The Maclaurin series for f(x) is:
f(x) = ∑[n=1 to infinity] (x^n)/n! * P_n(0)
= ∑[n=1 to infinity] (x^n)/n! * n!/n^8
= ∑[n=1 to infinity] (x^n)/n^8
To find the Maclaurin series of f(x), we can repeatedly differentiate f(x) and evaluate it at x=0 to find the coefficients of the series.
f(0) = 0
f'(x) = 1/(1 + x^7)
f''(x) = -7x^6/(1 + x^7)^2
f'''(x) = (42x^5 + 49x^13)/(1 + x^7)^3
f''''(x) = (-210x^4 - 637x^12 - 343x^20)/(1 + x^7)^4
and so on. The general formula for the nth derivative of f(x) is given by:
f^(n)(x) = P_n(x)/(1 + x^7)^(n+1)
where P_n(x) is a polynomial of degree at most 6n-1. We can find the coefficients of P_n(x) using the formula for the nth derivative and evaluating it at x=0:
P_n(0) = n!f^(n)(0) = n!/(1+0^7)^(n+1) = n!/n^8
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The meclauren series for the function f with f(0)=0 and derivative [tex]f'(x) = \frac{1}{1 + x⁷}[/tex], is equals to [tex]f(x) = x - \frac{ x⁸}{8} + \frac{x¹⁵}{15} - \frac{x²²}{22} + .... [/tex].
The Maclaurin series represents a function as an infinite sum of terms, each term being a derivative of the function evaluated at x = 0, 1,... Formula is written, [tex]\sum_{n= 0}^{\infty}\frac{ f^{n}(0)}{n!} x^n[/tex]
where fⁿ(0) --> derivatives of f(x) at x = 0
n --> real numbers
We have a function, f(x) such that f(0) = 0 and derivative of f(x), i.e, [tex]f'(x) = \frac{1}{1 + x⁷}[/tex].
We have to determine the meclauren series of function f(x). First we determine the value of f(x), so, expand the [tex]\frac{1}{1 + x⁷}[/tex] as meclauren series. The meclauren series for [tex]\frac{1}{1 + x}[/tex] is written, [tex] \frac{1}{1 + x} = 1 - x + x² - x³ + ......[/tex]
Replace the x by x⁷, we result
[tex] \frac{1}{1 + x^{7} } = 1 - {x}^{7} + {x}^{14} - {x}^{21} + ......[/tex]
Now, integrating the above series expansion, [tex]\int f'(x) dx= \int ( 1 - x⁷ + x¹⁴ - x²¹ + ......) dx[/tex]
[tex]f(x) = x - \frac{ x⁸}{8} + \frac{x¹⁵}{15} - \frac{x²²}{22} + .... + c \\ [/tex]
Using f(0) = 0
=> f(0) = 0 = 0 + 0 + 0 +.... + c
=> c = 0
Hence, required series is [tex]f(x) = x - \frac{ x⁸}{8} + \frac{x¹⁵}{15} - \frac{x²²}{22} + .... [/tex].
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Select the correct answer from the drop-down menu.
Compare the steps for constructing a perpendicular line to line ABthrough point C using a reflective device and using paper.
.C
A
When using paper to construct the perpendicular line, the paper is folded so that
line ABwill lie exactly on top of each other.
When using a reflective device to construct a line perpendicular to line AB, set one end of the device to coincide with
then swing the other end of the device so the reflection of line ABcoincides with
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and the two sides of
and
The C is perpendicular to AB
When using paper, the paper perpendicular te construct to is folded so that AB is divided equally and the two sides of. line AB will lie exactly on top of each other.
When using a reflective device to construct a live perpendicular to line AB, set One end of the device to coincide with point C, and then swing the other end of the device so that the reflection of line AB coincides with the midpoint of AB
The question is incomplete hence the complete question is shown here
Select the correct answer from the drop-down menu. Compare the steps for constructing a perpendicular line to line through point using a reflective device and using paper. , When using paper to construct the perpendicular line, the paper is folded so that and the two sides of line will lie exactly on top of each other. When using a reflective dewice to construct a line perpendicular to line , set one end of the device to coincide with , and then swing the other end of the device so the reflection of line coincides with.
