The sum of the given series is: [tex]∑(-1)^n * 4^n * x^(8n) / n![/tex]= coefficient of [tex]x^(8n)[/tex] in [tex]e^(-4x^8)[/tex]
The given series is:
[tex]∑(-1)^n * 4^n * x^(8n) / n![/tex]
To find the sum of this series, we can use the Maclaurin series expansion for the exponential function, which states:
[tex]e^x[/tex] = ∑(n=0 to infinity)[tex](x^n / n!)[/tex]
Comparing this with the given series, we see that it closely resembles the Maclaurin series for [tex]e^(-4x^8)[/tex]. Therefore, we can rewrite the series as:
[tex]∑(-1)^n * (4x^8)^n / n![/tex]
Using the formula for the Maclaurin series of [tex]e^(-4x^8)[/tex], we can substitute [tex](-4x^8)[/tex] for x in the series expansion of [tex]e^x[/tex]:
[tex]e^(-4x^8)[/tex] = ∑(n=0 to infinity) [tex]((-4x^8)^n / n!)[/tex]
Now, we can see that the series we need to find the sum for is the coefficient of [tex]x^(8n)[/tex] in the series expansion of [tex]e^(-4x^8)[/tex]. Therefore, the sum of the given series is:
[tex]∑(-1)^n * 4^n * x^(8n) / n![/tex]= coefficient of [tex]x^(8n)[/tex] in [tex]e^(-4x^8)[/tex]
Therefore, to find the sum of the series, we need to determine the coefficient of[tex]x^(8n)[/tex]in the series expansion of [tex]e^(-4x^8).[/tex]
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given that 4x-y=5 4y-z=7 and 4z-x=18 what is the value of x+y+z
Answer:
x + y +z= 10
Step-by-step explanation:
4x - y = 5 ---------(I)
4y - z = 7 ---------(II)
4z - x = 18 ---------(III)
Add all the three equations,
4x - y + 4y - z + 4z - x = 5 + 7 + 18
4x - x -y + 4y -z + 4z = 30
3x + 3y + 3z = 30
Divide the entire equation by 3
[tex]\sf \dfrac{3x}{3}+\dfrac{3y}{3}+\dfrac{3z}{3} = \dfrac{30}{3}[/tex]
x + y + z = 10
1. Factor and simplify: cos² xcsc² x-cos²xcot² x
Answer:
Cos² x
Step-by-step explanation:
Trigonometry:[tex]\sf Cos^2 \ x *Csc^2 \ x-Cos^2 \ x *Cot^2 \ x = Cos^2 \ x (Csc^2 \ x - Cot^2 \ x)[/tex]
[tex]\sf = Cos^2 \ x \left(\dfrac{1}{Sin^2 \ x} - \dfrac{Cos^2 \ x}{Sin^2 \ x}\right)\\\\= Cos^2 \ x \left(\dfrac{1-Cos^2 \ x}{Sin^2 \ x}\right)\\\\\boxed{\bf Indentity: \ 1 - Cos^2 \ x = Sin^2 \ x}\\\\\\= Cos^2 \ x \left(\dfrac{Sin^2 \ x}{Sin^2 \ x}\right)\\\\= Cos^2 \ x[/tex]
5 Jack works as a part-time waiter. He earns $60 per weekday and $98 per day on weekends. (a) On a particular week, he works from Tuesday to Sunday. How much was he paid for that week? (b) Find his average wage rate per day.
Find the equation of the line perpendicular to 2x+3y-6=0 and passing through (-1,1)
Answer: [tex]y-1=\frac{3}{2}(x+1)[/tex]
Step-by-step explanation:
[tex]2x+3y-6=0\\\\3y=-2x+6\\\\y=-\frac{2}{3}x+2[/tex]
So, the slope of the given line is -2/3, meaning the slope of the line we want to find is 3/2.
Substituting into point-slope form, we get the equation of the line is
[tex]y-1=\frac{3}{2}(x+1)[/tex]
How does the graph of g(x) = (x − 2)3 + 6 compare to the parent function of f(x) = x3?
