Recall the well-known series
[tex]\displaystyle 2 \arcsin^2(x) = \sum_{n=1}^\infty \frac{(2x)^{2n}}{n^2 \binom{2n}n}[/tex]
Replace x with √x :
[tex]\displaystyle 2 \arcsin^2(\sqrt x) = \sum_{n=1}^\infty \frac{(4x)^n}{n^2 \binom{2n}n}[/tex]
Differentiate both sides:
[tex]\displaystyle -\frac{\arcsin(\sqrt x)}{2\sqrt{x-x^2}} = \sum_{n=1}^\infty \frac{(4x)^n}{n \binom{2n}n}[/tex]
Multiply by 4x :
[tex]\displaystyle -\frac{4x \arcsin(\sqrt x)}{2\sqrt{x-x^2}} = \sum_{n=1}^\infty \frac{(4x)^{n+1}}{n \binom{2n}n}[/tex]
All the series I've mentioned converge for |x| < 1, so we can take x = -1/4 to find the value of the sum we want.
[tex]\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n \binom{2n}n} = -\frac{4 \times \left(-\frac14\right) \arcsin\left(\sqrt{-\frac14}\right)}{2\sqrt{-\frac14-\left(-\frac14\right)^2}} = -\frac{\arcsin\left(\frac i2\right)}{\frac{\sqrt5\,i}2}} = \boxed{\frac2{\sqrt5} \mathrm{arsinh}\left(\frac12\right)}[/tex]
where
[tex]\arcsin\left(\frac i2\right) = z \iff 2\sin(z) = -2i\sinh(iz) = i \implies z = i \, \mathrm{arsinh}\left(\frac12\right)[/tex]
The points plotted below are on the graph of a polynomial. In what range of x- a values must the polynomial have a root? Check all that apply. A. -2 to -1 B. -1 to 0 C. -3 to-2 D. 1 to 2 E. -4 to -3 F. O to 1
4sin(-x)cos(-x)= -2sin2x
true or false
[tex]\text{L.H.S}\\\\=4\sin(-x) \cos(-x)\\\\=-4\sin x \cos x\\\\=-2 \cdot 2 \sin x \cos x\\\\=-2 \sin 2x\\\\=\text{R.H.S}\\\\\text{The identity is true.}[/tex]
on
Find the distance between -1/2 and 1 2/3
on
a number line.
units
Answer:
Step-by-step explanation:
1 2/3 -(-1/2)
=5/3+1/2
[tex]=\frac{10+1}{6} \\=\frac{11}{6} \\=1\frac{5}{6}[/tex]
[tex] \rm \int_{0}^2 \left( \sqrt[3]{ {x}^{2} + 2x} + \sqrt{1 + {x}^3 } \right )dx \\ [/tex]
If f(x) has an inverse on [a, b], then integrating by parts (take u = f(x) and dv = dx), we can show
[tex]\displaystyle \int_a^b f(x) \, dx = b f(b) - a f(a) - \int_{f(a)}^{f(b)} f^{-1}(x) \, dx[/tex]
Let [tex]f(x) = \sqrt{1 + x^3}[/tex]. Compute the inverse:
[tex]f\left(f^{-1}(x)\right) = \sqrt{1 + f^{-1}(x)^3} = x \implies f^{-1}(x) = \sqrt[3]{x^2-1}[/tex]
and we immediately notice that [tex]f^{-1}(x+1)=\sqrt[3]{x^2+2x}[/tex].
So, we can write the given integral as
[tex]\displaystyle \int_0^2 f^{-1}(x+1) + f(x) \, dx[/tex]
Splitting up terms and replacing [tex]x \to x-1[/tex] in the first integral, we get
[tex]\displaystyle \int_1^3 f^{-1}(x) \, dx + \int_0^2 f(x) \, dx = 2 f(2) - 0 f(0) = 2\times3+0\times1=\boxed{6}[/tex]
Melanie goes to a shelter to buy a dog. She has settled on two dogs, an Alsatian and a bulldog. To decide between them, she plans to flip a coin. Before she does, she flips the coin 50 times to check for fairness of the model. Her results are shown in the table.
Coin Toss Result Times Result Occurred
heads 24
tails 26
The relative frequency of landing on heads is ____. The relative frequency of landing on tails is
____.
Using it's concept, it is found that the relative frequency of landing on heads is 0.48 and of landing on tails is 0.52.
What is a relative frequency?A relative frequency is given by the number of desired outcomes divided by the number of total outcomes.