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Answer:
I toke the test, I think the first one is A and B coincide
Step-by-step explanation:
Tom has x bags of apples. he buys 8 times more and sells 12. write the amount he has left as an algebraic expression:
Answer:
Step-by-step explanation:
x times 8 +12
8x+x=9x-12
Answer = 9x-12
A legal document contains pages numbered 1 through 50. What is the probability that the lawyer will flip to a page number that is not a multiple of 4
Using its concept, it is found that the probability that the lawyer will flip to a page number that is not a multiple of 4 is given by:
p=19/25
Probability is a field of mathematics that deals with the numerical description of the likelihood that an event will occur or that a statement is true. The probability of an event is a number between 0 and 1, with approximately 0 indicating the impossibility of the event and 1 indicating certainty.
A probability is given by the number of desired outcomes divided by the number of total outcomes.
In this problem, there are 50 numbers, of which 50 - (48/4) = 38 are not multiples of 4, hence the probability is given by:
p=38/50 = 19/25
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Tas
The three points marked on the graph are
the vertices (corners) of a parallelogram.
10
9
y
8
7
6
5
4
3
2
1
A
1 2
C
2
B
3 4 5 6 7 8 9
10
What are the coordinates of the fourth
vertex?
Answer: (8,10)
Step-by-step explanation:
To go from A to C, we go right 2 units and up 6 units.
So, if we let the fourth vertex be D, then to go from B to D, we must also go right 2 units and up 6 units from B.
Therefore, the answer is (8, 10).
Find the polar coordinates in radians from the given rectangular coordinates (-11, -3).
the polar coordinates are:
(√130, 285.3°)
How to find the polar coordinates?For rectangular coordinates (x, y), the polar coordinates are:
[tex]R = \sqrt{x^2 + y^2}[/tex]
For the angle θ we need to notice that if both coordinates are negative, like in this case, we would be on the fourth quadrant, then:
θ = arctan(y/x) + 270°
In this case, we have:
x = -11
y = -3
Then:
[tex]R = \sqrt{(-11)^2 + (-3)^2} = \sqrt{130} \\\\\theta = arctan(-3/-11) + 270\° = 285.3\°[/tex]
Then the polar coordinates are:
(√130, 285.3°)
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The expression 40x²-65x + 50 represents the sum of the interior angles of a regular pentagon in degrees. If the
interior angles of the pentagon are equal, which expression represents the measure of two angles?
Answer:
[tex]16x^2-26x+20[/tex]
Step-by-step explanation:
Since it represents the sum of interior angles, the angle is essentially: [tex]angle + angle + angle + angle + angle[/tex], and there are 5 angles since a pentagon has 5 interior angles. To find what one interior angle is, we simply divide the polynomial by 5, to do this we can just divide each coefficient by 5 (distributing the division): [tex]\frac{40x^2-65x+50}{5} = \frac{40x^2}{5} - \frac{65x}{5} + \frac{50}{5}[/tex] this simplifies to: [tex]8x^2-13x+10[/tex]. This represents what one of the interior angles is. Since we want an expression measuring two of the angles, we simply multiply this polynomial by 2, to do this we just multiply each coefficient by 2 (distributing the 2): [tex]2(8x^2-13x+10) = 16x^2-26x+20[/tex] which is the solution
You want to retire on the equivalent of $50,000 per year in today's money. inflation is expected to be 3%. you will retire in 30 years. you will earn 10% annually on your investments from not till retirement and you will earn 8% during retirement (which is expected to last 20 years). how much do you need to save per month in order to fund this retirement?
$662.18 needs to be saved per month.
$90,000 now at 3% inflation will become 90000*1.03^30 after 30 years.
i.e. $218,454 is required in retirement years (for 20 years)
Calculating the PV of these payments at 7% is calculated as below.
FV = 0
1/y = 7%
n=20; PMT = 218454; calculate PV = $2,314,305
Now we want to get $2,314,305 at the start of retirement by saving per year. Enter the following in the financial calculator to get the per month amount.