The graph of g(x) is the graph of f(x) translated 2 units to the right and 6 units up.
How does the graph of g(x) compare to the one of f(x)?
Here we have:
[tex]f(x) = x^3\\\\g(x) = (x - 2)^3 + 6[/tex]
You can notice that if we take f(x), and we shift it 2 units to the right, we have:
g(x) = f(x - 2)
Then if we apply a shift upwards of 6 units, then we have:
g(x) = f(x - 2) + 3
Replacing f(x) by the cubic parent function, we have:
[tex]g(x) = (x - 2)^3 + 6[/tex]
So we conclude that the graph of g(x) is the graph of f(x) translated 2 units to the right and 6 units up.
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Consider the function represented in this table. f(x) -1 -21 0 -9 1 3 2 15 Which of these tables could represent the inverse of function f? X X -1 0 1 2 q(x) 21 9 -3 -15
If f(-1)=-21,f(0)=-9,f(1)=3,f(2)=15 then the inverse of function f(x) is p(-21)=-1,p(-9)=0,p(3)=1,p(15)=2 which is option d.
Given that f(-1)=-21,f(0)=-9,f(1)=3,f(2)=15 and we are required to find the inverse if the function.
Function is relationship between two or more variables expressed in equal to form. In a function each value of x must have corresponding value of y.
The given function is as under:
f(-1)=-21,
f(0)=-9,
f(1)=3
f(2)=15
In the inverse of the function the value of x becomes the value of y and the value of y becomes the value of x. So the inverse of the function is as under:
p(-21)=-1,
p(-9)=0,
p(3)=1,
p(15)=2
Hence if f(-1)=-21,f(0)=-9,f(1)=3,f(2)=15 then the inverse of function f(x) is p(-21)=-1,p(-9)=0,p(3)=1,p(15)=2 which is option d.
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A sample of college students was asked how much they spent monthly on cell phone plans. approximate the standard deviation for the
cost.
monthly cell phone plan cost (5)
number of students
10.00-19.99
7
20.00-29.99
20
30.00-39.99
28
40.00-49.99
12
50.00-59.99
7
The mean and the standard deviation of the following data are 35.75 and 11.6765 respectively.
First, we'll find the mean (X) of the data:
[tex]X =\dfrac{(\sum x \times f) } {\sum f}[/tex]
So, the approximate standard deviation for the monthly cell phone plan cost is 9.87.
The mean is given by
[tex]\mu =\frac{\sum xf}{\sum f}[/tex]
Where x is the midpoint of the class and f is the frequency.
The calculation of mean and the standard deviation is shown in the below table:
Hence, the mean is calculated as:
mean is [tex]\mu =\frac{2359.67}{66}[/tex]
=35.75
The standard deviation is calculated as:
Standard deviation =[tex]\sigma[/tex] =[tex]\sqrt{\Dfrac{\sum f(x-\bar{x})^2}{n-1}}[/tex]
=[tex]\sqrt{\dfrac{8862.12121}{65}}[/tex]
=11.6765
Thus, the mean and the standard deviation are 35.75 and 11.6765 respectively.
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The table of standard deviation is attached below.
What is the perimeter of the given quadrilateral?
y
0000
20 units
10 units
50 units
25 units
A
D
B
C
Answer:
20 units
Step-by-step explanation:
As you know that the quadrilateral is a square, you’ll just have to find the length of one side and multiply it by four. We can count and see that one side is 5 units.
5 * 4 = 20 units
You can also add the length of each of the four sides.
5 + 5 + 5 + 5 = 20
Hope that helps you! Please let me know if you need more help with this question. Have a great day!
She needs to know the length of each side so she can buy fencing. What is the length of line segment FJ
Using the Pythagorean theorem, the length of line segment FJ is: 42 ft.
How to Apply the Pythagorean Theorem?To find the length of line segment FJ, we need to apply the Pythagorean theorem to find the length of the segment from F to the point, say point O, on FJ where a right triangle FOG will be gotten.
