Answer:
\((x - y)(x + y + 1)\)
Step-by-step explanation:
\(x(x + 1) - y(y + 1)\)
\(=> x^{2} + x - y^{2} - y\)
Lets re-arrange it.
\(=> x^{2} - y^{2} + x - y\)
Now , lets expand x² - y².
\(=> (x + y)(x - y) + (x - y)\)
Taking ( x - y ) common ,
\(=> (x - y)(x + y + 1)\)
Answer:
\(x(x + 1) - y(y + 1) \\ → {x}^{2} + x - {y}^{2} - y \\→( {x}^{2}- {y}^{2}) + (x - y) \\→ (x - y)(x + y) + (x - y) \\→ \boxed{ (x - y)(x + y + 1)}✓\)
(x-y)(x+y+1) is the right answer.Which equation could be used to create the graph? y=x−2 y=x−4 y=x3 y=3x
Answer:
y=3x
Step-by-step explanation:
If I am wrong sorry.
Answer:
y=3x
Step-by-step explanation:
i just took the test
If the observations have weights of 2, 3 and 1 respectively, solve these equations for the most probable values of A and B using weighted least squares method. Solve the problem using both algebraic approach and matrices and compare your results.
A+2B=10.50+V1
2A-3B=5.55+V2
2A-B=-10.50+V3
The results obtained using the algebraic approach and the matrix approach should be the same. Both methods are mathematically equivalent and provide the most probable values of A and B that minimize the sum of squared weighted residuals.
To solve the system of equations using the weighted least squares method, we need to minimize the sum of the squared weighted residuals. Let's solve the problem using both the algebraic approach and matrices.
Algebraic Approach:
We have the following equations:
A + 2B = 10.50 + V1 ... (1)
2A - 3B = 5.55 + V2 ... (2)
2A - B = -10.50 + V3 ... (3)
To minimize the sum of squared weighted residuals, we square each equation and multiply them by their respective weights:
\(2^2 * (A + 2B - 10.50 - V1)^2\)
\(3^2 * (2A - 3B - 5.55 - V2)^2\\1^2 * (2A - B + 10.50 + V3)^2\)
Expanding and simplifying these equations, we get:
\(4(A^2 + 4B^2 + 10.50^2 + V1^2 + 2AB - 21A - 42B + 21V1)\\9(4A^2 + 9B^2 + 5.55^2 + V2^2 + 12AB - 33A + 16.65B - 11.1V2)\\(A^2 + B^2 + 10.50^2 + V3^2 + 2AB + 21A - 21B + 21V3)\\\)
Now, let's sum up these equations:
\(4(A^2 + 4B^2 + 10.50^2 + V1^2 + 2AB - 21A - 42B + 21V1) +\\9(4A^2 + 9B^2 + 5.55^2 + V2^2 + 12AB - 33A + 16.65B - 11.1V2) +\\(A^2 + B^2 + 10.50^2 + V3^2 + 2AB + 21A - 21B + 21V3)\int\limits^a_b {x} \, dx\)
Simplifying further, we obtain:
\(14A^2 + 31B^2 + 1113 + 14V1^2 + 33V2^2 + 14V3^2 + 14AB - 231A - 246B + 21V1 - 11.1V2 + 21V3 = 0\)
Now, we have a single equation with two unknowns, A and B. We can use various methods, such as substitution or elimination, to solve for A and B. Once the values of A and B are determined, we can substitute them back into the original equations to find the most probable values of A and B.
Matrix Approach:
We can rewrite the system of equations in matrix form as follows:
| 1 2 | | A | | 10.50 + V1 |
| 2 -3 | | B | = | 5.55 + V2 |
| 2 -1 | | -10.50 + V3 |
Let's denote the coefficient matrix as X, the variable matrix as Y, and the constant matrix as Z. Then the equation becomes:
X * Y = Z
To solve for Y, we can multiply both sides of the equation by the inverse of X:
X^(-1) * (X * Y) = X^(-1) * Z
Y = X^(-1) * Z
By calculating the inverse of X and multiplying it by Z, we can find the values of A and B.
