determine the degree of the maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than 0.001.f(x) = sin(x), approximate f(0.5)
Answers
the answer is degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than 0.001 is 7.
To determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than 0.001, given f(x) = sin(x), we need to approximate f(0.5).The formula to calculate the degree of Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than the given amount is:$$R_n(x) = \frac{f^{n+1}(c)}{(n+1)!}(x-a)^{n+1}$$where c is a value between a and x, and Rn(x) is the remainder function.Then, to find the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than 0.001, we use the inequality:$$|R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1}$$where M is an upper bound for the $(n+1)^{th}$ derivative of f on an interval containing a and x.To approximate f(0.5), we use the formula for the Maclaurin series expansion of sin(x):$$\sin(x) = \sum_{n=0}^{\infty}(-1)^n \frac{x^{2n+1}}{(2n+1)!}$$Thus, for f(x) = sin(x) and a = 0, we have:$$f(x) = \sin(x)$$$$f(0) = \sin(0) = 0$$$$f'(x) = \cos(x)$$$$f'(0) = \cos(0) = 1$$$$f''(x) = -\sin(x)$$$$f''(0) = -\sin(0) = 0$$$$f'''(x) = -\cos(x)$$$$f'''(0) = -\cos(0) = -1$$$$f^{(4)}(x) = \sin(x)$$$$f^{(4)}(0) = \sin(0) = 0$$$$f^{(5)}(x) = \cos(x)$$$$f^{(5)}(0) = \cos(0) = 1$$Thus, M = 1 for all values of x, and we have:$$|R_n(x)| \leq \frac{1}{(n+1)!}|x|^n$$To make this less than 0.001 when x = 0.5, we need to find n such that:$$\frac{1}{(n+1)!}0.5^{n+1} \leq 0.001$$Dividing both sides by 0.001 gives:$$\frac{1}{0.001(n+1)!}0.5^{n+1} \leq 1$$Taking the natural logarithm of both sides gives:$$\ln\left(\frac{0.5^{n+1}}{0.001(n+1)!}\right) \leq 0$$Using a calculator, we can find that the smallest value of n that satisfies this inequality is n = 7. Therefore, the degree of the Maclaurin polynomial required for the error in the approximation of sin(0.5) to be less than 0.001 is 7.
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The degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than 0.001 is 3.
Given the function f(x) = sin(x) and we need to determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than 0.001.
We need to approximate f(0.5).
Maclaurin Polynomial: The Maclaurin polynomial of order n for a given function f(x) is the nth-degree Taylor polynomial for f(x) at x = 0. It is given by the formula:
\(Pn(x) = f(0) + f'(0)x + f''(0)x²/2! + ... + fⁿ⁽ᶰ⁾(0)xⁿ/ⁿ!\)
Where fⁿ⁽ᶰ⁾(0) denotes the nth derivative of f(x) evaluated at x = 0.
To determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than 0.001, we use the error formula.
Error formula:
\(|f(x) - Pn(x)| <= M(x-a)^(n+1)/(n+1)!\)
where M = max|fⁿ⁽ᶰ⁾(x)| over the interval containing x and a. For f(x) = sin(x)
and a = 0, we have f(0) = sin(0) = 0, f'(x) = cos(x), f''(x) = -sin(x), f'''(x) = -cos(x), f⁽⁴⁾(x) = sin(x), f⁽⁵⁾(x) = cos(x), f⁽⁶⁾(x) = -sin(x), ...
Thus, \(|f⁽ⁿ⁾(x)| <= 1\) for all n and x.
Therefore, \(M = 1.|x-a| = |0.5-0| = 0.5\)
Thus,\(|f(x) - Pn(x)| <= M(x-a)^(n+1)/(n+1)!\)
=> \(|sin(x) - Pn(x)| <= 0.5^(n+1)/(n+1)!\)
We need \(|sin(0.5) - Pn(0.5)| <= 0.001\).
So, \(0.5^(n+1)/(n+1)! <= 0.001\)
n = 3 (Minimum value of n to satisfy the condition).
Using the Maclaurin polynomial of degree 3, we have
\(P₃(x) = sin(0) + cos(0)x - sin(0)x²/2! - cos(0)x³/3! = x - x³/3\)
Therefore, the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than 0.001 is 3.
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Related Questions
Amber wants to put carpet on the floor of her bedroom. The floor is a square with an area of 7.29 square feet.
How long should the piece of carpet be on each side?
