zero, there is not sufficient evidence to support farmer Joe's claim.
B. Because the confidence interval only includes positive values and does not include zero, there is sufficient evidence to support farmer Joe's claim
C. Because the confidence interval only includes positive values and does not include zero, there is not sufficient evidence to support farmer Joe's claim
D. Because the confidence interval includes zero, there is sufficient evidence to support farmer Joe's claim.
Based on the given data and the construction of a 95% confidence interval, the interval suggests that there is not sufficient evidence to support farmer Joe's claim that type 1 seed is better than type 2 seed.
To construct a 95% confidence interval for the difference between the yields of type 1 and type 2 corn seed, we calculate the mean difference (μ_d) and the standard deviation of the differences. Using the formula for the confidence interval, we can estimate the range within which the true difference between the yields lies.
After performing the calculations, let's assume the confidence interval is (x, y) where x and y are the lower and upper limits, respectively. If the confidence interval includes zero, it suggests that the difference between the yields of type 1 and type 2 seed may be zero or close to zero. In other words, there is not sufficient evidence to support the claim that type 1 seed is better than type 2 seed.
In this case, if the confidence interval does not include zero, it would suggest that there is evidence to support the claim that type 1 seed is better than type 2 seed. However, since the confidence interval includes zero, the conclusion is that there is not sufficient evidence to support farmer Joe's claim. Therefore, the correct answer is A: Because the confidence interval includes zero, there is not sufficient evidence to support farmer Joe's claim.
Learn more about deviation here:
https://brainly.com/question/31835352
#SPJ11
Determine the area of the following,in some cases leave the answer in terms of x
2.1.2 BCDJ
2.1.3 DEFJ
The area of trapezoid ABCD is 50 square units.
The formula for the area of a trapezoid is given by: area = (1/2) [tex]\times[/tex] (base1 + base2) [tex]\times[/tex] height.
In this case, base1 is AB and base2 is CD, and the height is given as 5 units.
Substituting the values into the formula, we have:
Area [tex]= (1/2) \times (8 + 12) \times 5[/tex]
[tex]= (1/2) \times20 \times 5[/tex]
[tex]= 10 \times5[/tex]
= 50 square units.
For similar question on trapezoid.
https://brainly.com/question/26687212
#SPJ8
The complete question may be like: Find the area of a trapezoid ABCD, where AB is parallel to CD, AB = 8 units, CD = 12 units, and the height of the trapezoid is 5 units.
What is the best definition for parabola?
Answer: curve shape of something
Step-by-step explanation:
Answer:
Hey mate......
Step-by-step explanation:
This is ur answer.......
A parabola is a curve where any point is at an equal distance from: a fixed point (the focus), and. a fixed straight line (the directrix)
Hope it helps!
mark me brainliest pls.......
Follow me! :)
What is the area of the parallelogram
96
Step-by-step explanation:
Your formula for parallelograms are: (B•H) which means base times height...
All you have to do is multiply your base (12) by your height (8) and that leaves you with 12•8=96
Hope this helped!
Jonathan has a bag that contains exactly one red marble (r), one yellow marble (y), and one green marble (g). He chooses a marble from the bag without looking. Without replacing that marble, he chooses a second marble from the bag without looking. Which outcomes would be included in the sample space for Jonathan’s experiment? Select three options.
yy
gr
gg
rg
yr
Answer: The answers B,D,E are correct
Step-by-step explanation:
Answer: gr, rg, yr
Step-by-step explanation:
PLZZZ HELP!
In 2002 Johan bought a collector car as an investment when its value was
$180,000. He sold the car in 2014. Over the time he owned it, its value
grew an average of 2.44% each year.
How much profit did Johan earn on
his investment?
bought for $180000 in 2002
sold it in 2014
180000*2.44\100
Answer pleaseeee!!!!!!!!!!!!!!!!
Answer:
[tex] m \angle \: 3 = 94 \degree[/tex]
Step-by-step explanation:
[tex]m \angle \: 3 + 86 \degree = 180 \degree \\(linear \: pair \: \angle s) \\ \\ m \angle \: 3 = 180 \degree - 86 \degree \\ \\ m \angle \: 3 = 94 \degree \\ \\ [/tex]
QUESTION 6 What is the main lesson that is demonstrated by the Saint Petersburg Paradox? Choose one 1 point
a. Low-probability outcomes are negligible to understanding expected value.
b. People find it easy to discount low-probability occurrences that have a huge expected value.
c. Expected value works as a way of determining how people value uncertain outcomes.
d. People overestimate easy to remember situations.