Thus, we have:
FO = √(GF² - GO²)
FO = √(30² - 24²)
FO = 18 ft.
FJ = FO + GH
Plug in the values
FJ = 18 + 24
FJ = 42 ft.
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Find the perimeter of a parallelogram with corner points at (2,1), (4,5), (7,5), and (5,1)
Perimeter =
THANK YOU SO MUCHHH
Answer:
14 units
Step-by-step explanation:
Oops, I was solving for the area in the picture. You can still use the picture to find the perimeter. When I change the shape into a rectangle, you can see that the top and bottom lengths are 3 and the side lengths are 4. 3+3+4+4=14
The value of a company’s stock is represented by the expression x2 – 2y and the company’s purchases are modeled by 2x 5y. the company’s goal is to maintain a stock value of at least $5,000, while keeping the purchases below $1,000. which system of inequalities represents this scenario? x2 – 2y > 5000 2x 5y < 1000 x2 – 2y > 5000 2x 5y ≤ 1000 x2 – 2y ≥ 5000 2x 5y < 1000 x2 – 2y ≤ 5000 2x 5y ≤ 1000
Answer:
answer is (c)
May it help you
Option C. The inequality can then be written as x² - 2y ≥ 5000 and 2x + 5y <1000
How to write the inequality
Stock = x² - 2y
Purchase = 2x + 5y
Stock is said to be at least 5000 dollars. We can write this as
x² - 2y ≥ 5000
Then the purchase is said to be below 1000. This is written as 2x + 5y <1000
The inequality can then be written as x² - 2y ≥ 5000 and 2x + 5y <1000
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Which classification best represents a triangle with side lengths 10 in., 12 in., and 15 in.?
acute, because 102+122>152
acute, because 122+152>102
obtuse, because 102+122>152
obtuse, because 122+152>102
Answer:
Option (1)
Step-by-step explanation:
See attached image.
The given measures form a acute angle triangle.
What is an obtuse angle triangle?An obtuse-angled triangle is a triangle in which one of the interior angles measures more than 90° degrees. In an obtuse triangle, if one angle measures more than 90°, then the sum of the remaining two angles is less than 90°.
Given that, a triangle with side lengths 10 in., 12 in., and 15 in.
We know that,
a²+b²=c² then triangle will be right angled
a²+b²>c² then triangle will be acute angled
a²+b²><c² then triangle will be obtuse angled
Here, a=10 inches, b=12 inches and c=15 inches
10²+12²>15²
100+144>225
244>225
Therefore, the given measures form a acute angle triangle.
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Volume = length x width x height
1.
? ft
? ft
? ft
If the volume of the cube is 1 ft³,
if we multiply or divide both sides of a linear equation with a non zero number, then the solution of the linear equation
hi
"does not change " or "remain the same "
25 POINTS!!!!
The graph of f(x) = (0.5)x is replaced by the graph of g(x) = (0.5)x − k. If g(x) is obtained by shifting f(x) down by 7 units, the value of k is ________??
The answer is k = 7.
As the graph has been shifted down 7 units, the translation can be represented as :
g(x) = f(x) - 7
(0.5)x - k = (0.5)x - 7
k = 7
Can someone help me out on this please having trouble on both and show work please !!
The difference of the expression is as follows:
(6y⁴ + 3y² - 7) - (12y⁴ - y² + 5) = - 6y⁴ + 4y² - 12
3(x - 5) - (2x + 4) = x - 19
How to find the difference of the expression?The difference of the expression can be found when we combine the like terms.
Therefore,
(6y⁴ + 3y² - 7) - (12y⁴ - y² + 5)
6y⁴ + 3y² - 7 - 12y⁴ + y² - 5
6y⁴ - 12y⁴ + 3y² + y² - 7 - 5
- 6y⁴ + 4y² - 12
3(x - 5) - (2x + 4)
3x - 15 - 2x - 4
3x - 2x - 15 - 4
x - 19
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Rebecca filled 28 glasses with orange juice for a camp breakfast. if each glass holds 1 cup of juice, how many quarts of orange juice did rebecca use?