Answers
Answer:
2.7 feet
Step-by-step explanation:
S = a²
S = 7.29
a = sqrt(7.29) = 2.7
Answer:
2.7
Step-by-step explanation:
Area is \(side^{2}\)
a= \(s^{2}\)
\(\sqrt{7.29}\) = 2.7
2.7 x 2.7 = 7.29
A sign at a zoo says that their baby elephant weighs 200 pounds. The weight was rounded to the nearest tenth. What could be the actual weight of the elephant before it was rounded?
Answers
Answer:
199.5lbs, 199.6lbs, 199.7lbs, 199.8lbs, 199.9lbs
You can choose any of the 5
To begin the graphing process, rewrite this inequality so y is by itself: 3x − 4y ≥ 24. Select the correct answer.y ≥ -3/4x, y ≤ -3/4x + 6 + 6, y ≥ 3/4x – 6, y ≤ 3/4x – 6
Answers
Answer:
d. y ≤ 3/4x -6
Step-by-step explanation:
3x -4y ≥ 24
-4y ≥ -3x +24 subtraction property
y ≤ 3/4x -6 division property (change the direction of the symbol)
Answer:
Part A
y ≤ 3/4x – 6
the last option
Part B
answer is in the attachment
Part C
(-2,-8)
(8,0)
(4,-4)
Step-by-step explanation:
Part A
Use the properties of inequalities to rewrite the inequality:
3x − 4y ≥ 24
-4y ≥ -3x + 24 subtraction property
y ≤ x – 6 division property (change the direction of the symbol)
Part B
Plot the boundary line, which will be solid because the inequality is non-strict. Use either the equation 3x − 4y = 24 or y=3/4x-6
Next, use a test point, such as (0,0), to determine where to shade:
3(0) − 4(0) ≥ 24
0 ≥ 24
The result is false, so shade on the opposite side of the boundary line.
Part C
Use the graph of the inequality to determine whether each point lies in the solution set.
For this non-strict inequality, any point in the shaded region or on the line is a solution. So (8,0), (4,-4), and (-2,-8) are all solutions to the inequality.
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solve for r
negative 13 equals r over 9 plus eight
Answers
Answer:
r = - 189
Step-by-step explanation:
Given
- 13 = \(\frac{r}{9}\) + 8 ( subtract 8 from both sides )
- 21 = \(\frac{r}{9}\) ( multiply both sides by 9 )
- 189 = r
Answer: r= -189
Step-by-step explanation:
Multiply. 6x^2/x(x−3)⋅(x−3)/10(x+2) Help!!
Answers
Answer:
\(\frac{3x}{5(x+2)}\)
Step-by-step explanation:
1) Simplify \(\frac{6{x}^{2}}{x(x-3)}\) to \(\frac{6x}{x-3}\)
\(\frac{6x}{x-3}\times \frac{x-3}{10(x+2)}\)
2) Cancel x - 3.
\(6x\times \frac{1}{10(x+2)}\)
3) Simplify.
\(\frac{6x}{10(x+2)}\)
4) Simplify.
\(\frac{3x}{5(x+2)}\)
Hi, I just want to say have a great day! Remember to eat if you haven’t yet, and I love you! God Bless You :)
Answers
Answer:
Aww! Thanks! Have a great day as well!
Step-by-step explanation:
Karen wants to advertise how many chocolate chip are in in each Big Chip cookie at her bakery. She randomly selects a sample of 55 cookies and finds that the number of chocolate chips per cookie in the sample has a mean of 15.4 and a standard deviation of 3.2. What is the 99% confidence interval for the number of chocolate chips per cookie for Big Chip cookies? Enter your answers to accurate to one decimal place
Answers
99% confidence interval for the number of chocolate chips per cookie is
15.4 ± 1.1115
What is confidence interval?The mean of sample plus and minus the range of that sample constitutes a confidence interval.
Within a specific level of confidence, this is the range of values anticipated to fall within, if the test is repeated.
Number of cookies in the sample n = 55
Mean μ = 15.4
Standard deviation s = 3.2
Confidence = 99%
For 99% confidence interval Z-value = 2.576
Confidence interval
= μ ± Zs/√n
= 15.4 ± 2.576×3.2/√55
= 15.4 ± (8.2432/7.4162)
= 15.4 ± 1.1115
Hence, 15.4 ± 1.1115 is the 99% confidence interval for the number of chocolate chips per cookie.
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A tree casts a shadow of 15 meters, while a 2-meter post nearby casts a shadow of 3 meters. How tall is the tree
Answers
The tree is 10 meters tall.
To find the height of the tree, we can use the concept of similar triangles. The ratio of the height of the tree to the length of its shadow should be equal to the ratio of the height of the post to the length of its shadow, since they are both located in the same plane and are being illuminated by the same light source.