Comparing Results:
The results obtained using the algebraic approach and the matrix approach should be the same. Both methods are mathematically equivalent and provide the most probable values of A and B that minimize the sum of squared weighted residuals.
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Which of the following is a counterexample for this conditional statement?
If a pet is furry, then it is a hamster.
A. Hamster
B. Frog
C. Lizard
D .Dog
Is the answer option b?
Answer:
yes, it is B
Step-by-step explanation:
every input value of x has an unique result value y.
there are no 2 values for x that produce the same result value y.
and there are no multiple result values possible for a subject x value.
bottom line, yes, it is a function, and a 1:1 function at that.
solve the following inequality-4<10+2x<=18
To solve the inequality we can subtract 10 as:
\(\begin{gathered} -4<10+2x\leq18 \\ -4-10<10+2x-10\leq18-10 \\ -14<2x\leq8 \end{gathered}\)Then, dividing by 2, we get:
\(\begin{gathered} -\frac{14}{2}<\frac{2x}{2}\leq\frac{8}{2} \\ -7Therefore, the solution is:-7 < x ≤ 4
Answer: -7 < x ≤ 4
Next score the six tests (1 to 7) below from Exhibit 4.6 to your recommendation in
Q1. Write 5 to 9 lines to justify your score of each of the six tests. See an example on page 131. You need to write in six bullet points of 75 to 150 words for each test. (60 points with 10 points for each test) )
Publicity Test Score and reason
The Moral Mentor Test and reason
The Admired Observer Test and reason
The Transparency Test and reason
The Person in the Mirror Test
The Golden Rule Test
The Golden Rule Test suggests that the decision embodies the principles of empathy and fairness, treating others as one would want to be treated. The Publicity Test indicates that the decision has been well received and is aligned with public expectations.
- The Publicity Test is designed to assess whether the decision or action can withstand public scrutiny and be disclosed openly.
- I score this test an 8 because the decision or action in question has already been made public and received positive feedback from various stakeholders.
- The organization has been transparent in its communication, engaging with the public and addressing concerns promptly.
- The decision aligns with the organization's values and mission, which resonates positively with the public.
- The organization has effectively managed any potential negative publicity by proactively addressing criticisms and providing clear justifications for the decision.
- Overall, the Publicity Test indicates that the decision has been well received and is aligned with public expectations.
The Moral Mentor Test Score: 6
- The Moral Mentor Test evaluates the decision or action by considering whether it reflects the guidance of a wise and ethical mentor.
- I score this test a 6 because while the decision is generally ethical, there are some potential ethical dilemmas that need to be carefully addressed.
- The decision has considered the welfare of various stakeholders and adheres to legal and regulatory frameworks.
- However, there might be some moral complexities associated with certain aspects of the decision that need further evaluation.
- It is important for the organization to seek guidance from ethical experts and consider potential long-term consequences to ensure the decision aligns with the principles of a moral mentor.
- Overall, the Moral Mentor Test suggests that the decision is on the right track but requires careful ethical analysis and considerations.
The Admired Observer Test Score: 9
- The Admired Observer Test assesses whether the decision or action would be approved by an admired and objective observer.
- I score this test a 9 because the decision demonstrates a high level of integrity and is likely to be approved by an objective observer.
- The decision is aligned with industry best practices and follows ethical guidelines.
- It takes into account the interests of various stakeholders and aims to maximize positive outcomes for all parties involved.
- The decision is well-reasoned, considering both short-term and long-term implications.
- Overall, the Admired Observer Test indicates that the decision is commendable and likely to be seen as fair and ethical by an objective observer.
The Transparency Test Score: 7
- The Transparency Test evaluates whether the decision-making process is transparent and accountable.
- I score this test a 7 because while the decision-making process has been relatively transparent, there is room for improvement.
