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What is 3/2? Do you have a calculator? Divide 3/2 and tell me what you got.

Only the last one. The sample space of the other two is too small I think.

Let F(x) be the real-valued function defined for all real x except for x = 0 and x = 1 and satisfying

the functional equation

\(F(x) + F\left(\frac{x-1}x\right) = 1+x.\)

F(x) + F\left(\frac{x-1}x\right) = 1+x.

Find the F(x) satisfying these conditions.

Write F(x) as a rational function with expanded polynomials in the numerator and denominator.

\(\begin{array}{|lrclcl|} \hline & F(x) + F\left(\frac{x-1}x\right) &=& 1+x \qquad (1) \\\\ \text{Set in (1) }x=\frac{x-1}{x}: & F\left(\frac{x-1}x\right) + F\left(\frac1{1-x}\right) &=& 1+\frac{x-1}{x} \qquad (2) \\\\ \text{Set in (1) }x=\frac1{1-x}: & F\left(\frac1{1-x}\right) + F(x) &=& 1+\frac1{1-x} \qquad (3) \\\\ \hline \\ (1) - (2) + (3): & F(x) + F\left(\frac{x-1}x\right) \\ & - \Big(F\left(\frac{x-1}x\right) + F\left(\frac1{1-x}\right) \Big) \\ & + F\left(\frac1{1-x}\right) + F(x) &=& 1+x -(1+\frac{x-1}{x}) + 1+\frac1{1-x} \\\\ & F(x) + F\left(\frac{x-1}x\right) \\ & -F\left(\frac{x-1}x\right) - F\left(\frac1{1-x}\right) \\ & + F\left(\frac1{1-x}\right) + F(x) &=& 1+x -(1+\frac{x-1}{x}) + 1+\frac1{1-x} \\\\ & 2F(x) &=& 1+x -(1+\frac{x-1}{x}) + 1+\frac1{1-x} \\ & 2F(x) &=& 1+x -1-\frac{x-1}{x} + 1+\frac1{1-x} \\ & 2F(x) &=& 1+x -\frac{x-1}{x} +\frac1{1-x} \\ & 2F(x) &=& 1+x +\frac{1-x}{x} +\frac1{1-x} \\\\ & 2F(x) &=& \dfrac{(1+x)x(1-x)+(1-x)^2+x}{x(1-x)} \\\\ & 2F(x) &=& \dfrac{(1-x^2)x+1-2x+x^2+x}{x(1-x)} \\\\ & 2F(x) &=& \dfrac{(1-x^2)x+1-x+x^2}{x(1-x)} \\\\ & 2F(x) &=& \dfrac{ x-x^3 +1-x+x^2}{x(1-x)} \\\\ & 2F(x) &=& \dfrac{ 1+x^2-x^3}{x(1-x)} \\\\ & F(x) &=& \dfrac{ 1+x^2-x^3}{2x(1-x)} \\\\ & \mathbf{F(x)} & \mathbf{=} & \mathbf{\dfrac{ 1+x^2-x^3}{2x - 2x^2}} \\ \hline \end{array}\)

graph:

Use logarithmic differentation to find the derivative of the function.

\(y=\sqrt{x}{e}^{({x}^{2})}({{x}^{2}+1})^{10}\)

y=\sqrt{x}{e}^{(x^2)}(x^2+1)^{10}

Anyone who knows how to solve for the answer and can write down the steps?

Formula

\(\begin{array}{|rclcl|} \hline \left(~\ln f(x)~ \right)' = \dfrac{f'(x)}{f(x)} \\ \hline \end{array} \)

1. logarithm of both sides

\(\begin{array}{|rcll|} \hline y &=& \sqrt{x}{e}^{(x^2)}(x^2+1)^{10} \quad & | \quad \text{$\ln()$ both sides} \\ \ln(y) &=& \ln(\sqrt{x})+\ln({e}^{(x^2)}) + \ln( (x^2+1)^{10} ) \\ \ln(y) &=& \ln(x^{\frac12})+\ln({e}^{(x^2)}) + \ln( (x^2+1)^{10} ) \quad & | \quad \text{Formula: $\ln(a^b) = b\ln(a) $} \\ \ln(y) &=& \frac12\ln(x)+x^2\ln(e) + 10\ln(x^2+1) \quad & | \quad \text{Formula: $\ln(e) = 1 $} \\ \ln(y) &=& \frac12\ln(x)+x^2 + 10\ln(x^2+1) \\ \hline \end{array}\)

