Answer:
(1, 4)
General Formulas and Concepts:
<u>Algebra I</u>
- Reading a Cartesian plane
- Coordinates (x, y)
- Solving systems of equations by graphing
Step-by-step explanation:
Where the 2 lines intersect is the solution set to the systems of equations.
Answer:
See Below.
Step-by-step explanation:
We are given the function:
![\displaystyle y=\sqrt{\sin x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%3D%5Csqrt%7B%5Csin%20x%7D)
And we want to show that:
![\displaystyle 4y^3\frac{d^2y}{dx^2}+y^4+1=0](https://tex.z-dn.net/?f=%5Cdisplaystyle%204y%5E3%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%2By%5E4%2B1%3D0)
Find the first derivative of <em>y</em> using the chain rule:
![\displaystyle \frac{dy}{dx} = \frac{1}{2\sqrt{\sin x}}\cdot \cos x = \frac{\cos x}{2\sqrt{\sin x}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B1%7D%7B2%5Csqrt%7B%5Csin%20x%7D%7D%5Ccdot%20%5Ccos%20x%20%3D%20%5Cfrac%7B%5Ccos%20x%7D%7B2%5Csqrt%7B%5Csin%20x%7D%7D)
And find the second derivative using the quotient and chain rules:
![\displaystyle \begin{aligned} \frac{d^2y}{dx^2} &= \frac{1}{2}\left(\frac{(\cos x)'(\sqrt{\sin x})-(\cos x)(\sqrt{\sin x})'}{(\sqrt{\sin x})^2}\right) \\ \\ &=\frac{1}{2}\left(\frac{-\sin x\sqrt{\sin x} - \left(\cos x\right) \left (\dfrac{\cos x}{2\sqrt{\sin x}}\right)}{\sin x}\right) \\ \\ & = \frac{1}{2}\left(\frac{ -\sin x(2\sin x) -\cos x(\cos x) }{\sin x \left(2\sqrt{\sin x}\right) }\right) \\ \\ &= -\frac{1}{2} \left(\frac{2\sin^2 x + \cos^2 x}{2\sin^{{}^{3}\!/\! {}_{2}}x}\right)\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%20%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%26%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cleft%28%5Cfrac%7B%28%5Ccos%20x%29%27%28%5Csqrt%7B%5Csin%20x%7D%29-%28%5Ccos%20x%29%28%5Csqrt%7B%5Csin%20x%7D%29%27%7D%7B%28%5Csqrt%7B%5Csin%20x%7D%29%5E2%7D%5Cright%29%20%5C%5C%20%5C%5C%20%26%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%28%5Cfrac%7B-%5Csin%20x%5Csqrt%7B%5Csin%20x%7D%20-%20%5Cleft%28%5Ccos%20x%5Cright%29%20%5Cleft%20%28%5Cdfrac%7B%5Ccos%20x%7D%7B2%5Csqrt%7B%5Csin%20x%7D%7D%5Cright%29%7D%7B%5Csin%20x%7D%5Cright%29%20%5C%5C%20%5C%5C%20%26%20%3D%20%5Cfrac%7B1%7D%7B2%7D%5Cleft%28%5Cfrac%7B%20-%5Csin%20x%282%5Csin%20x%29%20-%5Ccos%20x%28%5Ccos%20x%29%20%20%7D%7B%5Csin%20x%20%20%5Cleft%282%5Csqrt%7B%5Csin%20x%7D%5Cright%29%20%7D%5Cright%29%20%20%5C%5C%20%5C%5C%20%26%3D%20-%5Cfrac%7B1%7D%7B2%7D%20%5Cleft%28%5Cfrac%7B2%5Csin%5E2%20x%20%2B%20%5Ccos%5E2%20x%7D%7B2%5Csin%5E%7B%7B%7D%5E%7B3%7D%5C%21%2F%5C%21%20%7B%7D_%7B2%7D%7Dx%7D%5Cright%29%5Cend%7Baligned%7D)