- The organization has provided some level of information and rationale behind the decision, but there may be areas where more transparency is required.
- It is essential for the organization to proactively communicate the decision, its underlying factors, and potential impacts to ensure stakeholders have a clear understanding.
- Increasing transparency can help build trust and mitigate any potential misunderstandings or mistrust.
- The Person in the Mirror Test assesses whether the decision or action aligns with the individual's values and principles.
- I score this test an 8 because the decision demonstrates alignment with the individual's values and principles.
- The decision-maker has considered the ethical implications and personal integrity while making the decision.
- The decision reflects a commitment to fairness, honesty, and accountability.
I score this test a 9 because the decision demonstrates a high level of empathy and fairness towards others.
- The decision considers the interests and well-being of various stakeholders, treating them with respect and dignity.
- It reflects a commitment to fairness, equality, and reciprocity in relationships.
- The organization has taken steps to ensure that the decision does not cause harm or disproportionately benefit any specific group.
- Overall, the Golden Rule Test suggests that the decision embodies the principles of empathy and fairness, treating others as one would want to be treated.
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Jim wants to buy a new tablet for his homework. His budget is $350. If the tablet costs $330 plus 4.5% sales tax, can he buy it for $350 or less?
Answer:
Yes, he can buy it for $350 or less
Step-by-step explanation:
330.00 x 4.5%
330.00 x 0.045 = 14.85
330.00 + 14.85 = 344.85
or
330.00 x 1.045 = 344.85
John F. Kennedy died in 1963 at the age of 46. In what
year was he born?
*im lazy *
Answer:
1917
Step-by-step explanation:
that was dummy ez lazy boy
Answer:
John Fitzgerald Kennedy May 29, 1917 Brookline, Massachusetts, U.S.
Died November 22, 1963 (aged 46) Dallas, Texas, U.S.
Step-by-step explanation:
check it periodt
h(x)=x−4h
What is the domain of h?
Answer:
h= x/ x+4
Step-by-step explanation:
Move all terms to the left side and set equal to zero, Then set each factor equal to zero.
On a recent day, 8 euros were worth $9 and 40 euros were worth $45. Enter an equation of the form y = kx to show the relationship between the number of euros and the value in dollars. Let y be the value In dollars and x be the number of euros. The equation is y =
Answer: y = 1.125x
Step-by-step explanation:
Divide the amount in dollars by the number of euros.
9/8 = 1.125
45/40 = 1.125
Since the constant change is 1.125, the input the 1.125 into the formula y=kx for k to get the equation y = 1.125x
The total income for the Mr. Johnson’s apartment building can be represented by the equation 5r-2c-3p, where r is the amount of rent paid by each tenant, c is the cost of the cable bill, and p is the cost of the phone bill. if the rent is $500, The cable bill is $80 in the phone bill is $40, what is the total income for Mr. Johnson?
The total income is represented by: 5r - 2c - 3p
The rent is $500 (r = 500)
The cable bill is $80 (c = 80)
The phone bill is $40 (p = 40)
Replacing these data into the equation, we get:
Income = 5(500) - 2(80) - 3(40)
Income = 2500 - 160 - 120
Income = $2220
EEEE
Using the line of random numbers to run one simulation,
what proportion of students tried out for a sport?
O 0.40
A study reported that about 40% of high school students
have tried out for a sport at their school. To find out if this
applies to Jessica's school, she surveyed an SRS of 20
students. Nine of them said that they had tried out for a
sport. To see if this result is surprising, a simulation is to
be conducted to estimate the probability of obtaining a
sample result as high as this one.
Let 0-3 represent students who tried out for a sport and
4-9 represent students who did not try out for a sport.
O 0.45
O 0.50
O 0.60
52644
52348
82462
66233
Answer:
pretty sure its .4
Step-by-step explanation:
52644
52348
82462
66233
2 2 3 2 2 2 3 3 being the # of people trying out so
8/total #of people (20) =.4
Using the line of random numbers to run one simulation, the proportion of students that tried out for a sport is .4
Calculation and Parameters:Given that 40% of students tried out for a sport in a school
The survey is an SRS of 20 students.