Let's use proportions to solve this problem. We can set up the proportion:
height of tree / length of tree's shadow = height of post / length of post's shadow
Let h be the height of the tree. Then we have:
h / 15 = 2 / 3
Cross-multiplying, we get:
3h = 30
h = 10
Therefore, the tree is 10 meters tall.
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At an online bookstore, Kendra bought 4 copies of the same book for the members of her book club. She got free shipping because her total was at least $50. What was the minimum price of each book?
Answers
Answer:
$12.50
Step-by-step explanation:
$50/4 = $12.50
if covariance between two variables is near 0, it implies that group of answer choices a positive relationship exists between the variables. none of the choices. the variables are not linearly related. the variables are negatively related. the variables are strongly related.
Answers
If covariance between two variables is near 0, it implies that the variables are not linearly related.
In probability theory and statistics, covariance is a measure of the combined variability of two random variables in probability theory. The covariance is positive if the larger values of one variable mostly correlate with the greater values of the other variable, and vice versa for the smaller values.It is given that the covariance between two variables is near zero.In mathematical modelling, statistical modelling, and experimental sciences, dependent and independent variables are variables. Dependent variables are so named because their values are explored in an experiment under the assumption or requirement that they are dependent, by some law or rule, on the values of other variables.It clearly implies that the variables are not linearly related.To learn more about covariance, visit :
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please help im confused
Answers
Hi there!
We are given two exterior angles. Don't panic! We know that exterior angles form a linear pair, which means they add up to 180 degrees.
180-130=50
180-96=84
Now that we have those angles, how can we work out the value of x?
Turns out that all the angles in any triangle add up to 180 degrees.
So
180-84-50=46 degrees.
So the value of x is 46 degrees.
Now let's work out the value of x in the next problem.
180-126=54
180-54-43=83 degrees
So the value of x in this problem is 83 degrees.
Let's move on to the third problem.
At first, it seems like we do not have enough information.
If we know that the right angle is 90 degrees (it is shown by a little square), then we have enough information.
180-90-43=47 degrees
So the value of x is 47 degrees.
Hope this helps you!
~Just an emotional teen who listens to her favorite song "Try Everything"
\(SilentNature\)
there are two charges q1= -5c and q2 = -8c placed 20 cm apart
Answers
The two charges q1=-5c and q2=-8c are placed 20 cm apart from each other. Given two charges, q1 = -5C and q2 = -8C, placed 20 cm apart, you might be interested in finding the electrostatic force between them.
Given two charges, q1 = -5C and q2 = -8C, placed 20 cm apart, you might be interested in finding the electrostatic force between them. To do this, we can use Coulomb's Law:
F = (k * |q1 * q2|) / r^2
Where:
- F is the electrostatic force
- k is the electrostatic constant (8.9875517923 × 10^9 N m²/C²)
- q1 and q2 are the charges (-5C and -8C)
- r is the distance between the charges (20 cm or 0.2 m)
Now, plug the values into the formula and calculate the force:
F = (8.9875517923 × 10^9 N m²/C² * |-5C * -8C|) / (0.2 m)^2
F = (8.9875517923 × 10^9 N m²/C² * 40 C²) / 0.04 m²
F = 8,987,551,792.3 N (approx.)
The electrostatic force between the two charges, 20 cm apart, is approximately 8,987,551,792.3 Newtons.
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Please help me with this
Answers
V= (4/3) π (0.5)^3
V= (π/6)
Explanation: The formula for finding the volume of a sphere is, (4/3) πr^3. Plug in your radius and solve with your answer left in terms of π.
according to your text, which of the following is the primary source of primate endangerment today? habitat destruction human diseases social stress and dislocation use of primates for pets
Answers
Based on the text, the primary source of primate endangerment today is habitat destruction.
Habitat destruction is considered the primary source of primate endangerment based on various factors mentioned in the text. Primates, like many other species, heavily rely on their natural habitats for survival, including food, shelter, and breeding. However, human activities such as deforestation, urbanization, and land conversion for agriculture or infrastructure development have significantly reduced and fragmented primate habitats.
Habitat destruction leads to the loss of critical resources and disrupts the ecological balance necessary for primate populations to thrive. As their habitats shrink, primates face increased competition for limited resources, reduced access to food and water sources, and decreased opportunities for successful reproduction and rearing of offspring. This loss of habitat also exposes primates to increased vulnerability to predation, disease transmission, and human-wildlife conflicts.
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1/5 de los animales en el zoológico son monos 5/7 de los monos son machos
¿Qué fracción de los animales en el zoológico son monos machos?