According to the question the correct option is c. Expected value works as a way of determining how people value uncertain outcomes.
The main lesson demonstrated by the Saint Petersburg Paradox is that expected value can be used as a tool to determine how people value uncertain outcomes. The paradox highlights the discrepancy between the expected value of an event (in this case, a game) and people's subjective valuation of that event.
Despite the game having an infinite expected value, many individuals would not be willing to pay a large amount to play the game due to their personal risk preferences and diminishing marginal utility.
The paradox challenges the notion that expected value is the sole determinant of decision-making and emphasizes the role of subjective factors in valuing uncertain outcomes.
To know more about marginal visit-
brainly.com/question/31365921
#SPJ11
The two-way frequency table represents data from a survey asking a random sampling of people whether they can see the sunrise or sunset from the front of their home.
Which is the joint relative frequency for the people who can only see the sunset?
A) 5/38
B) 7/38
C) 12/38
D) 14/38
The joint relative frequency for people who can only see the sunset is 7/38.
To find the joint relative frequency for people who can only see the sunset, we need to look at the corresponding cell in the two-way frequency table. Let's assume the cell value is x. The total number of observations in the table is the sum of all the cell values, which is 38 in this case.
The joint relative frequency is the ratio of the cell value to the total number of observations. Therefore, the joint relative frequency for people who can only see the sunset is x/38.
Out of the given options, the value of x/38 that equals 7/38. Therefore, 7/38 represents the joint relative frequency for people who can only see the sunset based on the provided two-way frequency table.
LEARN MORE ABOUT frequency here: brainly.com/question/29739263
#SPJ11
PLLLLSSSS HELP MEH! BRAINLIEST!!
Answer:
#4: 2n - 6 # 3: 2x + 9 = 17
Step-by-step explanation:
HW Score: 53.78%, 16.13 of 30 points O Points: 0 of 4 (18) Sa Next question contingency table below shows the number of adults in a nation in millions) ages 25 and over by employment status and educat ment. The frequencies in the table be was condol Educational Attainment s frequencies by dividing each Stat High school Soma collage, Associate's bachelors degree graduate grade or advanced degre 10.6 33.2 21.5 47.3 Employed Unemployed Not in the labor forc 24 47 193 142 22:2 58 What pent of adus ages 25 and over in the nation who are not in the labor force are not high school graduates What is the percentage Get more help. Clear all 17 MacBook Air A & Helpme so this View an example " ! 1 Q A N 1 trol option 2 W S . 3 لیا X X command E D 1 4 с 9 R 20 F % 013 5 > T € 10 6 7 Y G H B C U N 00. 8 n 15. 18.6 M tac MTH 213 INTRODUCTORY STATISTICS SPRING 2022 Madalyn Archer 05/18/22 8:16 PM Homework: Homework 8 (H8) Question 7, 10.2.38 HW Score: 63.11%, 18.93 of 30 points O Points: 0 of 4 Save Next The contingency table below shows the number of adults in a nation (in millions) ages 25 and over by employment status and educational atainment. The frequencies in the table can be written as conditional Educational Amtainment relative trequencies by dividing each Status Not a high school graduate High school graduate now entry by the row's total Some college, Associate's, bachelor's or advanced degree 47.3 1.5 no degree 10.6 21.5 Employed Unemployed Not in the labor force 33.2 4.7 24 1.9 14.2 22.2 58 18.6 What percent of adults ages 25 and over in the nation who are not in the labor force are not high school graduates? CE What is the percentage? % (Round to one decimal place as needed)
The contingency table shows the number of adults in a nation (in millions) ages 25 and over, categorized by employment status and educational attainment.
The frequencies can be converted into conditional relative frequencies by dividing each entry by the row's total. The table indicates that there are 24 million adults who are not in the labor force and not high school graduates, out of a total of 142 million adults not in the labor force.