Hence, to find the proportion of students that tried out for a sport with the use of a line of random numbers would be:
52644523488246266233Where # is the number of people trying out is 8
2 2 3 2 2 2 3 3
Therefore,
8/total #of people (20)
=.4
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a process of observation that has an uncertain outcome is referred to as a(n) ________.
Answer: experiment
Step-by-step explanation:
A course of perception that has an unsure result is alluded to as an experiment.
What is an experiment?The study of mathematical objects by computing in order to discover their properties and patterns is known as experimental mathematics. It has been described as "that mathematical theory that concerns itself ultimately with both the convention and transmission of insights within the mathematical neighborhood through the use of experimental investigation of conjectures and also more irregular beliefs and just a careful analysis of the information acquired in this endeavor (in either the Galilean, Baconian, Aristotelian or Kantian sense).
The nineteenth century saw the reemergence of experimental mathematical concepts as a distinct field of study as the range of calculations that could be performed was greatly expanded by the development of the electronic computer, which could perform calculations at speeds and with levels of precision that were unimaginable to earlier generations of mathematicians.
A course of perception that has an unsure result is alluded to as an experiment.
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Bro this makes no sense can someone explain
Answer:
\(x=110^o\)
Step-by-step explanation:
If AB = AD, then you have that triangle ABD is an isosceles triangle (it has two equal sides). If it is isosceles, then the angles opposite to angles those equal sides must be equal. Then angle ADB = \(35^o\). and by the property of all internal angles of a triangle have to add to \(180^o\), we get that:
\(x=180^o-35^o-35^o=110^o\)
Answer:
x = 110 degrees, y = 49 degrees.
Step-by-step explanation:
We have adjacent isosceles triangles here, so
x = 180 - 2(35) = 110 degrees, and
2y + 82 = 180
so y = (180 - 82) / 2 = 49 degrees.
Ryan is making a fruit drink. The directions say to mix 3 cups of water with 1 1/2 scoops of powdered fruit mix.
1. how many cups of water should he use with 6 scoops of fruit mix?
2. Rate needed?
3. Unit rate?
4. Interpretation of unit rate?
For per unit scoops of powdered 2 cups of Water is Needed.
What is the average rate of change?It is a measure of how much the function changed per unit, on average, over that interval. It is derived from the slope of the straight line connecting the interval's endpoints on the function's graph.
Given, Ryan is making a fruit drink. The directions say to mix 3 cups of water with 1 1/2 scoops of powdered fruit mix.
We have to find how many cups of water should he use with 6 scoops of fruit mix
Let "n" be the number of cups required for 6 scoops of fruit mix
3 cups of water ⇒ 1.5 scoops of fruit mix.
Then, n cups of water ⇒ 6 scoops of fruit mi
So, by cross multiplication,
6 * 3 = n *1.5
n = 12
Ryan has to use 12 cups of water for 6 scoops.
thus,
For per unit scoops of powdered 2 cups of Water is Needed.
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a random variable is normally distributed with a mean of 50 and a standard deviation of 5. 1. what is the probability that the random variable will assume a value between 42.5 and 57.5 (round up your answer to 4 decimals)? 2. what is the probability that the random variable will assume a value more than 65 (round up your answer to 4 decimals)?
1) The probability that the random variable will assume a value between 42.5 and 57.5 is 0.8664, rounded up to 4 decimals.
2) The probability that the random variable will assume a value more than 65 is 0.0013, rounded up to 4 decimals.
Using the standard normal distribution table, we need to convert the values of interest into z-scores, which represent the number of standard deviations away from the mean. The formula to calculate the z-score is:
z = (x - μ) / σ
where x is the value of interest, μ is the mean, and σ is the standard deviation.