2. derivation of both sides

\(\begin{array}{|rcll|} \hline \ln(y) &=& \frac12\ln(x)+x^2 + 10\ln(x^2+1) \quad & | \quad \text{derivate both sides} \\ \Big(\ln(y)\Big)' &=& \left(\frac12\ln(x) \right)' + \left(x^2 \right)' + \left(10\ln(x^2+1) \right)' \\\\ \dfrac{y'}{y} &=& \frac12\cdot \frac1x + 2x + 10\cdot \frac{2x}{x^2+1} \\\\ \dfrac{y'}{y} &=& \frac{1}{2x} + 2x + \frac{20x}{x^2+1} \quad & | \quad \cdot y \\\\ y' &=& y\cdot \left( \frac{1}{2x} + 2x + \frac{20x}{x^2+1} \right) \quad & | \quad y = \sqrt{x}e^{(x^2)}(x^2+1)^{10} \\\\ y' &=& \sqrt{x}e^{(x^2)}(x^2+1)^{10} \cdot \left( \frac{1}{2x} + 2x + \frac{20x}{x^2+1} \right) \\\\ y' &=& x^{\frac12}e^{(x^2)}(x^2+1)^{10} \cdot \left( \frac{1}{2x} + 2x + \frac{20x}{x^2+1} \right) \\\\ y' &=& \frac{ x^{\frac12}{e}^{(x^2)}(x^2+1)^{10} }{2x} + x^{\frac12}e^{(x^2)}(x^2+1)^{10} \cdot 2x \\ &+& x^{\frac12}e^{(x^2)}(x^2+1)^{10} \cdot \frac{20x}{x^2+1} \\\\ y' &=& \frac{ e^{(x^2)}(x^2+1)^{10} }{2x^{1-\frac12}} + 2{e}^{(x^2)}x^{\frac12+1}(x^2+1)^{10}\\ &+& 20e^{(x^2)} x^{\frac12+1}(x^2+1)^{10} \cdot \frac{1}{(x^2+1)^1} \\\\ y' &=& \frac{ e^{(x^2)}(x^2+1)^{10} }{2x^{\frac12}} + 2{e}^{(x^2)}x^{\frac32}(x^2+1)^{10}\\ &+& 20e^{(x^2)} x^{\frac32}(x^2+1)^{10-1} \\\\ y' &=& \frac{ e^{(x^2)}(x^2+1)^{10} }{2\sqrt{x}} + 2{e}^{(x^2)}x^{\frac32}(x^2+1)^{10}\\ &+& 20e^{(x^2)} x^{\frac32}(x^2+1)^{9} \\ \hline \end{array} \)

\(\begin{array}{rcll} \mathbf{ y' } & \mathbf{=} & \mathbf{ \dfrac{ e^{(x^2)}(x^2+1)^{10} }{2\sqrt{x}} + 2{e}^{(x^2)}x^{\frac32}(x^2+1)^{10} + 20e^{(x^2)} x^{\frac32}(x^2+1)^{9} } \\ \end{array} \)

There 

You could return to them more easily if you were a member as all your posts would sit there fully accessable.

You should just have inserted this with the image link. It is easy to do.  Practice and you will find this out.

Ok thank you, I'll wait for a second response :) 

Total: 2*2=4, 2*4=8, 2*7=14, 2*9=18

4*2=8. 4*4=16, 4*7=28, 4*9=36

7*2= 14, 7*4=28, 7*7=49, 7*9=63

9*2=18, 9*4=36, 9*7=63 9*9=81

2*7 works, 2*9 works, 4*4 works, 7*2, and 9*2 works.

Thus, \(\boxed{5}\) values.

Total 100:

More than 3 broken bones, as shown in the diagram: 10+15=25

Thus, 25/100=1/4=\(\boxed{0.25}\)

Clarity is really bad, can't read.

Is this what you're trying to say? 

35+30s-45t35+30s-45t35+30-s-45?

(a) x+y=25, both are addends that make up the sum.

(b) x=6, y=19

x=10, y=15

x=16, y=9

(c) Graph it on a data line.

Probably, number 3 is distributed, unless if there's something special on that day.