FV = 2314305; PV = 0; n= 12*30; 1/y = 12%/12; calculate PMT = $662.18
So $662.18 needs to be saved per month.
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I need help solving this problem
Answer:
Please see the picture below.
Step-by-step explanation:
A teacher with 10 students has 30 lesson times available. She will teach exactly one of her students during each lesson time. How many ways are there for her to decide which student she will teach during each lesson time if she must teach each student exactly 3 times
There are [tex]\frac{30!}{(3!)^{10} }[/tex] ways for the teacher to decide which student she will teach during each lesson time if she must teach each student exactly 3 times. Here, "!" represents the factorial.
A number's factorial is the result of multiplying the integer by each natural number below it. Factorial can be symbolized by the letter "!". Thus, n factorial is denoted by n! and is the result of the first n natural numbers.
A whole number's "n" factororial is the sum of that number and each whole number up to one.
When a question asks you to determine how many different ways you can arrange or order a given number of items, you use a factorial.
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A sector has a radius of 12 centimeters and an angle 65 degrees. find its arc length
If the radius of sector is 12 centimeters and angle of 65 degrees then the arc's length is 13.6 cm.
Given a sector has a radius of 12 cm and an angle 65 degrees.
We have to calculate arc's length.
We know that arc's length is equal to Θr where Θ is in radians.
We have been given angle in degrees ,so we have to convert degrees in radians.
Angle=65*π/180=13π/36
length of arc =Θr
Putting the values in the formula above :
length =13π/36*12
=13*3.14/3
=40.82/3
=13.6
Hence if the radius of sphere is 12 cm and angle 65 degrees then the length of arc of sphere is 13.6 cm.
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1. If 8x + 4y = 44, what is the value of 6x + 3y?
The value of the expression 6x+3y is 33
System of equationsSystem of equations consists of two or more equation with unknown variables.
Given the equation below
8x + 4y = 44,
Factor out 4 from the equation to have:
4(2x+y) = 44
Divide both sides by 4
2x+y = 11
Determine the value of 6x+3y
6x+3y = 3(2x+y)
6x+3y = 3(11)
6x+3y =33
Hence the value of the expression 6x+3y is 33
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15 athletes are assigned to one of three teams. Each team will have five athletes. The teams are different, so it matters who gets assigned to which team. How many ways are there to assign the athletes to the teams
Answer:
There is only one way to assign the athletes to the team by giving 5 athletes to each team
If triangle abc is reflected over the x-axis, reflected over the y-axis, and rotated 180 degrees, where will point b' lie?
Answer:
Step-by-step explanation:
The triangle would end up back where it started. It is hard to explain without a graph. If you have graph paper, you might want to try drawing this out. Say that the original points are at A (1,2) B (2,0) and C(0,0). Now, when we reflect the points over the x axis, they will be the same distance below the x that the points original were about the x axis. Since A was 2 units above the axis, it will now be 2 units below at (1, -2). Points B and C will stay on the x axis and will remain in place at B(2.0) and C(0,0). Since these points are on the line, they were not above the x axis, so they will now not be below the x axis.
Now, we are going to reflect the triangle over the y axis. Since C (0,0) is already on the y axis, it will not move. It will remain there. Since B(2,1) is two units to the right of the y axis, when we flip it, it will now be 2 units to the left of the y axis B (-2,0). Point C will move one unit to the left of the y axis to become (-1,2).
The last thing left it to rotate this final triangle 180 degrees. Since a circle is 360 degrees and 180 is half of a circle, it does not matter if we rotate clockwise or counter-clockwise. If you could trace our new triangle and put a plus sign at the origin (0,0). You would put your pencil on the origin and rotate the two turns at the plus sign. This would put your triangle right back to the beginning. So the original value of B would be the same. In this case C ((2,0)
The point B' would be at point B. The position of point won't change.
What is a reflection?A reflection is a type of transformation that involves flipping a shape, known as the preimage, over a line, known as the line of reflection, to produce a new shape (called the image). You can picture what would happen if you flipped the form over the line in order to graph a reflection.