For the lower limit of 42.5, the z-score is:
z = (42.5 - 50) / 5 = -1.5
For the upper limit of 57.5, the z-score is:
z = (57.5 - 50) / 5 = 1.5
Using the standard normal distribution table, we can find the area under the curve between the z-scores of -1.5 and 1.5, which is 0.8664.
To answer this question, we need to find the area under the normal distribution curve to the right of the value of interest, which represents the probability of the random variable assuming a value greater than 65.
Using the z-score formula, the z-score for 65 is:
z = (65 - 50) / 5 = 3
Using the standard normal distribution table, we can find the area under the curve to the right of the z-score of 3, which is 0.0013.
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1.Make an industry analysis using either PESTEL or Five forces
model.
2. Prepare a strategic group map using updated information. Use
three parameters-for x axis, for y axis and for the diameter of th
Industry Analysis using five forces model which are Political, Economic, Social, Technological, Environmental, Legal. A strategic group map visually represents the competitive positioning of companies within an industry.
1. Industry Analysis using PESTEL Model:
The PESTEL analysis examines the external factors that impact an industry:
Political: Government regulations, stability, and policies affecting the industry.
Economic: Economic growth, inflation, exchange rates, and consumer purchasing power.
Social: Demographic trends, cultural factors, and consumer behavior.
Technological: Technological advancements, innovation, and automation in the industry.
Environmental: Environmental regulations, sustainability practices, and climate change impact.
Legal: Legal frameworks, industry-specific regulations, and intellectual property protection.
By conducting a PESTEL analysis, one can gain insights into the industry's overall environment, identify opportunities and threats, and understand the factors influencing its growth and competitiveness.
2. Strategic Group Map:
A strategic group map visually represents the competitive positioning of companies within an industry. It uses parameters to plot companies on an x and y axis, and the diameter of the circle represents their market share or another relevant metric.
Parameters for x-axis: Price range (e.g., low to high)
Parameters for y-axis: Product differentiation (e.g., basic to premium)
Diameter of the circle: Market share (e.g., small to large)
By plotting companies based on these parameters, the strategic group map helps identify market segments, competitive dynamics, and potential areas for differentiation or strategic alliances.
3. Reconstructed Vignette 5: Cost of Operation for GP (2019 and 2020):
In 2019, the cost of operation for the GP (General Practitioner) increased due to rising expenses such as rent, salaries, and medical supplies. This was influenced by factors such as inflation and increased demand for healthcare services.
In 2020, the COVID-19 pandemic significantly impacted the cost of operation for GPs. The costs surged due to additional expenses related to personal protective equipment (PPE), sanitation measures, and telehealth infrastructure. Simultaneously, some costs decreased as patient visits reduced temporarily.
The increased costs challenged GPs' profitability, especially for independent practitioners or smaller clinics with limited resources. Adapting to new operational requirements and investing in technology further added to the financial burden.
4. Agreement with the Idea in the Case:
As the case or specific idea isn't provided, it's challenging to agree or disagree without context. Please provide more information or details about the case or idea so that I can offer a justified answer based on logic or data.
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COMPLETE QUESTION - 1.Make an industry analysis using either PESTEL or Five forces model.
2. Prepare a strategic group map using updated information. Use three parameters-for x axis, for y axis and for the diameter of the circle.
3. Reconstruct Vignette 5: Cost of operation for GP. Make it for 2019 and 2020.
4. Do you agree with the idea described in the case? Justify your answer in brief (you may use logic or data in support of your answer).
Neville and his friends are going to a local peach festival. They plan to go on rides all day. The tickets for the rides are $0.50 each. Find the domain and range with words and inequalities.
The domain of x will be a set of whole numbers greater than 2 which is; x > 2
The range of y will be a set of rational numbers greater than or equal to 1.5 which is; y ≥ $1.5
What is the domain and range?
The domain of a function is defined as the set of all possible input values that make the function possible while range is the set of all possible output values that are possible from the input values.