Wait till someone else posts a full solution.

In the triangle with a height of 5.5 inches, the top RHS angle will be:

Sin(Top RHS) =5.5 / 11 =1 / 2

Top RHS angle = arcsin(1/2) = 30 degrees.

In the triangle with height h, the top angle is complementary =90 - 30 = 60 degrees.

So, this triangle is: 30, 60, 90 triangle.

Sin(30) = h / 9.9

1 / 2 = h / 9.9

h =9.9 x 1/2

h =4.95 inches.

x = 368.5 feet

Thanks for the response! I didn't think anybody was going to see my question this late.

I also thought that  x ≠ -1  should be a restriction, but since it is not listed in the restrictions in the answer option I wanted to make sure... 

Use Law of Sines as follows:

Sin(63.4) / 339.4 =Sin(right center) / 90

0.894154 /339.4 =Sin(right center) / 90      cross multiply

80.474 = 339.4 x sin(right center)               divide both sides by 339.4

0.23711 = sin(right center)

Right center angle =arcsin(0.23711)

Right center angle =13.716 degrees

180 - [63.4 + 13.716] =102.884 degrees - the angle at 3rd base.

x / sin(102.884) = 339.4 / sin(63.4)

x / 0.974823 = 339.4 / 0.894154                 cross multiply

0.894154x =330.855                                   divide both sides by 0.894154

x =~370 feet - the length of x

The circle travels outward at a speed of 60 cm/s, which means the radius of the circle is increasing at a rate of 60 cm per second.

radius after  0  sec  =  0

radius after  1  sec  =  60

radius after  2  sec  =  60 + 60  =  60(2)

radius after  3  sec  =  60 + 60 + 60  =  60(3)

radius after  s  sec  =  60s

Or we can say...

radius  =  60s    , where   s  is the number of seconds after the stone hit the water.

Let   s  =  the number of seconds after the stone hit the water   and   a  =  the area of the circle

We know that the equation for the area of a circle is...

a  =  pi (radius)²

                            Substitute  60s  in for radius.

a  =  pi( 60s )²

a  =  3600 pi s²

This equation tells us the area, a , at any  s . But we want to know the rate of change in  a  per change in  s  at any  s . So take  d / ds  of both sides of the equation.

da / ds  =  d / ds [ 3600 pi s² ]

da / ds  =  ( 3600 )( pi )( d/ds s² )

da / ds  =  ( 3600 )( pi )( 2s )

da / ds  =  7200 pi s

To find the rate at which the area is increasing after 1 second, plug in  1 for  s  and solve for  da / ds .

(a)   when  s  =  1 ,          da / ds   =   7200 pi (1)   =   7200 pi     (sq cm per second)

(b)   when  s  =  3 ,          da / ds   =   7200 pi (2)   =   14400 pi

I'll let you do part  (c)  .

Notice that the bigger the number of seconds, the bigger the rate at which the area is increasing.

Pequino and I have answered these questions:

For your convenience, here is the solutions copied and pasted:

1.  (Peqino)

in decimal form im pretty positive it would be .30 or .3

now in percentage form the answer would be 30%

if you want it in fraction form it would be 30/100 or 3/10

2. (Me)

Hey Hev!

There are a total of 4*4 = 16 different outcomes. 

Only the product of:

(2, 6) ; (3, 4) ; (3, 5) ; (4, 3) ; (5, 3) is satisfactory to \(P(12\le{X}\le15)\)

5 out of the 16 outcomes work, 

5/16 is your final answer. 

I hope this helped,

Gavin

Hey Hev!

There are a total of 4*4 = 16 different outcomes. 

Only the product of:

(2, 6) ; (3, 4) ; (3, 5) ; (4, 3) ; (5, 3) is satisfactory to \(P(12\le{X}\le15)\)

5 out of the 16 outcomes work, 

5/16 is your final answer. 

I hope this helped,

Gavin

Hey Hev123!

I’m not familiar with this form of representing data, but I can try. 

If the right side is the amount of outcomes, one can draw this conjecture:

A total of 4 + 12 + 21 + 32 + 12 = 80 trials have been made. 

A total of 32 attempts have been satisfactory to be placed in the 4th row. 

Therefore the probability of another attempt being in the 4th row is:

32/80 = 0.4 = 2/5 = 40%

I hope this helped,

Gavin