Given:
Triangle ABC is reflected over the x‐axis, reflected over the y‐axis, and rotated 180 degrees.
Let (x, y) be the coordinates of B.
To find B':
If a figure reflected over x-axis, then
(x, y) → (x, -y)
So, let point B₁ is (x, -y).
If a figure reflected over y-axis, then
(x, y) → (-x, y)
So, (x, -y) → (-x, -y)
So, let point B₂ is (-x, -y).
If a figure rotated 180 degrees about the origin, then
(x, y) → (-x, -y)
(-x, -y) → (x, y)
So the B' is (x, y).
Therefore, the position of point won't change.
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If fish are sold at 10 for $864. what will be the cost to the nearest dollar of 16 fish
If fish are sold at [tex]10[/tex] for $[tex]864[/tex]. the cost to the nearest dollar of [tex]16[/tex]fish is $ [tex]1382[/tex]
How can we find the cost of the [tex]16[/tex] fish ?
The cost of the selling [tex]10[/tex] fish = $[tex]864[/tex]
So the cost of the [tex]1[/tex] fish is
[tex]= \frac{864}{10} \\=86.4[/tex]
So the cost of the selling [tex]16[/tex] fish
[tex]= 16* 86.4\\=1382 $[/tex]
$[tex]1382[/tex] is the selling cost of [tex]16[/tex] fish.
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determine which two of the three given triangles are similar. Find the scale factor for the pair.
Since the ratio of their similar sides are equal hence the triangles are similar
Scale factorsTwo figures are are known to be similar if the ratio of their similar sides are equal to a constant known as the scale factor.
For the triangles given, take the ratio of the sides;
12/2 = 6
30/5 = 6
36/6 = 6
Since the ratio of their similar sides are equal hence the triangles are similar
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HELP! what is this? what’s the answer?!
D. not continuous at X=0
if you trace the function you'll find a jump in the graph. To be continuous the line can't have any breaks in it. In this example, it's discontinuous when X=0
PLS HELP ASAP
does anyone know these answers
Answer: i know it
Step-by-step explanation: i know
The table displays The scores of students on the recent exam. Find the mean of the scores to the nearest 10th.
Answer:
80
Step-by-step explanation:
To find the mean we add up all the numbers and then divide that number by the amount of numbers we added up.
So in this case we added up 65, 70, 75, 80, 85, 90 & 95. This gives us 560. Now we divide 560 by 7 as that is the amount of numbers we added up. This gives us 80 which means the answer is 80.
I hope this helped! :)
During the summer, jody earns $10 per hour babysitting and $15 per hour doing yardwork. this week she worked 34 hours and earned $410. if x represents the number hours she babysat and y represents the number of hours she did yardwork, which system of equations models this situation? a. x y = 34 10x 15y = 410 b. x y = 410 10x 15y = 34 c. x y = 34 15x 10y = 410 d. x y = 410 15x 10y = 34
The option A is correct, the linear equation x + y = 34. and 10x + 15y = 410 represent the the statement.
According to the statement
we have given that the
Jody earns $10 per hour babysitting And jody earns $15 per hour doing yardwork.
And she worked 34 hours and earned $410
and we have to express in the linear equation terms.
So, For this purpose,
Let x represents the number hours she babysat
Let y represents the number of hours she did yardwork
And the equations become
x + y = 34.
10x + 15y = 410
And with the help of these linear equations we find the hours which are x and y.
So, The above written equations perfectly express the given conditions in the statement.
The option A is correct, the linear equation x + y = 34. and 10x + 15y = 410 represent the the statement.
