The Taylor Remainder Theorem states that for a function f(x) and its nth-degree Taylor polynomial approximation Pn(x), the remainder Rn(x) is given by:
Rn(x) = f(x) - Pn(x) = (1/(n+1)) * f^(n+1)(c) * (x-a)^(n+1)
where f^(n+1)(c) is the (n+1)-th derivative of f evaluated at some value c between a and x.
In this case, to find the values of x for which the approximation is within 0.00447 of f(x), we need to find the values of x such that |Rn(x)| ≤ 0.00447.
Since the problem limits |x| ≤ π/2, we can use the Taylor series expansion centered at a = 0 (Maclaurin series) to approximate f(x).
Let's consider the approximation up to the 4th degree Taylor polynomial:
P4(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + (f''''(0)/4!)x^4
To determine the values of x for which |R4(x)| ≤ 0.00447, we need to find the maximum value of the (n+1)-th derivative in the interval [-π/2, π/2] to satisfy the Taylor remainder inequality.
The 5th derivative of f(x) is f^(5)(x) = 24x^(-7), which is decreasing as x approaches 0 from either side. Therefore, the maximum value of the 5th derivative occurs at the boundaries of the interval [-π/2, π/2], which are x = ±π/2.
Substituting x = ±π/2 into the remainder formula, we get:
|R4(±π/2)| = (1/5!) * |f^(5)(c)| * (±π/2)^5
To find the values of c that make the remainder within 0.00447, we solve the inequality:
(1/5!) * |f^(5)(c)| * (π/2)^5 ≤ 0.00447
Simplifying, we have:
|f^(5)(c)| ≤ (0.00447 * 5!)/(π^5/2^5)
|f^(5)(c)| ≤ 0.00447 * (2^5/π^5)
We can now find the values of c for which the inequality holds. Note that f^(5)(c) = 24c^(-7).
|24c^(-7)| ≤ 0.00447 * (2^5/π^5)
Solving for c, we have:
c^(-7) ≤ (0.00447 * (2^5/π^5))/24
Taking the 7th root of both sides, we get:
|c| ≥ [(0.00447 * (2^5/π^5))/24]^(1/7)
Now we can calculate the right-hand side of the inequality to find the values of c:
|c| ≥ 0.153
Therefore, the values of x for which the approximation is within 0.00447 of f(x) in the interval |x| ≤ π/2 are x = ±π/2.
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What expression is equivalent to (square root)
Answer:
C.) [tex]x^{\frac{3}{2} } y^{\frac{19}{2} }[/tex]
Step-by-step explanation:
To simplify the expression, you need to:
1.) Rewrite [tex]\sqrt{xy^{3} }[/tex]
-----> [tex]\sqrt{x} = x^{\frac{1}{2} }[/tex]
2.) Distribute the exponents
-----> When an exponent is raised to another exponent, they should be multiplied
3.) Create common denominators
4.) Combine the variables
-----> When two variables with exponents are being multiplied, the exponents should be added
Solve the equation by completing the square
Answer:
Below.
Step-by-step explanation:
x^2 + 4x = 18
x^2 + 4x + (2)^2 = 15 + 2^2
x^2 + 4x + 4 = 18 + 4
(x + 2)^2 = 22
x + 2 = +/- √22
x = -2 + √22 and x = -2 + √22
x = 2.7 and -6.7 to nearest tenth.
The water was pumped out of a backyard pond. What is the domain of this graph?
quick answer (A)
ALL real numbers less than or equal to 8
Alek is taking an inventory of styles of compression bandages for work. Here is the data he has collected.
The individuals in this data set that can be gotten is B) Compression bandage styles and this data set contains A) 3 variables, 1 of which is categorical.
What is categorical variable ?A categorical variables serve as the variable type that posses two or more categories.
This categories could be nominal variable, however considering the data , we can see that the first two are quantitative data and the data are:
WidthTotal lengthColorTHE COMPLETE QUESTION IS:
Alek is taking an inventory of styles of compression bandages for work. Here is the data he has collected.
The individuals in this data set are:Choose 1 answer:
A) Blood Centers
B) Compression bandage styles
C) Nurses
This data set contains:Choose 1 Answer:
A) 3 variables, 1 of which is categorical
B) 3 variables, 3 of which are categorical
C) 6 variables, 1 of which categorical
D) 6 variables, 3 of which are categorical
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Consider a circle whose equation is x2 + y2 – 2x – 8 = 0. Which statements are true? Select three options.
The radius of the circle is 3 units.
The center of the circle lies on the x-axis.
The center of the circle lies on the y-axis.
