If a car goes from 0 to 60 mph in 8.0 seconds, its acceleration would be 60/8 which is equal to 7.5mph/sec
Answer:
x<6 this is the answer because the circle is hallow (not including) and going left (less than)
![\dfrac{\mathrm dy}{\mathrm dx}=\cos(x+y)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%3D%5Ccos%28x%2By%29)
Let
, so that
:
![\dfrac{\mathrm dv}{\mathrm dx}=\cos v+1](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dv%7D%7B%5Cmathrm%20dx%7D%3D%5Ccos%20v%2B1)
Now the ODE is separable, and we have
![\dfrac{\mathrm dv}{1+\cos v}=\mathrm dx](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dv%7D%7B1%2B%5Ccos%20v%7D%3D%5Cmathrm%20dx)
Integrating both sides gives
![\displaystyle\int\frac{\mathrm dv}{1+\cos v}=\int\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20dv%7D%7B1%2B%5Ccos%20v%7D%3D%5Cint%5Cmathrm%20dx)
For the integral on the left, rewrite the integrand as
![\dfrac1{1+\cos v}\cdot\dfrac{1-\cos v}{1-\cos v}=\dfrac{1-\cos v}{1-\cos^2v}=\csc^2v-\csc v\cot v](https://tex.z-dn.net/?f=%5Cdfrac1%7B1%2B%5Ccos%20v%7D%5Ccdot%5Cdfrac%7B1-%5Ccos%20v%7D%7B1-%5Ccos%20v%7D%3D%5Cdfrac%7B1-%5Ccos%20v%7D%7B1-%5Ccos%5E2v%7D%3D%5Ccsc%5E2v-%5Ccsc%20v%5Ccot%20v)
Then
![\displaystyle\int\frac{\mathrm dv}{1+\cos v}=-\cot v+\csc v+C](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20dv%7D%7B1%2B%5Ccos%20v%7D%3D-%5Ccot%20v%2B%5Ccsc%20v%2BC)
and so
![\csc v-\cot v=x+C](https://tex.z-dn.net/?f=%5Ccsc%20v-%5Ccot%20v%3Dx%2BC)
![\csc(x+y)-\cot(x+y)=x+C](https://tex.z-dn.net/?f=%5Ccsc%28x%2By%29-%5Ccot%28x%2By%29%3Dx%2BC)
Given that
, we find
![\csc\left(0+\dfrac\pi2\right)-\cot\left(0+\dfrac\pi2\right)=0+C\implies C=1](https://tex.z-dn.net/?f=%5Ccsc%5Cleft%280%2B%5Cdfrac%5Cpi2%5Cright%29-%5Ccot%5Cleft%280%2B%5Cdfrac%5Cpi2%5Cright%29%3D0%2BC%5Cimplies%20C%3D1)
so that the particular solution to this IVP is
![\csc(x+y)-\cot(x+y)=x+1](https://tex.z-dn.net/?f=%5Ccsc%28x%2By%29-%5Ccot%28x%2By%29%3Dx%2B1)
<h3>Answer:</h3>
D
<h3>Explanation:</h3>
The graph of |x| opens upward. Your graph is vertically stretched by a factor of 3 and translated downward 1 unit.
Graphs B and D open upward, but only graph D is translated downward (not upward, as in B).
The smallest natural number is 1. The largest natural number does not exists because there will always be one even larger. But if you are asking the number of all natural numbers or the size of natural number set then
is your answer.