Find <em>y</em>³: <em> </em>
![\displaystyle y^3 = \left((\sin x)^{{}^{1}\!/\!{}_{2}}\right) ^3= \sin^{{}^{3}\! / \! {}_{2} }x](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%5E3%20%3D%20%5Cleft%28%28%5Csin%20x%29%5E%7B%7B%7D%5E%7B1%7D%5C%21%2F%5C%21%7B%7D_%7B2%7D%7D%5Cright%29%20%5E3%3D%20%5Csin%5E%7B%7B%7D%5E%7B3%7D%5C%21%20%2F%20%5C%21%20%7B%7D_%7B2%7D%20%7Dx)
And find <em>y</em>⁴:
![\displaystyle y^4 = \left((\sin x)^{{}^{1}\!/\!{}_{2}}\right)^4 = \sin^2 x](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%5E4%20%3D%20%5Cleft%28%28%5Csin%20x%29%5E%7B%7B%7D%5E%7B1%7D%5C%21%2F%5C%21%7B%7D_%7B2%7D%7D%5Cright%29%5E4%20%3D%20%5Csin%5E2%20x)
Substitute:
![\displaystyle 4\left( \sin^{{}^{3}\! / \! {}_{2} }x\right)\left(-\frac{1}{2}\left(\frac{2\sin ^2x + \cos ^2 x}{2\sin^{{}^{3}\!/ \! {}_{2}}x}\right)\right)+\left(\sin ^2 x\right) + 1= 0](https://tex.z-dn.net/?f=%5Cdisplaystyle%204%5Cleft%28%20%5Csin%5E%7B%7B%7D%5E%7B3%7D%5C%21%20%2F%20%5C%21%20%7B%7D_%7B2%7D%20%7Dx%5Cright%29%5Cleft%28-%5Cfrac%7B1%7D%7B2%7D%5Cleft%28%5Cfrac%7B2%5Csin%20%5E2x%20%2B%20%5Ccos%20%5E2%20x%7D%7B2%5Csin%5E%7B%7B%7D%5E%7B3%7D%5C%21%2F%20%5C%21%20%7B%7D_%7B2%7D%7Dx%7D%5Cright%29%5Cright%29%2B%5Cleft%28%5Csin%20%5E2%20x%5Cright%29%20%2B%201%3D%200)
Simplify:
![-\left(2\sin^2 x + \cos^2 x\right) + \sin ^2 x + 1=0](https://tex.z-dn.net/?f=-%5Cleft%282%5Csin%5E2%20x%20%2B%20%5Ccos%5E2%20x%5Cright%29%20%2B%20%5Csin%20%5E2%20x%20%2B%201%3D0)
Distribute:
![-2\sin ^2 x - \cos^2 x + \sin ^2 x + 1=0](https://tex.z-dn.net/?f=-2%5Csin%20%5E2%20x%20-%20%5Ccos%5E2%20x%20%2B%20%5Csin%20%5E2%20x%20%2B%201%3D0)
Simplify:
![-\sin ^2 x - \cos^2 x + 1= 0](https://tex.z-dn.net/?f=-%5Csin%20%5E2%20x%20%20-%20%5Ccos%5E2%20x%20%2B%201%3D%200)
Factor:
![-(\sin ^2 x + \cos^2 x ) + 1=0](https://tex.z-dn.net/?f=-%28%5Csin%20%5E2%20x%20%2B%20%5Ccos%5E2%20x%20%29%20%2B%201%3D0)
Pythagorean Identity:
![-(1)+1=0\stackrel{\checkmark}{=}0](https://tex.z-dn.net/?f=-%281%29%2B1%3D0%5Cstackrel%7B%5Ccheckmark%7D%7B%3D%7D0)
Q.E.D.
Answer:
F = 1
Step-by-step explanation:
3(f-1) + f =
3(1 - 1) + 1 =
3(0) + 1 =
0 + 1 =
1
Therefore F = 1 hope this helped! Please give me a 5 star rating????
40 x ( $84 / 8 ) = ( 40 / 8) x $84 = 5 x $84 = $420
<u><em>Answer:</em></u>
7 1/2=7.5
<u><em>Step-by-step explanation:</em></u>
712=7+12
7+(1÷2)
7+0.5=7.5
Hope This Helps! Have A Nice Night!!