The standard form of the equation is (x – 1)² + y² = 3.
The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
The center of the circle lies on the x-axis
The radius of the circle is 3 units.
The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
How to determine the statements
The standard equation of a circle is expressed as:
[tex]x^2 + y^2 + 2gx + 2fy + C = 0[/tex]
Where
Centre is (-g, -f)radius = √g²+f²-CGiven a circle whose equation is
Let's get the center of the circle
2gx = -2x
2g = -2
We have that;
g = -1
Similarly, 2fy = 0
given that
f = 0
Centre = (-(-1), 0) = (1, 0)
This explains that the center of the circle lies on the x-axis
We move further to the radius
r = radius = √g²+f²-C
radius = √1²+0²-(-8)
radius =√9
= 3 units
The radius of the circle is 3 units.
For the circle x² + y² = 9, the radius is expressed as:
r² = 9
r = 3 units
Hence the radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
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PLEASE HELP IM STUCK
Step-by-step explanation:
[tex] \frac{y2 - y1 }{x2 - x1} = \frac{3 - - 2}{5 - 0} = \frac{5}{5} = 1[/tex]
An account pays 8% per year simple interest. in year 1, the amount in the account is $850. how much is in the account in year 6?
Answer:
Step-by-step explanation:
$1,248.92
terrell watched 300 dogs being groomed. He found that 0.35 of the groomed dogs tried to bite the groomer. what percent of dogs did not try to bite the groomer?
The percentage of dogs who didn't try to bite the groomer is; 65%.
What is the percentage of dogs that did not try to bite the groomer?If follows from the task content that the number of dogs is 300 of which, 0.35 faction of the whole tried to bite the groomer.
Hence, the faction that did not try to bite the groomer is; 0.65 which corresponds to 65% when expressed as a percentage.
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In a math class of $30$ students, $12$ out of $15$ girls are freshmen and $11$ out of $15$ boys are freshmen. What is the probability that in a randomly selected group of five students from the class, there will be two freshmen girls and three freshmen boys
The probability of selecting five students from the class, where there will be two freshmen girls and three freshmen boys is 0.23 approximately.
What is Probability ?Probability is the game of chance. Probability of an event is the ratio of the required outcome to the total possible outcome.
Given that In a math class of 30 students, 12 out of 15 girls are freshmen and 11 out of 15 boys are freshmen. In Probability, the word "and" means multiplication
The probability that in a randomly selected group of five students from the class, there will be two freshmen girls and three freshmen boys can be expressed as follow
12/15 x 11/14 x 11/15 x 10/14 x 9/13
132/210 x 990/2730
130680/573300
0.2279
Therefore, the probability that in a randomly selected group of five students from the class, there will be two freshmen girls and three freshmen boys is 0.23 approximately.
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Determine the missing coordinate in the ordered pair (?,5) so that it will satisfy x+6y=18
x/2+y=4/5. x+y/2=7/10
Answer:
x = 2/5
y = 3/5
Step-by-step explanation:
**Disclaimer** Hi there! I assumed the question to be a system of equations. The following answer is a method for solving a system of equations. Thus, if it is not, please let me know and I will modify my answer.
Given information:
[tex]\frac{x}{2}+y=\frac{4}{5}[/tex]
[tex]x+\frac{y}{2} =\frac{7}{10}[/tex]
Eliminate fractions by multiplying 10 on both sides of the first equation:
[tex]10*\frac{x}{2}+10*y=10*\frac{4}{5}[/tex]
[tex]5x+10y=8[/tex]
Eliminate fractions by multiplying 10 on both sides of the second equation:
[tex]10*x+10*\frac{y}{2} =10*\frac{7}{10}[/tex]
[tex]10x+5y=7[/tex]
Current system:
[tex]5x+10y=8[/tex]
[tex]10x+5y=7[/tex]
Multiply the second equation by 2:
[tex]5x+10y=8[/tex]
[tex]20x+10y=14[/tex]
Subtract the first equation from the second equation:
[tex](20x+10y)-(5x+10y)=14-8[/tex]
[tex]20x+10y-5x-10y=6[/tex]
[tex](10y-10y)+20x-5x=6[/tex]
[tex]0+15x=6[/tex]
[tex]\boxed{x=\frac{2}{5} }[/tex]
Substitute the x value back to one of the equations to get the y value:
[tex]x+\frac{y}{2} =\frac{7}{10}[/tex]
[tex](\frac{2}{5}) +\frac{y}{2} =\frac{7}{10}[/tex]
[tex](\frac{2}{5}) +\frac{y}{2}-\frac{2}{5} =\frac{7}{10}-\frac{2}{5}[/tex]
[tex]\frac{y}{2} =\frac{3}{10}[/tex]
[tex]\frac{y}{2}*2 =\frac{3}{10}*2[/tex]
[tex]\boxed{y=\frac{3}{5} }[/tex]
Hope this helps!! :)
Please let me know if you have any questions
The answer is x = 2/5, y = 3/5 or (2/5, 3/5)
First, in order to avoid fractional values during calculation, multiply both equations by 10 to simplify.