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1. Find the length of each side of the triangle if its perimeter is 86cm the sides are. 3a-2, 5a+3, 6a+1
Hello,
3a - 2 + 5a + 3 + 6a + 1 = 86
14a + 2 = 86
14a = 86 - 2
14a = 84
a = 84/14
a = 6
lengths :
3a - 2 = 3 × 6 - 2 = 18 - 2 = 16 cm
5a + 3 = 5 × 6 + 3 = 30 + 3 = 33 cm
6a + 1 = 6 × 6 + 1 = 36 + 1 = 37 cm
How many minutes are equal to 2 hours 20 minutes? how many minutes are equal to 2 hours 20 minutes? 100 120 220 240 none of these
Answer:
None of the answers
Step-by-step explanation:
There are 60 minutes in one hour so to find the number of minutes in two hrs we multiply 2 by 60: 2*60 = 120 minutes
We then add the extra 20 minutes: 120+20 = 140 minutes
None of the answers include 140 minutes.
The area of Marc's square garden is 225 ft².
What are the dimensions of his garden?
F. 5 ft by 45 ft
H. 15 ft by 15 ft
G. 12 ft by 12 ft
J. not given
Answer:
H. 15 ft by 15 ft
Step-by-step explanation:
area of square = side²
225 ft² = side²
side = √(225 ft²)
side = 15 ft
Answer: H. 15 ft by 15 ft
"The average speed of a vehicle and the distance it covers in a period " Is this a direct proportion
Answer:
Yes
Step-by-step explanation:
As the Average speed increases, the amount of distance covered will increase. As the average speed decreases, the amount of distance traveled will decrease as well over a certain period.
Please i really need help
for both qeutions
100 points
Answer:
7. a) slope = 5
7. b) slope = 1/2
8. slope = 1
Step-by-step explanation:
slope = rise/run
rise is the vertical change, run is the horizontal change
7. a)
Pick any two points on your line. (0,0) and (1,5) are easy to work with.
The vertical change between the two points is 5
The horizontal change between those points is 1
so slope = 5/1 = 5
7. b)
Again, you can use (0,0) as one of your points. The point (2,4) is good as the second.
slope = 4/2 = 1/2
8. We don't have a graph, but we are given two points to work with, (1,4) and (5,8). Use the formula given...
m = slope = (y2-y1) / (x2-x1)
= (8 - 4) / (5 - 1)
= 4/4
= 1
I hope that helped!
Complete the sentence using the following formula. Enter your answer using digits.
area
The length of a rectangle can be found using the following formula: length width
The length of a rectangle that has a width of 2.5 cm and an area of 20.5 cm² is __ cm.
Answer: 8.2 cm
Step-by-step explanation:
Area of a rectangle = l x w
[tex]A= 25 cm^{2}[/tex]
[tex]l= ?[/tex]
[tex]w= 2.5 cm[/tex]
[tex]25cm^{2} = l(2.5cm)[/tex]
Rearrange the equation to get Length by itself[tex]\frac{25cm^{2} }{2.5cm}= l[/tex]
[tex]8.2 cm=l[/tex]
Which of the statements below is true for the following set of numbers?42,10,36,51,70,28
Answer:
statement 1(one)
Step-by-step explanation:
SOMEONE PLEASE HELP!!! Find the area of the sector. Round your answer to the nearest tenth.
Answer:
398.2 cm^2
Step-by-step explanation:
[tex]\frac{270}{360}[/tex] * [tex]169\pi[/tex] ≈398.2 cm^2
what is 23,578 times 39,344
Answer:927652832
Step-by-step explanation:
using long multiplication step by step multiply.
how to simplify cos(6x) + 1
Which set of numbers is included in the solution set of the compound inequality?
The set of numbers is included in the solution set of the compound inequality is {-7, 5, 18, 24, 32}
Number line and inequalitiesInequalities are expressions not separated by an equal sign.
From the given number line, the equivalent inequality expression are
x > 22 (22 s excluded since it is an open circle) and x≤ 18 (18 is included since the circle is shaded)
Hence the set of numbers is included in the solution set of the compound inequality is {-7, 5, 18, 24, 32}
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The set of numbers that is been included in the solution set of the compound inequality is [tex]{-7, 5, 18, 24, 32}[/tex]
What is inequalities and set of numbers?The inequality can be referred to as the relationship between two values which are not equal and cannot be separated by an equal sign.
From the open circle, Checking the number set, [tex]x > 22[/tex], 22 cannot be involved, from the shaded circle, x≤ 18 , 18 can be included.
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