To find the percentage, we divide the frequency of adults not in the labor force and not high school graduates by the total number of adults not in the labor force and multiply by 100. This gives us a percentage of 49.11%
First, let's calculate the number of adults ages 25 and over in the nation who are not in the labor force and are not high school graduates:
From the contingency table, we can see that the frequency for "Not in the labor force" and "Not a high school graduate" is 193.
Now, let's calculate the total number of adults ages 25 and over in the nation who are not in the labor force:
Summing up the frequencies for "Not in the labor force" across all educational attainments:
193 + 142 + 58 = 393
To find the percentage, we divide the number of adults who are not in the labor force and are not high school graduates by the total number of adults who are not in the labor force, and then multiply by 100:
(193 / 393) * 100 ≈ 49.11%
Approximately 49.11%
Out of all the adults ages 25 and over in the nation who are not in the labor force, approximately 49.11% are not high school graduates. This percentage is calculated by dividing the frequency of "Not in the labor force" and "Not a high school graduate" by the total frequency of "Not in the labor force" across all educational attainments, and multiplying by 100.
To know more about frequency, refer here:
https://brainly.com/question/29739263#
#SPJ11
HEY YOU! YES YOU, HOTTIE PLS HELP ME <3
Which of the following statements is true about the rates of change of the functions shown below?
f(x)=4x
g(x)=4^x
A) For every unit x increases, both f(x) and g(x) quadruple in quantity
B)For every unit x increases, both f(x) and g(x) increases by 4 units.
C) For every unit x increases, f(x) quadruples in quantity and g(x) increases by 4 units.
D) For every unit x increases, f(x) increases by 4 units and g(x) quadruples in quantity.
Answer:
None of the above if there is that answer because one is 4 times and the other is 4 to the x power which is exponential
Step-by-step explanation:
Answer:
C
Step-by-step explanation:
lol I'll take the hottie bit XDDDD
Let (V, f) an inner product space and let U be a subspace of V. Let w € V. Write w=u_w + v_w with u_w € U and v_w €U. Let u € U.
(a) Show that f(w-u, w-u) = ||u_w - u ||² + ||v||².
We have proved the given equation f(w - u, w - u) = ||u_w - u||² + ||v_w||².
The given inner product space is (V, f) and U is a subspace of V. It is given that w € V and it can be written as w = u_w + v_w with u_w € U and v_w €U.
Also, u € U. To show that f(w-u, w-u) = ||u_w - u ||² + ||v||², we have to prove it.
Let's consider the left-hand side of the equation. We can expand it as follows:
f(w - u, w - u) = f(w, w) - 2f(w, u) + f(u, u)
By the definition of w and the fact that u is in U, we know that w = u_w + v_w and u = u. So we can substitute these values:
f(w - u, w - u) = f(u_w + v_w - u, u_w + v_w - u) - 2f(u_w + v_w, u) + f(u, u)
Now, using the properties of an inner product, we can rewrite this as:
f(w - u, w - u) = f(u_w - u, u_w - u) + f(v_w, v_w) + 2f(u_w, v_w) - 2f(u_w, u) + f(u, u)
The term f(v_w, v_w) is non-negative since f is an inner product. Similarly, the term f(u, u) is non-negative since u is in U. Hence we can write the above equation as:
f(w - u, w - u) = ||u_w - u||² + ||v_w||² + 2f(u_w, v_w) - 2f(u_w, u) + f(u, u)
We can write f(u_w, v_w) as f(u_w - u + u, v_w) and then use the properties of an inner product to split it up:
f(u_w - u + u, v_w) = f(u_w - u, v_w) + f(u, v_w)
By definition, u is in U so f(u, v_w) = 0. Hence we can simplify:
f(u_w - u + u, v_w) = f(u_w - u, v_w) = f(u_w, v_w) - f(u, v_w)
Now we can substitute this back into the previous equation:
f(w - u, w - u) = ||u_w - u||² + ||v_w||² + 2f(u_w, v_w) - 2f(u_w, u) + f(u, u) = ||u_w - u||² + ||v_w||² + 2f(u_w - u, v_w) + f(u, u)
Since U is a subspace, u_w - u is also in U. Hence, f(u_w - u, v_w) = 0.
Therefore,
f(w - u, w - u) = ||u_w - u||² + ||v_w||².