10 (x/2 + y) = 10 (4/5) ⇒ 5x + 10y = 810 (x + y/2) = 10 (7/10) ⇒ 10x + 5y = 7Multiply the 1st equation by 10 and 2nd equation by 5.
3. 10 (5x + 10y) = 10 (8) ⇒ 50x + 100y = 80
4. 5 (10x + 5y) = 5 (7) ⇒ 50x + 25y = 35
Subtract : 3rd equation - 4th equation
50x + 100y - 50x - 25y = 80 - 3575y = 45y = 3/5Now, substitute for x in the 1st equation to find y.
x/2 + 3/5 = 4/5x/2 = 1/5x = 2/5What three-dimensional object would be generated by the rotation of the semi circle about the x axis ?
If a semicircle is rotated about the x axis of the graph then the resulting three dimensional shape will be a sphere.
Given a semicircle on a graph.
A semicircle is a half circle .A sphere is a geometrical object that is a three dimensional shape , It is the set of points that are all at the same distance r from given point in three dimensional shape.
Rotation is basically an action of rotating around an object or an axis.
If a semicircle is rotated about x axis then the resulting figure will be a sphere. This is because of the fact that when the semi circle is rotated about the x axis then all the points will be at equal distance in all directions from the center.
Hence if a semicircle is rotated about x axis then the resulting figure will be a sphere.
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The function f has the property that, for all x, 5(f)x=f(5x). If f(20)=40, what is the value of f(4)?
Using the given property, we conclude that:
f(4) = 8
What is the value of f(4)?
We know that our function has the property:
5*f(x) = f(5*x)
Now, we also know that:
f(20) = 40
Notice that we can rewrite:
f(20) = f(5*4)
Using the above property, we know that:
f(20) = f(5*4) = 5*f(4) = 40
Using the last part:
5*f(4) = 40
f(4) = 40/5 = 8
f(4) = 8
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What numbers are zeros of g(x) = x^2 - 2x - 4? take your time if you need to!
Answer:
[tex]x = 1 + \sqrt5, x = 1 - \sqrt5[/tex]
Step-by-step explanation:
Hello!
We can solve the quadratic by using the quadratic formula.
Standard form of a quadratic: [tex]ax^2 + bx + c = 0[/tex]
Quadratic Formula: [tex]x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}[/tex]
Given our Equation: [tex]g(x) = x^2 - 2x - 4[/tex]
a = 1b = -2c = -4Plug the values into the equation and solve.
Solve[tex]x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}[/tex][tex]x = \frac{-(-2)\pm\sqrt{(-2)^2 - 4(1)(-4)}}{2(1)}[/tex][tex]x = \frac{2\pm\sqrt{4 +16}}{2}[/tex][tex]x = \frac{2\pm\sqrt{20}}{2}[/tex][tex]x = \frac{2\pm\sqrt{4 * 5}}{2}[/tex][tex]x = \frac{2\pm(\sqrt4 * \sqrt5)}{2}[/tex][tex]x = \frac{2\pm2\sqrt5}{2}[/tex][tex]x = 1 + \sqrt5, x = 1 - \sqrt5[/tex]what is the circumference?
(4x⁴-4x²-x-3) ÷ (2x²-3)
so far i have 2x²+1+?/2x²-3
what is the question mark?
