An irrational number is one that can’t be expressed as a simple fraction.
For instance, the first few digits of the square root of two is written as 1.414213562373095... The digits keep going and cannot be expressed as a fraction. But think of 0.33333... That can easily be written as one-third. The distinguishing feature is that there’s no pattern in the digits for the square root of two.
The first two options are integer fractions. We rule those out immediately. The square root of four is tempting, but realize that it is just equal to two. We come to π (pi).
Arguably the most famous irrational number is π, which starts off as 3.14159265358979... Here, there is again no pattern and the digits extend forever. This meets our definition of our irrational.
Answer: 2.14 %
Step-by-step explanation:
Given : pH measurements of a chemical solutions have
Mean : ![\mu=6.8](https://tex.z-dn.net/?f=%5Cmu%3D6.8)
Standard deviation : ![\sigma=0.02](https://tex.z-dn.net/?f=%5Csigma%3D0.02)
Let X be the pH reading of a randomly selected customer chemical solution.
We assume pH measurements of this solution have a nearly symmetric/bell-curve distribution (i.e. normal distribution).
The z-score for the normal distribution is given by :-
![z=\dfrac{x-\mu}{\sigma}](https://tex.z-dn.net/?f=z%3D%5Cdfrac%7Bx-%5Cmu%7D%7B%5Csigma%7D)
For x = 6.74
![z=\dfrac{6.74-6.8}{0.02}=-3](https://tex.z-dn.net/?f=z%3D%5Cdfrac%7B6.74-6.8%7D%7B0.02%7D%3D-3)
For x = 6.76
![z=\dfrac{6.76-6.8}{0.02}=-2](https://tex.z-dn.net/?f=z%3D%5Cdfrac%7B6.76-6.8%7D%7B0.02%7D%3D-2)
The p-value =![P(6.74](https://tex.z-dn.net/?f=P%286.74%3Cx%3C6%2C76%29%3DP%28-3%3Cz%3C-2%29)
![P(z](https://tex.z-dn.net/?f=P%28z%3C-2%29-P%28z%3C-3%29%3D0.0227501-%200.0013499%3D0.0214002%5Capprox0.0214)
In percent, ![0.0214\times=2.14\%](https://tex.z-dn.net/?f=0.0214%5Ctimes%3D2.14%5C%25)
Hence, the percent of pH measurements reading below 6.74 OR above 6.76 = 2.14%
Answer:
![\int \frac{3e^x}{e^{2x}+2e^x+1}dx=-\frac{3}{e^x+1}+C](https://tex.z-dn.net/?f=%5Cint%20%5Cfrac%7B3e%5Ex%7D%7Be%5E%7B2x%7D%2B2e%5Ex%2B1%7Ddx%3D-%5Cfrac%7B3%7D%7Be%5Ex%2B1%7D%2BC)
Step-by-step explanation:
To find this integral
you must:
1. Take the constant out:
![3\cdot \int \frac{e^x}{e^{2x}+2e^x+1}dx](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cint%20%5Cfrac%7Be%5Ex%7D%7Be%5E%7B2x%7D%2B2e%5Ex%2B1%7Ddx)
2. Factor ![{e^{2x}+2e^x+1}=(e^{x}+1)^2](https://tex.z-dn.net/?f=%7Be%5E%7B2x%7D%2B2e%5Ex%2B1%7D%3D%28e%5E%7Bx%7D%2B1%29%5E2)
![3\cdot \int \frac{e^x}{\left(e^x+1\right)^2}dx](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cint%20%5Cfrac%7Be%5Ex%7D%7B%5Cleft%28e%5Ex%2B1%5Cright%29%5E2%7Ddx)
3. Apply u-substitution ![u=e^x+1](https://tex.z-dn.net/?f=u%3De%5Ex%2B1)
![3\cdot \int \frac{e^x}{\left(u\right)^2}dx](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cint%20%5Cfrac%7Be%5Ex%7D%7B%5Cleft%28u%5Cright%29%5E2%7Ddx)
![u=e^x+1\\du=e^xdx\\dx=\frac{du}{e^x}](https://tex.z-dn.net/?f=u%3De%5Ex%2B1%5C%5Cdu%3De%5Exdx%5C%5Cdx%3D%5Cfrac%7Bdu%7D%7Be%5Ex%7D)
![3\cdot \int \frac{e^x}{\left(u\right)^2}\frac{du}{e^x} \\\\3\cdot \int \frac{1}{u^2}du](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cint%20%5Cfrac%7Be%5Ex%7D%7B%5Cleft%28u%5Cright%29%5E2%7D%5Cfrac%7Bdu%7D%7Be%5Ex%7D%20%5C%5C%5C%5C3%5Ccdot%20%5Cint%20%5Cfrac%7B1%7D%7Bu%5E2%7Ddu)
![3\cdot \int \:u^{-2}du](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cint%20%5C%3Au%5E%7B-2%7Ddu)
4. Apply the Power Rule ![\int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1](https://tex.z-dn.net/?f=%5Cint%20x%5Eadx%3D%5Cfrac%7Bx%5E%7Ba%2B1%7D%7D%7Ba%2B1%7D%2C%5C%3A%5Cquad%20%5C%3Aa%5Cne%20-1)
![3\cdot \frac{u^{-2+1}}{-2+1}](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cfrac%7Bu%5E%7B-2%2B1%7D%7D%7B-2%2B1%7D)
5. Substitute back ![u=e^x+1](https://tex.z-dn.net/?f=u%3De%5Ex%2B1)
![3\cdot \frac{\left(e^x+1\right)^{-2+1}}{-2+1}=3\cdot -\left(e^x+1\right)^{-1}=-\frac{3}{e^x+1}](https://tex.z-dn.net/?f=3%5Ccdot%20%5Cfrac%7B%5Cleft%28e%5Ex%2B1%5Cright%29%5E%7B-2%2B1%7D%7D%7B-2%2B1%7D%3D3%5Ccdot%20-%5Cleft%28e%5Ex%2B1%5Cright%29%5E%7B-1%7D%3D-%5Cfrac%7B3%7D%7Be%5Ex%2B1%7D)
6. Add a constant to the solution
![-\frac{3}{e^x+1}+C](https://tex.z-dn.net/?f=-%5Cfrac%7B3%7D%7Be%5Ex%2B1%7D%2BC)
Answer:
<em>for the function </em>
Step-by-step explanation:
The function <u>y=x^2-8x-16 </u>has a___ value of ___
- 1) maximum or minimum
- 2) -8,-4,-32,4
- minimum value of -32 - u can just plug the equation into a graphing calculator!