The number of quarts of orange juice did Rebecca use is 7
Given that 28 glasses with orange juice are filled for a camp breakfast
Each glass holds 1 cup of juice
We need to find the number of quarts of orange juice did Rebecca use
We know that a quarts has 4 cups
We need to calculate the quarts from 28 glasses
Let the number of the quarts be x
So the equation formed is
4x = 28
Where x is number of quarts and 28 are the number of glasses
x = 7
Hence the number of quarts is 7
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Drag the tiles to the correct boxes to complete the pairs.
Match the descriptions of cross sections of three-dimensional figures to their corresponding shapes.
Answer:
B, C, A
Step-by-step explanation:
Don't really know how to explain this.
determine which two of the three given triangles are similar. Find the scale factor for the pair
Answer:
Triangles MNP and QRS
Step-by-step explanation:
This is due to each corresponding line follows the same ratio of 1.5:1
e.g.
NP=6
RS=4
6:4
1.5:1
You can verify this by dividing 6 by 4 (6/4) which is 1.5
Which is the completely factored form of 12x4 + 39x3 + 9x2?
3x2(x – 3)(4x – 1)
3x2(x + 3)(4x + 1)
3x(x + 3)(4x + 1)
3x(x – 3)(4x + 1)
Answer:
The second option, 3x^2(x+3)(4x+1)
Step-by-step explanation:
Factor out the common term, which is 3x^2
This then becomes 3x^2(4x^2+13x+3)
Factor the inside term and it becomes
(3x^2)(x+3)(4x+1)
Select the correct answer from the drop-down menu.
Jaden and his family are using a map to drive down to Napa Valley. On the map, the distance is given by the equation c = 1.3m, where c is the number of centimeters on the map and m is the corresponding real-world distance in miles.
If they cover a distance of 18 miles, that would mean a distance of
centimeters on the map.
Answer:
13.846153846
Step-by-step explanation:
Its easy Just 18:1.3=13.846153846
The length of tr is 17 units. what are the lengths of sv and qt? sv = units qt = units
Applying the properties of kite, the measures are: SV = 41 units; QT = 21 units.
What is a kite?A kite is a quadrilateral that has 2 pairs of congruent adjacent sides, and has two diagonals where the longest one bisects the shorter one and divides it into equal segments.
The diagram given is a kite. Where TR = 17 units. Thus we would solve for x using the equation as shown below:
TR = RV [congruent adjacent sides of the kite]
Plug in the values
17 = 3x + 2
17 - 2 = 3x
15 = 3x
15/3 = x
x = 5
SV = TS = 9x - 4 [congruent adjacent sides of the kite]
Plug in the value of x
SV = 9(5) - 4
SV = 41 units
The length of segment QT is 41 units.
QT = QV = 4x + 1 [congruent adjacent sides of the kite]
Plug in the value of x
QT = 4(5) + 1
QT = 21 units
The length of segment QT is: 21 units.
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Answer:
SV: 41
QT:21
Step-by-step explanation:
A graduate school entrance exam has scores that are normally distributed with a mean of 560 and a standard deviation of 90. What percentage of examinees will score between 600 and 700
26.8% of examinees will score between 600 and 700.
This question is based on z score concept.
A Z-score is a numerical measurement that describes a value's relationship to the mean of a group of values. Z-score is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score.
[tex]z={x-\mu \over \sigma }[/tex]
where:
μ is the mean
σ is the standard deviation of the population
Given:
μ = 560
σ = 90
For
600≤ X≤700
for x = 700
Z score =x - μ/σ
=(700 - 560)/90
= 1.55556
P-value from Z-Table:
P(560<x<700) = P(x<700) - 0.5 = 0.44009
for x = 600
Z score =x - μ/σ
=(600 - 560)/90
= 0.44444
P-value from Z-Table:
P(560<x<600) = P(x<600) - 0.5 = 0.17164
∴ P(600<x<700) = P(560<x<700) - P(560<x<600)
= 0.44009 - 0.17164
=0.26845
∴26.8% percentage of examinees will score between 600 and 700.