Answer:
[tex]2x^2+1-\dfrac{x}{2x^2-3}[/tex]
Step-by-step explanation:
Use the long division method to divide polynomials:
[tex]\large \begin{array}{r}2x^2\phantom{)))))}+1\phantom{)}\\2x^2-3{\overline{\smash{\big)}\,4x^4-4x^2-x-3\phantom{)}}}\\\underline{-~\phantom{(}(4x^4-6x^2)\phantom{-b))))))}}\\0+2x^2-x-3\phantom{)}\\\underline{-~\phantom{()}(2x^2\phantom{))))}-3)}\\-x\phantom{))))))}\end{array}[/tex]
Therefore:
[tex]\dfrac{4x^4-4x^2-x-3}{2x^2-3}=2x^2+1-\dfrac{x}{2x^2-3}[/tex]
Can someone help me please
The simplified form of the expression is x-4 + 2/x-2
Synthetic division of equationGiven the following fraction below
f(x) = x^2 + 2x - 6/x-2
Using the long division as shown below;
x + 4
x-2 | x^2 + 2x - 6
x^2 - 2x
_________________
4x - 6
4x - 8
___________________
2
Hence the simplified form of the expression is x-4 + 2/x-2
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Speedy Swift is a package delivery service that serves the greater Atlanta, Georgia, metropolitan area. To maintain customer loyalty, one of Speedy Swift's performance objectives is on-time delivery. To monitor its performance, each delivery is measured on the following scale: early (package delivered before the promised time), on-time (package delivered within 15 minutes of the promised time), late (package delivered more than 15 minutes past the promised time), or lost (package never delivered). Speedy Swift's objective is to deliver 99% of all packages either early or on-time.
Speedy collected the following data for last month's performance:
On-time On-time Early Late On-time On-time On-time On-time Late On-time
Early On-time On-time Early On-time On-time On-time On-time On-time On-time
Early On-time Early On-time On-time On-time Early On-time On-time On-time
Early On-time On-time Early Early Early On-time Loss On-time Early
Late Late Late On-time On-time On-time On-time On-time On-time On-time
On-time Late Early On-time Early On-time Lost On-time On-time On-time
Early Early On-time On-time Late Early Early On-time On-time On-time
On-time On-time Early On-time Early On-time Early On-time Late On-time
On-time Early On-time On-time On-time Late On-time Lost On-time On-time
On-time On-time On-time On-time On-time Early Early On-time On-time On-time
(a) What scale is used to measure delivery performance? What kind of variable is delivery performance?
(b) Construct a frequency table for delivery performance for last month.
(c) Construct a relative frequency table for delivery performance last month.
(a) The scale used in the analysis is qualitative as the data is not assigned any numeric value.
The daily performance is an ordinal variable, as the values assigned are ranked according to an order of the delivery time.
(b) The frequency table of the given data is:
Variable Frequency
Early 23
On-time 65
Late 9
Lost 3
Total 100.
(c) The relative frequency table for the given data is:
Variable Frequency Relative frequency
Early 23 23/100 = 0.23 = 23%
On-time 65 65/100 = 0.65 = 65%
Late 9 9/100 = 0.09 = 9%
Lost 3 3/100 = 0.03 = 3%
(a) The scale used in the analysis is qualitative as the data is not assigned any numeric value.
The daily performance is an ordinal variable, as the values assigned are ranked according to an order of the delivery time.
(b) The frequency table is a table showing frequency of each variable for the number of times they appear in the data.
The frequency table of the given data is:
Variable Frequency
Early 23
On-time 65
Late 9
Lost 3
Total 100.
(c) The relative frequency for a variable is the ratio of its frequency to the total frequency.
The relative frequency table for the given data is:
Variable Frequency Relative frequency
Early 23 23/100 = 0.23 = 23%
On-time 65 65/100 = 0.65 = 65%
Late 9 9/100 = 0.09 = 9%
Lost 3 3/100 = 0.03 = 3%
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Answer:
C.
Step-by-step explanation:
Which expression is equivalent to startfraction 3 x superscript negative 6 baseline y superscript negative 3 baseline over 15 x squared y superscript 10 baseline endfraction? assume x not-equals 0, y not-equals 0
The equivalent expression of [tex]\frac{3x^{-6}y^{-3}}{15x^2y^{10}}[/tex] is [tex]\frac{1}{5x^{8}y^{13}}[/tex]
How to determine the equivalent expression?The expression is given as:
[tex]\frac{3x^{-6}y^{-3}}{15x^2y^{10}}[/tex]
Divide 3 and 15 by 3
[tex]\frac{x^{-6}y^{-3}}{5x^2y^{10}}[/tex]
Apply the law of indices
[tex]\frac{1}{5x^{2+6}y^{10+3}}[/tex]
Evaluate the sum
[tex]\frac{1}{5x^{8}y^{13}}[/tex]
Hence, the equivalent expression of [tex]\frac{3x^{-6}y^{-3}}{15x^2y^{10}}[/tex] is [tex]\frac{1}{5x^{8}y^{13}}[/tex]
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What are the solutions of the equation (2x 3)2 8(2x 3) 11 = 0? use u substitution and the quadratic formula to solve.
Answer:
Step-by-step explanation:
How would I solve this?? My math course didn't teach me anything related to this sort of problem..