In this problem, out of 50 flips, 24 landed on heads and 26 on tails, hence:
Fh = 24/50 = 0.48.
Ft = 26/50 = 0.52.
The relative frequency of landing on heads is 0.48. The relative frequency of landing on tails is 0.52.
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how many part time and full time employees does the company have?
Answer:
there are 36 part time employees and there are 144 full time employees
Step-by-step explanation: multiply 45 by 4
then multiply 180 by 0.2 to get 36 and 180 by 0.8 and get 144
Melanie invested $63,000 in an account paying an interest rate of 7\tfrac{1}{8}7
8
1
% compounded quarterly. Lillian invested $63,000 in an account paying an interest rate of 7\tfrac{3}{8}7
8
3
% compounded continuously. After 15 years, how much more money would Lillian have in her account than Melanie, to the nearest dollar?
Write the value of (2 +√ 2) (2 -√3)
[tex]\left(2 + \sqrt 2 \right)\left(2-\sqrt 3 \right)\\\\=4- 2\sqrt 3 + 2 \sqrt 2 - \sqrt 6\\\\=0.91484.....[/tex]
What is the mean (average) of
a, a, b, b, b,c
Answer:
Ans might be a+b+c
Step-by-step explanation:
Hope the picture will help
Becket needs to solve the system of equations using elimination.
-2x+4y=-2
(6x-y=28)
Which correctly describes the first step Becket should take?
A. Multiply each term in the 1st equation by -3
B. Multiply each term int he 1st equation by 3
C. Multiply each term in the 2nd equation by -4
D. Both B and C would work
Answer:
B. Multiply each term in the 1st equation by 3
Step-by-step explanation:
A is incorrect because multiplying the entire 1st equation by -3 will give you 6x - 12y = 6. This does not cancel with either x or y term in the second equation.B is correct because by multiplying the entire 1st equation by 3 will give you -6x + 12y = -6. The -6x term will cancel (or eliminate) the 6x term in the second equation.C is incorrect because multiplying the entire 2nd equation by -4 will give you -24x + 4y = -112. This does not cancel with either x or y term in the 1st equation.D is incorrect because C was proven to be incorrect.Step-by-step explanation:
step 1
Multiply each term in the second equation with 4
step 2
Add equation 1 to equation 2
step 3
make x subject of formula and obtain value of X
step 4
substitute value of X in equation 1 and find Y
find the equation (in terms of x) of the line through the points (-4,2) and (3,-5)
Y=
==================================================
Explanation:
First we need the slope
[tex](x_1,y_1) = (-4,2) \text{ and } (x_2,y_2) = (3,-5)\\\\m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\\\\m = \frac{-5 - 2}{3 - (-4)}\\\\m = \frac{-5 - 2}{3 + 4}\\\\m = \frac{-7}{7}\\\\m = -1\\\\[/tex]
Then we can find the y intercept using the coordinates of (x,y) = (-4,2)
y = mx+b
2 = -1(-4)+b
2 = 4+b
2-4 = b
-2 = b
b = -2
Alternatively, you can use the other point (x,y) = (3,-5) and m = -1.
Because m = -1 and b = -2, we go from y = mx+b to y = -1x+(-2) which simplifies to y = -x-2
If line segment AB is 400%, what is the length of a line segment that is 100%?
the length of the line would be 1 inch
Drew and Beau each brought cylindrical water bottle to football practice. Drew's water bottle has a diameter of 3 inches and a height
of 7 inches. Beau's water bottle has a radius of 2 inches and a height of 9 inches. Whose water bottle holds more water and how much more does it hold? Round to the nearest hundredth.
The bottle that holds more water has the larger volume
Beau's water bottle holds more water
How to determine the bottle with more water?The given parameters are:
Drew
Diameter = 3 inchesHeight = 7 inchesBeau
Radius = 2 inchesHeight = 9 inchesThe volumes of their bottles is calculated using:
V = πr^2h
So, we have:
Drew = 3.14 * (3/2)^2 * 7
Drew = 49.46
Beau = 3.14 * (2)^2 * 9
Beau = 113.04
By comparison, 113.04 is greater than 49.46
Hence, Beau's water bottle holds more water
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A pizza is divided into 12 slices. If there are eight slices left, what fraction of the pizza is
remaining?