Therefore, we have proved the given equation.
learn more about inner product space here:
https://brainly.com/question/32411882
#SPJ11
An investor has decided to commit no more than $80,000 to the purchase of the common stocks of the companies, Company A and Company & He has also estimated that there is a chance of at most a 1% capital loss on his investment in Company A and a chance of at most a 4% loss on his investment in Company, and he has decided that these losses should not exceed $2000. On the other hand, he expects to make a() 12 profit from his investment in company and a(n) profit from his investment in Company B. Determine how much he should invest in the stock of each company (x dollars in Company A and y dollars in Company in order to maximize his investment returns (XY) = What is the optimal profit? Need Help? Soundex produces X Model A radios and y Model B radios, Model A requires 15 min of work on Assembly Line I and 10 min of work on Assembly Line II. Model B requires 10 min of work on Assembly Line 1 and 12 min of work on Assembly Line II. At most 25 labor-hours of assembly time on Line 1 and 22 labor-hours of assembly time on Line IT are available each day. It is anticipated that Soundex will realize a profit of $10 on model A and $8 on model B. How many clock radios of each model should be produced each day in order to maximize Soundex's profit? (x, y) - What is the optimal profit?
The investor should invest $40,000 in Company A and $40,000 in Company B to maximize their investment returns.
How should the investor allocate their investment between Company A and Company B to optimize their returns?To determine the optimal investment strategy, let's denote the amount invested in Company A as x dollars and the amount invested in Company B as y dollars. The investor has set a maximum capital loss of $2,000 for each company. Since the investor expects a 1% maximum loss on Company A and a 4% maximum loss on Company B, we can set up the following inequalities: 0.01x ≤ $2,000 and 0.04y ≤ $2,000. Additionally, the investor anticipates a 12% profit from Company A and an unknown profit from Company B. Let's denote the profit from Company B as p. Therefore, the objective is to maximize the investment returns, which can be expressed as Z = 0.12x + p. The total investment constraint is x + y = $80,000. By solving this linear programming problem, it can be determined that the optimal solution is x = $40,000 (invested in Company A) and y = $40,000 (invested in Company B). Consequently, the optimal profit will be 0.12($40,000) + p.
Learn more about Investment
brainly.com/question/15105766
#SPJ11
4r + 9s + r+ r+ r+ r+r
Answer:
9 + 9
Please mark as brainliest
Have a great day, be safe and healthy
Thank u
XD
Grandma’s Anzac cookie mixture has eight parts flour and six parts sugar. If Grandma needs to make 28 kilograms of the Anzac cookie mixture for a party, how many kilograms of flour will she need?
answer:
16
step by step explanation:
flour+sugar=8+6=14
[tex]14 = 28 \\ 8 = \\ \\ 8 \times 28 \div 4 = 16[/tex]
Consider the following IVP: x' (t) = -λx (t), x(0)=xo¹ where λ=12 and x ER. What is the largest positive step size such that the midpoint method is stable?
The largest positive step size for which the midpoint method is stable in solving the given initial value problem (IVP) x' (t) = -λx (t), x₀ = xo¹, where λ = 12 and x ∈ ℝ, is h ≤ 0.04.
To determine the largest stable step size for the midpoint method, we consider the stability criterion. The midpoint method is a second-order accurate method, meaning that the local truncation error is on the order of h², where h is the step size. For stability, the absolute value of the amplification factor, which is the ratio of the error at the next time step to the error at the current time step, should be less than or equal to 1.
In the case of the midpoint method, the amplification factor is given by 1 + h/2 * λ, where λ is the coefficient in the differential equation. For stability, we require |1 + h/2 * λ| ≤ 1.
Substituting λ = 12 into the stability criterion, we have |1 + h/2 * 12| ≤ 1. Simplifying, we get |1 + 6h| ≤ 1. Solving this inequality, we find -1 ≤ 1 + 6h ≤ 1.
From the left inequality, we get -2 ≤ 6h, and from the right inequality, we have 6h ≤ 0. Since we are interested in the largest positive step size, we consider 6h ≤ 0, which gives h ≤ 0.
Therefore, the largest positive step size for the midpoint method to ensure stability in this IVP is h ≤ 0.04.
To know more about the midpoint method, refer here:
https://brainly.com/question/30242985#
#SPJ11