Answers
1/7 of the animals in the zoo are male monkeys.
What fraction of the animals in the zoo are male monkeys? Explain with workings.
To find the fraction of animals in the zoo that are male monkeys, we have to calculate the product of the fractions representing the proportion of monkeys and the proportion of male monkeys among them.
Given that 1/5 of the animals in the zoo are monkeys, we will then represent this as:
= 1/5
= 5/25.
And 5/7 of the monkeys are male which is written as 5/7.
To get fraction of male monkeys, we will multiply these two fractions:
= (5/25) * (5/7)
= 25/175
= 1/7.
Full question:
1/5 of the animals in the zoo are monkeys 5/7 of the monkeys are male. What fraction of the animals in the zoo are male monkeys?
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Find the area surface generated by revolving the curves about the x-axis (round to two decimal places).
y = sin x, 0≤x≤π
Answers
The area surface generated by revolving the curves about the x-axis is 9.87 square units
To find the surface area generated by revolving the curve y=sin(x) about the x-axis, we use the formula:
S = 2π∫aᵇ y ds
where a and b are the limits of integration, y is the function being revolved, and ds is the differential element of arc length.
To find ds, we use the formula:
ds = √(1 + (dy/dx)²) dx
where dy/dx is the derivative of y with respect to x.
Taking the derivative of y=sin(x), we get:
dy/dx = cos(x)
Substituting into the formula for ds, we get:
ds =√(1 + cos²(x)) dx
Now we can substitute y=sin(x) and ds into the formula for surface area and integrate from x=0 to x=π:
S = 2π \(\int\limits^\pi_0\) sin(x) √t(1 + cos²(x)) dx
This integral cannot be evaluated in terms of elementary functions, so we must use numerical methods to approximate the value. Using a numerical integration tool, we get:
S ≈ 9.87
Therefore, the surface area generated by revolving the curve y=sin(x) about the x-axis is approximately 9.87 square units.
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If I played at 2am and for my friend his time 8am so now if it 6pm for me now , what time is it for him? Hint For him will be Am
Answers
Answer:
12am
Step-by-step explanation:
Add 6 hrs to 6pm, you will get 12am
Answer: 12
Step-by-step explanation:
I NEED HELP ASAP (Only answer b )
Amani is calculating the slope of the trendline in the scatterplot below. How can she use the trendline to
check the reasonableness of her answer in other words how can she verify that her trendline fits the
data point
Answers
Answer:
Step-by-step explanation:
A factorial design has ____________________________________________________________. a. two or more dependent variables. b. two or more independent variables. c. one independent variable and one dependent variable. d. one independent variable and two or more dependent variables.
Answers
Answer:
b. 2 or more independent variables
Step-by-step explanation:
factorial designs give an opportunity to figure out what factorial design affect dependent variables.
A local little league ha a total of 60 player, of whom 20% are right-handed. How many right-handed player are there?
Answers
There are 12 players of right handed, according to the question.
What do you mean by percentage?
A percentage in mathematics is a number or ratio that is expressed as a fraction of 100. The abbreviations "pct.", "pct.", and occasionally "pc" are also used to indicate it, though the percent sign, "%," is most frequently used. A percentage has no dimensions and no standard measurement.
According to data in the given question,
We have given data:
A local little league has a total of 60 players of whom 20% are right-handed.
Let x be the total number of players that are right-handed.
The value of x can be found as follow:
x = 20% of 60
x = (20/100)60
x = 0.2(60)
x = 12
Therefore, the total number of players that are right-handed is 12, if the local little league has a total of 60 players of whom 20% are right-handed.
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Josue wraps a gift box in the shape of a cube. The figure below shows a net for the gift
box.
5-3 cm
How much wrapping paper did he use, in square
centimeters?
Answers
Josue used 150 square centimeters of wrapping paper to cover the cube-shaped gift box
To determine the amount of wrapping paper used by Josue for the cube-shaped gift box, we need to calculate the surface area of the cube.
The net of the gift box consists of six squares, each representing a face of the cube. The dimensions provided in the figure show that each side of the squares is 5 cm.
Since a cube has six faces, we need to calculate the total surface area by multiplying the area of one square face by six.
The formula to find the area of a square is side length squared, so we can calculate the area of one square face as 5 cm * 5 cm = 25 cm².
Now, to find the total surface area, we multiply the area of one face by six: 25 cm² * 6 = 150 cm².
Therefore, Josue used 150 square centimeters of wrapping paper to cover the cube-shaped gift box.
It's worth noting that this calculation assumes that there is no overlap or excess wrapping paper used while wrapping the gift box. Additionally, we have assumed that the dimensions provided are accurate and refer to the side length of the squares in the net.