The function is; y = 0.5x
Since the value of x represents the number of people that purchased tickets for the rides including Neville, then we can say that the domain of x will be a set of whole numbers greater than 2 because we are told it is him and his friends which is plural. Thus;
The domain is; x > 2
The range will be;
y ≥ 3(0.5)
y ≥ $1.5
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Solve for xxx.
x=x=x, equals
Answer: The answer is of course, 3.
Step-by-step explanation:
answer to all these in order please !
Answer:
?? 6363
Step-by-step explanation:
my braincells are disappearing help me
Answer:
Last option
Step-by-step explanation:
In order to manipulate a, you have to divide by 3 on both sides.
en una granja hay comida para alimentar a 300 conejos durante 60 días. Cuántos conejos hay que vender si se quieren alimentar durante 15 días más?
The number of rabbits that must be sold if they want to be fed for 15 more days is 150 rabbits.
In a farm, there is food to feed 300 rabbits for 60 days.
Let x be the number of rabbits that must be sold if they want to be fed for 15 more days.
From the given data, we have the following equation:
300 * 60 = (300 - x) * 75
The equation represents the amount of food for 300 rabbits that can last 60 days is equal to the amount of food for 300 - x rabbits that can last 75 days.
Solving for x, we get:
x = 150 rabbits
Hence, the number of rabbits that must be sold if they want to be fed for 15 more days is 150 rabbits.
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Use half-angle and double-angle identities to solve the trigonometric expressions and match them with their values.
sin 67.5°
cos 105°
tan 165°
√2-√3
2
1+√2
√2+√2
√3-2
tan 67.5°
Therefore , the solution of the given problem of trigonometry comes out to be - (√2-+√6 ) / 4.
What exactly does trigonometry mean?The connection between trigonometric and square spline The field is believed to have been founded in the late 1700s, but it really began in the third-century BC when the mathematical and astronomy sciences were combined. Numerous geometric equations can be solved or the results of calculations using them can be computed using precise mathematical methods. Trigonometry is the study of the six basic trigonometric operations.
Here,
Given :
cos(x + y)
cos(x) cos(y) - sin(x) sin(y) by the cosine addition identity.
We receive:
If we look just at unit circle, we should notice that cos(x) = √2/2 instead than sin(x) = √2/2.
Additionally, we are told that cos(y)= -1/2,
but if we look at the unit circle, we should find that sin(y)=√3/2.
Use both of them to the following: cos(x + y).
cos(x) sin - cos(y) (x) sin(y) via the cosine addition identity.
=> √2/2 - 1/2 - √2/2*√3/2
=> -√2/4 - √6/4
=> -√2- √6 / 4
=> - (√2-+√6 ) / 4
Therefore , the solution of the given problem of trigonometry comes out to be - (√2-+√6 ) / 4.
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please help factorise 7ab+a
Answer:
= a(7b + 1)
Step-by-step explanation:
7ab + a
a(7b + 1).
The linear function FX equals 0. 5 X +80 represents the average test score in your math class, where x is the number of test taken. The linear Function GX represents the average test score in your science class, where x is the number of test taken
Part A Determine the test average for your math class after completing test 2?
Part B Determine the test average for your science class after completing test 2?
Part C Which class had the higher average after completing tasks 4? show work to support your answer
The tests average for your math class after completing test 2 is f(2)=81
The test average for your science class after completing test 2 is g(2)=83
The test average for maths class after test 2 is greater
Given linear function f(x)= 0.5x+80
x↔g(x)
1↔81
2↔83
3↔85
Solving (a): f(2)
We have:
f(x)= 0.5x+80
f(2)= 0.5*2+80
f(2)=81
Solving (b): g(2)
From the table:
g(x)=83 when x=2
So g(2)=83
Solving (c): Which is greater f(2) or g(2)
In (a) and (b),
f(2)=81
g(2)=83
Hence, the test average for maths class is greater
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5000000
bi 5000
Section B
1. Factorise the following numbers and e
32 b) 81 128 dj 243
Express the following numbers as the
Make's absolutely no sense. Please rephrase this as a question.