Answer:
when 12 pieces say =1/12
so 8 pieces say = 1/8
Step-by-step explanation:
Answer:
2/3
Step-by-step explanation:
fraction of remaining part=8/12 (divided with 4)
=2/3
A library had 6422 music CDs stored on 26 shelves.If the same number of CDs were stored on each shelf how many CDs were stored on each shelf
one shell stores CDs:6422/26
246
A shipping company will ship a package for $7.50 when the volume is no more than 15,000 cubic centimeters. Grace needs to ship a package that is 3x-5 centimeters long, 2x centimeters wide, and x + 20 centimeters tall. A.Write a polynomial expression to represent the situation if Grace plan as to spend a maximum of $7.50
B. Write and solve a system of equations
C. What should the dimensions of the package be to have the maximum volume?
The maximum volume is at x = 0.85 cm. The inequality is 6x³ + 110x² - 200x ≤ 15000
What is an inequality?
An inequality is an expression that shows the non equal comparison between two or more numbers and variables.
A shipping company will ship a package for $7.50 when the volume is no more than 15,000 cubic centimeters. Hence, if Grace plan as to spend a maximum of $7.50:
Volume (V) = (3x - 5)(2x)(x + 20) = 6x³ + 110x² - 200x
The inequality is:
6x³ + 110x² - 200x ≤ 15000
6x³ + 110x² - 200x - 15000 ≤ 0
x = 10
The maximum volume is at:
V' = 0
V' = 18x² + 220x - 200 = 0
x = 0.85 cm
The maximum volume is at x = 0.85 cm. The inequality is 6x³ + 110x² - 200x ≤ 15000
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Plsss help mee I’m not sure what the answer is
Answer:
D. would be the correct answer
Hey ! Can anyone help me solve these problems please. Thank you !
[tex] \qquad\begin{gathered} \begin{array}{ ||c|| c} \hline \\ \textsf { \huge{ \underline{lets\: \: solve}}} \\ \\ \: \: \sf6. \: \underline{ find \: measure \: of \: angle \: TSR}\: \: \\ \\ \textsf {As we know, diameter subtends 90° } \\ \textsf{ at an arc of same circle.} \\ \textsf{ hence we can conclude that :} \\ \\ \rm{ \therefore\angle \:TSR = 90 \degree } \\ \\ \\ \sf7. \: \underline{ find \: measure \: of \: angle \: subtended }\\ \: \underline{ \sf by \: arc \: ST \: at \: the \: centre \: of \: circle}\: \\ \\ \\ \textsf{Angle subtended by an arc at centre} \\ \textsf{ of the circle is double the angle} \\ \textsf{ subtended by the same arc on other } \\ \textsf{ part of circle, hence we can} \\ \textsf{conclude that :} \\ \\ \sf\angle SOT =2 \angle SRT \\ \\ \angle \sf SOT =2 \times 41\degree \\ \\ \sf \therefore\angle SOT = 82 \degree \\ \\ \\ \\ \hline \end{array} \end{gathered}[/tex]
What does Opie create with simple geometric shapes and lines?
A. Facial features
B. Eyes and ears
C. The portrait's hair
D. The shape of the bust
Opie created facial features with simple geometric shapes and lines. Option A
Who is Julian Opie?
Julian Opie is a known as a renowned British artist known for his distinctive depictions of figures, portraits, and landscapes.
His earliest works was the painted steel sculptures, domestic appliances, architectural structures and geometrical shapes showing the relationship between visual and spatial observation
He created distinct facial shapes and forms on a flat surface by means of lines, colors and shadings.
Thus, Opie created facial features with simple geometric shapes and lines. Option A
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Subtract 55 from the sum of 234 and 8
A Goodyear Blimp is positioned at a height of 1,500 feet over the M&T
Bank Stadium to broadcast the Baltimore Ravens game. The blimp is
observed by a car of fans as they look upward at an angle of 22º.
Another car closer to the stadium can see the blimp if they look
upward at an angle of 65º.
DEOOV
כשע
Figure not
drawn to scale
1500 ft
65
22
x ft.
Determine x, the distance between the two cars.
The distance between two cars comes to be 4412.06 feet.
Find the attached diagram to visualize it better.
From the attached diagram
The Distance between two cars is x+y
The angle of elevation for the first car = 22 degrees
The angle of elevation for the second car = 65 degrees.
Height of the Goodyear blimp = 1500 feet
What is the tangent of an angle?The tangent of an angle is the ratio of the opposite side to the adjacent side of that angle.
So, tan 22° = 1500/x
So, x= 1500/tan 22°
x = 3712.63 feet
tan 65° = 1500/y
So, y = 1500/tan 65°
y = 699.46 feet
x+y = 4412.06 feet
So, the distance between two cars = 4412.06 feet
Therefore, the distance between two cars comes to be 4412.06 feet.