Answer:
See below
Step-by-step explanation:
The larger is
7-6 = 1 unit bigger than the smaller....and this unit = 24 gallons
each unit is 24 gallons
larger barrel holds 24 * 7 = 168 gal
smaller barrel holds 24 * 6 = 144 gal
Answer:
Small barrel=144 large barrel=168
Step-by-step explanation:
The ratio is 6 to 7 so lets assign x as a factor.
6x=Small barrel (S)
7x=Large barrel (L)
7x-6x=x
x is the difference so x is 24
x=24
6*24=144=S
7*24=168=L
HELPPP PLEASEEEE
Other than translation, what transformation is used to create this frieze pattern?
horizontal reflection
glide reflection
vertical reflection
90° rotation
Answer:
I think glide reflection
Step-by-step explanation:
a certain video on yt had explained the examples of similar look alike this picture
A store sells about 30 pants each week for Nu 800 each. The owner expects to lose one sale each week for every increase in price of Nu 40. a) Write an expression to represent the i) new price of a pant after n Nu 40 price increases. ii) expected number of pants that will be sold weekly after n price increases. b) Write a function to represent the expected weekly sales as a function of the number of price increases of Nu 40. c) Use the function to determine the price that will maximize totally sales.
The new price after the price increase of Nu 40 will be Nu 840.
How to calculate the price?The price of a pant after n Nu 40 price increases will be:
= 800 + 40
= 840
The expected weekly sales as a function of the number of price increases will be:
= n - 1) × 800
= (30 - 1) × 800
= 29 × 800
= 23200
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there are three red marbles 5 green marbles 2 blue marbles in a bag which of the follwing are true
Answer:
4 and 1
Step-by-step explanation:
how do you graph y=12/x
Answer: [tex]\frac{12}{x}[/tex] is an asymptote
Step-by-step explanation:
Apply a table, and plug in values of x into the equation. For example f(2)=[tex]\frac{12}{2} =6[/tex]. That's one of the points on the graph that we just found (2,6)
help please its khan academy
30 points please help me
The area of the shaded region = area of larger circle - area of smaller circle = 423.9 yd².
What is the Area of a Circle?Area = πr²
Area of the shaded region = area of larger circle - area of smaller circle
Area of larger circle = πr² = (3.14)(16²) = 803.84 yd²
Area of smaller circle = πr² = (3.14)(11²) = 379.94 yd²
Area of the shaded region = 803.84 - 379.94 = 423.9 yd².
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Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options.
x < 5
–6x – 5 < 10 – x
–6x + 15 < 10 – 5x
A number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.
A number line from negative 3 to 3 in increments of 1. An open circle is at negative 5 and a bold line starts at negative 5 and is pointing to the left.
The correct statements are –6x + 15 < 10 – 5x and a number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.
Inequality expressionInequality are expressions not separated by an equal sign. Given the inequality expression:
–3(2x – 5) < 5(2 – x)
Expand
-6x +15 < 10 - 5x
Collect the like terms
-6x+5x < 10 - 15
-x <-5
x > 5
Hence the correct statements are –6x + 15 < 10 – 5x and a number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right.
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Which system of equations can you use to find the roots of the equation 2x3 4x2 – x 5 = –3x2 4x 9? y = 2x3 x2 3x 5 y =9 y = 2x3 x2 y = 3x 14 y = 2x3 4x2 – x 5 y = –3x2 4x 9
Finding the roots of an equation is done using the equation systems [tex]y=2x^{3}+4x^{2} -x+5[/tex] & [tex]y= -3x^{2} +4x+9[/tex] .
Determining the roots of the equationThis equation seems to have the following form:[tex]2x^{3}+4x^{2} -x+5= -3x^{2} +4x+9[/tex]
We must assess the appropriate set of equations that may be utilized to identify the equation's roots.
By drawing the equations' graph, we may determine the equation's root. We can suppose that the left and right sides of the following equations are two independent equations for the sake of graphing.
Therefore,
[tex]y=2x^{3}+4x^{2} -x+5[/tex]
[tex]y= -3x^{2} +4x+9[/tex]
The graphs of these two equations are now shown in the identical coordinate plane. The roots of the equation given will be the crossing point(s).
The equation system that we may utilize to get the root of the formula given is thus,
[tex]y=2x^{3}+4x^{2} -x+5[/tex]
[tex]y= -3x^{2} +4x+9[/tex]
So, the correct answer is option C.
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Calculate the first difference between 7, and where 7,-688. 2
The difference between the numbers given is 6.882.
How to illustrate the information?Based on the information given, it should be noted that difference means subtraction.
The difference between the numbers given will be:
= 7 - (7 - 6.882)
= 7 - 0.118
= 6.882
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