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Students in a ninth grade class measured their heights, h, in centimeters. The height of the shortest student was 155 cm, and the height of the tallest student was 190 cm. Which inequality represents the range of heights?
A. h > 155 or h < 190
B. 155 ≤ h ≤ 190
C. 155 < h < 190
D. h ≥ 155 or h ≤ 190
Answer:
a h>155 or h<190
Step-by-step explanation:
The height of the shortest student is 155 cm and the height of the tallest student is 190 cm, then the inequality shows the range is 155 ≤ h ≤ 190. Hence, option B is correct.
What is an inequality?Inequalities are mathematical expressions where neither side is equal. In inequality, as opposed to equations, we compare the two values. Less than (or less than and equal to), larger than (or greater or equal to), or not similar to signs are used in place of the equal sign in between.
As per the provided data in the question,
The height of the shortest student is 155 cm and,
The height of the tallest student is 190 cm.
It means that the height rage is starting from 155 cm and ending at 190 cm.
The shortest height should be at least equal to 155 cm and the greater height is maximum 190 cm.
Therefore, the inequality according to this is 155 ≤ h ≤ 190.
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What comes next in the following series?
The next number in the series is 30.
What is series?It should be noted that series simply means the operation of adding values together in a data set.
In this case, it can be seen that the numbers are interchangeable between 30 and 31.
In this case, since 31 was the last number, the next number will be 30.
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Identify which property you would use first to solve the following equation.
3(2y+5) = 8y - 23
Answer:
Distributive property
Step-by-step explanation:
In order to solve this equation we first need to distribute 3 into 2y+5, we use the distributive property to do this.
Graph by hand or using a calculator to determine the solution to the given system. Put in slope-intercept form first if necessary.
The solution to the system of equations is (8, -4)
How to solve the system?The system of equations is given as;
y = -1/4x - 2
y = 3/8x - 7
See attachment for the graph of the equations
From the attached graph, we have the point of intersection to be
(x, y) = (8, -4)
Hence, the solution to the system of equations is (8, -4)
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Instructions: Find the missing side. Round your answer to the nearest tenth.
x=
Check
30
X
17
Sine rule:
30/sin(17) = x/sin(90)
x sin(90)
30 x sin(90) / sin(17) = x
x = 102.6091086
So, x = 102.6 to the nearest tenth.
Hope this helps!
You are deciding a social hall that will seat no more than 1600 people. You want at least 500 seats to be priced at $20 each, and at least 800 seats to be priced at $30 each. Those investing in the social hall would also like to bring in at least $2800 in revenue from the ticket sales. Let X represent the number of $20 seats, and let Y represent the number of $30 seats. Which system of inequalities represents this situation?
x ≥ 0
y ≥ 0
x + y ≥ 1600
20x + 30y ≤ 2800
x ≥ 500
y ≥ 800
x + y ≤ 1600
20x + 30y ≥ 2800
x ≥ 800
y ≥ 500
x + y ≤ 1600
20x + 30y ≤ 2800
x ≥ 500
y ≥ 800
x + y ≤ 1600
20x + 30y ≤ 2800
Answer: Option B
Just trust me that its right.
What are the roots of the polynomial equation x superscript 4 baseline x squared = 4 x cubed minus 12 x 12? use a graphing calculator and a system of equations. round noninteger roots to the nearest hundredth.
The roots of the polynomial equation are -1.73, 1.73 and 0
How to determine the roots?The polynomial equation is given as:
[tex]x^4 + x^2 = 4x^3 - 12x + 12[/tex]
Next, we split the equation as follows:
[tex]y = x^4 + x^2[/tex]
[tex]y= 4x^3 - 12x + 12[/tex]
Next, we plot the equations on a graphing calculator
From the graph, we have the x-coordinates of the intersection points to be:
x = -1.73, 1.73 and 0
Hence, the roots of the polynomial equation are -1.73, 1.73 and 0
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