Overall, the amount of wrapping paper used can vary based on the wrapping technique employed and any additional decorations or folds added to the gift box.
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Which figures can the composite figure be broken into? A. a square pyramid and a square prism B. a square pyramid and a cube C. a triangular pyramid and a square prism D. a triangular prism and a cube
Answers
Answer:
c
Step-by-step explanation:
the base dimensons of the bottom portion are not the same therefore it cannot be a cube. so it is c a triangular pyramid and square prism
Answer:
its C i'm taking the test
Step-by-step explanation:
There are 1,354 Animals in one barn There are 574 goats, 346 cows, and the rest are horses If 89 horses were sold how many horses are left in that barn?
Answers
Answer: 345
Step-by-step explanation: 1354-574-346=434-89= 345
The relationship between the count of each animal is an illustration of arithmetic expressions.
There are 345 horses left in the barn
The given parameters are:
\(Animals = 1354\)
\(Goats = 574\)
\(Cows =346\)
Start by calculating the number of horses in one barn using the following equation
\(Animals = Goats+ Cows + Horses\)
So, we have:
\(1354 = 574+ 346+ Horses\)
\(1354 = 920+ Horses\)
Subtract 920 from both sides
\(434= Horses\)
Rewrite the equation as:
\(Horses = 434\)
When 89 horses were sold, the expression for the number of horses remaining is:
\(Horses = 434 - 89\)
So, we have:
\(Horses = 345\)
Hence, there are 345 horses left in the barn
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Please help me with these two problems
Answers
Distribute the -4 into the numbers inside giving you
-4n + 20 -8 = 30
Then combine your like terms giving you
-4n + 12 = 30
Then move the number to one side so the variable and a number aren’t on the same side giving you
-4n = 18
Then you have to divide -4 by both sides since that is the number next to the variable giving you
N = -4.5
Which of the following is equivalent to the complex number i^7
Answers
Answer:-i
Step-by-step explanation: :)
Jada swims 4 laps, which is 2/5
of the number of laps she plans to swim.
How many laps does Jada plan to swim? Use x
for the number of laps Jada plans to swim.
Equation:
Solution:
Answers
Let x be the number of laps Jada plans to swim.
According to the problem, 4 laps is 2/5 of the number of laps she plans to swim, so we can write:
4 = 2/5 * x
To solve for x, we can first multiply both sides by the reciprocal of 2/5, which is 5/2:
4 * 5/2 = x
Simplifying the left side:
10 = x
Therefore, Jada plans to swim 10 laps.
0.5y=-1.25 grade 8th linear equation
Answers
Answer:
y = - 2.5
Step-by-step explanation:
Given
0.5y = - 1.25 ( divide both sides by 0.5 )
y = \(\frac{-1.25}{0.5}\) = - 2.5
Answer:
y=-2.5
Step-by-step explanation:
0.5/0.5 y=-1.25/0.5
y= -2.5
I am so confused with this can someone please help me?
Answers
The required coordinate of point H is given as (d+c, b), as per the given condition.
Coordinate, is represented as the values on the x-axis and y-axis of the graph. while the coordinate x is called abscissa and the coordinate of the y is called ordinate.
Here,
As point H is collinear with point G so the y coordinate of the Point H is equal to y coordinate point G,
So
Y coordinate of point H = b
Now,
x coordinate of point H is given as,
Line GH is parallel with LINE FE
So, x-coordinate = FE = d-(-c) = d+ c
Thus, the required coordinate of point H is given as (d+c, b), as per the given condition.
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the ratio between the measure of the interior angles of the triangle is 3:4:7, find the degree measure of each angle
Answers
180 / 14 = 90/7
The three angles are:
3 x 90/7 = 270/7 = 38.6 degrees (1 d.p.)
4 x 90/7 = 360/7 = 51.4 degrees (1 d.p.)
7 x 90/7 = 90 degrees
PLEASE HELP I BEG IMMA FAILLLLLLLLL
An isosceles triangle has two sides that measure 5.6 in. and a side that measures 4.2 in. If the triangle is enlarged such that the two sides measure 20 in, what is the length of the third side?
Answers
Answer:
15
Step-by-step explanation:
If the bigger triangle has the two congruent sides measuring 20, then the ratio of the bigger to the smaller is 20/5.6 This means that the ratio between the third side of the bigger to the third side of the smaller is 20/5.6
Setting a ratio:
20/5.6=x/4.2
Cross multiplying: 84=5.6x
x = 84/5.6 -->12/0.8 --> 15
x = 15