Casey wants to buy some instruments for his niece and nephew. He has $40 to spend. Based on the prices listed below, what is the estimated total, rounded to the nearest dollar? Which explains why he can afford the tambourine and xylophone? Tambourine: $13.95
Xylophone $23.50
The estimated total to the nearest dollar is $38, so he can afford them. The estimated total to the nearest dollar is $37, so he can afford them. The estimated total to the nearest dollar is $36, so he can afford them. The estimated total to the nearest dollar is $37.45, so he can afford them.
Answer:
It’s the estimate tidal to the nearest dollar is $38 so he can offered them
Step-by-step explanation:
Please answer!?! Thank you!?!
Answer:
3(4x - 4) - 7x
Step-by-step explanation:
Answer
The first expression has a simplified expression of 5x-12.
Explanation
Distribute all of the expressions and subtract 7x.
Expression 1
3(4x-4)-7x
12x-12-7x
5x-12
Expression 2
-3(4x-4)-7x
-12x+12-7x
-19x+12
Expression 3
3(-4x+4)-7x
-12x+12-7x
-19x+12
Expression 4
-3(-4x-4)-7x
12x+12-7x
5x+12
Let f be a function that has derivatives of all orders for allreal numbers.
Assume f(1) = 3, f '(1) = -2. f ''(1) = 2, and f '''(1) =4.
a) Write the second-degree Taylor polynomial for f about x = 1and use it to approximate f(0.7).
b) Write the third-degree Taylor polynomial for f about x = 1and use it to approximate f(1.2).
c) Write the second-degree Taylor polynomial for f ', thederivative of f , about x = 1 and use it to approximate f'(1.2).
a) The second-degree Taylor polynomial for f about x=1 is 1.69
b) The third-degree Taylor polynomial for f about x=1 is 0.867
c) The second-degree Taylor polynomial for f' about x=1 is -0.4
a) The second-degree Taylor polynomial for f about x = 1 is:
f(x) ≈ f(1) + f'(1)(x-1) + (f''(1)/2)(x-1)^2
Plugging in the given values, we get:
f(x) ≈ 3 - 2(x-1) + (2/2)(x-1)^2
f(x) ≈ 1 - 2(x-1) + (x-1)^2
To approximate f(0.7), we substitute x = 0.7 in the polynomial:
f(0.7) ≈ 1 - 2(0.7-1) + (0.7-1)^2
f(0.7) ≈ 1 + 2(0.3) + 0.09
f(0.7) ≈ 1.69
Therefore, f(0.7) ≈ 1.69 using the second-degree Taylor polynomial.
b) The third-degree Taylor polynomial for f about x = 1 is:
f(x) ≈ f(1) + f'(1)(x-1) + (f''(1)/2)(x-1)^2 + (f'''(1)/6)(x-1)^3
Plugging in the given values, we get:
f(x) ≈ 3 - 2(x-1) + (2/2)(x-1)^2 + (4/6)(x-1)^3
f(x) ≈ 1 - 2(x-1) + (2/3)(x-1)^3
To approximate f(1.2), we substitute x = 1.2 in the polynomial:
f(1.2) ≈ 1 - 2(0.2) + (2/3)(0.2)^3
f(1.2) ≈ 0.867
Therefore, f(1.2) ≈ 0.867 using the third-degree Taylor polynomial.
c) The second-degree Taylor polynomial for f', the derivative of f, about x = 1 is:
f'(x) ≈ f'(1) + f''(1)(x-1)
Plugging in the given values, we get:
f'(x) ≈ -2 + 2(x-1)
f'(x) ≈ -2 + 2x - 2
f'(x) ≈ 2x - 4
To approximate f'(1.2), we substitute x = 1.2 in the polynomial:
f'(1.2) ≈ 2(1.2) - 4
f'(1.2) ≈ -0.4
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