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if f(-3)=14 what would be the equation for f(x)
Answer:
Step-by-step explanation:
if quadratic
(-3)²+5=9+5=14
so f(x)=x²+5
or
if linear
(-3)+17=14
f(x)=x+17
if cubic
(-3)³-2(-3)²-3(-3)+14=14
f(x)=x³-2x²-3x+14
so on.
Find the value of the expression.
Answer:
0
Step-by-step explanation:
((3/5 ^0)^-2)
3/5^0=0
0^-2= 0
PLEASE ANSWER FAST WILL MARK BRAINLIEST
[tex]\sf y=50\left(1.8\right)^{x}[/tex]
=================================
the first term : 50
crosses point : (0, 50), (1, 90)
solve for y,
[tex]\sf y=50\left(1.8\right)^{1}[/tex]
[tex]\sf y=90[/tex]
Thus D is correct option.Graph:
john had thrice as many stickers as Ken. After Ken had given away 12 stickers to his brother, John had 5 times as many stickers as Ken. How many stickers did John and Ken have in the end?
The number of stickers John and Ken have in the end is 6 and 18 respectively.
How to write and solve equation?let
Ken stickers = xJohn stickers =3xNew stickers
Ken = x - 12John = 5xx + x - 12 = 3x + 5x
2x - 12 = 8x
-12 = 8x - 6x
-12 = -2x
x = 12/2
x = 6
So,
Ken stickers = x
= 6 stickers
John stickers = 3x
= 3 × 6
= 12 stickers
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Find all the factors of 12.
A) 2, 3, 4, 6
B) 2, 3, 4, 6, 12
C) 1, 2, 3, 4, 6, 12
Answer:
1, 2, 3, 4, 6, and 12, because each of those divides 12 without leaving a remainder (or each of those is a counting number that can be multiplied by another counting number to make 12).
Step-by-step explanation:
hope this helps.
b^2 – 4ac if a = 2, b = 9, and c = 4
Answer:
49
Step-by-step explanation:
[tex](b)^{2} - 4(a)(c) \\(9)^{2} - 4(2)(4) \\81 - 4(2)(4) \\81 - 32 \\49[/tex]
Please solve with explanation
Try this option, all the details are in the attachment. Note the equation a) is performed with green, the equation b) with red line.
With break green line is the cureve y= -x²; with break red line is y=2x².
Compared with the graph of the parent function,
which equation shows a vertical stretch by a factor
of 6, a shift of 7 units right, and a reflection over the
X-axis?
Step-by-step explanation:
as I don't see any equations or graphs here I can only answer in general :
f(x) is the patent function.
g(x) is the transformed function.
g(x) = -6f(x-7)
the "-" reflects the curve over the x-axis (what was positive is now negative and vice versa).
the factor 6 stretches the curve up and down by the factor 6 (everything is now 6 times larger or smaller).
and the argument "x-7" relates x (for the new function) with "x-7", meaning that now things are happening at x, that were originally happening at x-7, so they happen now "later" (more to the right on the x-axis) than before - hence a shift to the right.
Answer:
its D
Step-by-step explanation:
took the test :)
mathematics pls help 4 questions will give brain to best answer
Answer:
1)ur first answer is 12
2)Your median is 9
3)9
4) range 7
Step-by-step explanation:
it’s the most common number
The middle numbers are 8 and 10.9 is the number in between them which mean it is the median
4) you subtract the biggest number with the smallest one 12-5=7
Answer:
[1] 12
[2] 9
[3] 9
[4] 5
Step-by-step explanation:
[1] What is the mode for the following set of data?
5,7,8,10,12,12
Answer = 12
Explanation: Mode is what most appears in data set. Thus, we see that 12 is the mode.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[2] What is the median for the following set of data?
5,7,8,10,12,12
Answer = 9
Explanation:
Median is basically what in the middle.
5,7,8,10,12,12
5,7,8,10,12,12
Since there are two number in the middle: Add them up and then divide by 2. 8 + 10 = 18/2 = 9
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[3] What is the mean for the following set of data?
5,7,8,10,12,12
Answer = 9
Explanation:
Mean is add all the number up and then divide by total number in data set
5 + 7 + 8 + 10 + 12 + 12 = 54
Total number in data set : 6
54/6 = 9
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
[4] What is the range for the following set of data?
5,7,8,10,12,12
Answer = 5
Explanation:
Range = IQR = Q3 - Q1
Knowing that;
Quartiles:
Q1 --> 7
Q2 --> 9
Q3 --> 12
Hence, 12 - 7 =5
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Lenvy~