Answer:
no you will not use a table
The answer is
g(x) = 4x
![g{x} = 4x { \: }^{2} = 8x](https://tex.z-dn.net/?f=g%7Bx%7D%20%3D%204x%20%7B%20%20%5C%3A%20%7D%5E%7B2%7D%20%20%3D%208x)
Answer:
20 games
Step-by-step explanation:
if anya one 5/7 games with a total of 28 games that means 5/7=x/28
5 times 4 = 20
4 times 7=28
so they one 20 games
Divide 72 by 8
Then get the answer 9
≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈
9 times 5 answer= 45 (d)
Answer
m= 1/2
Step-by-step explanation:
Formula for slope = change in y divided by the change in x
Y2 - Y1 = 6 - 7 = -1
X2 - X1 = 2 - 4 = -2
m= -1 divided by -1 = 1/2 or 0.5
I suppose you mean
![f(x)=\cos(x^2)](https://tex.z-dn.net/?f=f%28x%29%3D%5Ccos%28x%5E2%29)
Recall that
![\cos x=\displaystyle\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}](https://tex.z-dn.net/?f=%5Ccos%20x%3D%5Cdisplaystyle%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7Bx%5E%7B2n%7D%7D%7B%282n%29%21%7D)
which converges everywhere. Then by substitution,
![\cos(x^2)=\displaystyle\sum_{n=0}^\infty(-1)^n\frac{(x^2)^{2n}}{(2n)!}=\sum_{n=0}^\infty(-1)^n\frac{x^{4n}}{(2n)!}](https://tex.z-dn.net/?f=%5Ccos%28x%5E2%29%3D%5Cdisplaystyle%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7B%28x%5E2%29%5E%7B2n%7D%7D%7B%282n%29%21%7D%3D%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7Bx%5E%7B4n%7D%7D%7B%282n%29%21%7D)
which also converges everywhere (and we can confirm this via the ratio test, for instance).
a. Differentiating the Taylor series gives
![f'(x)=\displaystyle4\sum_{n=1}^\infty(-1)^n\frac{nx^{4n-1}}{(2n)!}](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cdisplaystyle4%5Csum_%7Bn%3D1%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7Bnx%5E%7B4n-1%7D%7D%7B%282n%29%21%7D)
(starting at
because the summand is 0 when
)
b. Naturally, the differentiated series represents
![f'(x)=-2x\sin(x^2)](https://tex.z-dn.net/?f=f%27%28x%29%3D-2x%5Csin%28x%5E2%29)
To see this, recalling the series for
, we know
![\sin(x^2)=\displaystyle\sum_{n=0}^\infty(-1)^{n-1}\frac{x^{4n+2}}{(2n+1)!}](https://tex.z-dn.net/?f=%5Csin%28x%5E2%29%3D%5Cdisplaystyle%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5E%7Bn-1%7D%5Cfrac%7Bx%5E%7B4n%2B2%7D%7D%7B%282n%2B1%29%21%7D)
Multiplying by
gives
![-x\sin(x^2)=\displaystyle2x\sum_{n=0}^\infty(-1)^n\frac{x^{4n}}{(2n+1)!}](https://tex.z-dn.net/?f=-x%5Csin%28x%5E2%29%3D%5Cdisplaystyle2x%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7Bx%5E%7B4n%7D%7D%7B%282n%2B1%29%21%7D)
and from here,
![-2x\sin(x^2)=\displaystyle 2x\sum_{n=0}^\infty(-1)^n\frac{2nx^{4n}}{(2n)(2n+1)!}](https://tex.z-dn.net/?f=-2x%5Csin%28x%5E2%29%3D%5Cdisplaystyle%202x%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7B2nx%5E%7B4n%7D%7D%7B%282n%29%282n%2B1%29%21%7D)
![-2x\sin(x^2)=\displaystyle 4x\sum_{n=0}^\infty(-1)^n\frac{nx^{4n}}{(2n)!}=f'(x)](https://tex.z-dn.net/?f=-2x%5Csin%28x%5E2%29%3D%5Cdisplaystyle%204x%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7Bnx%5E%7B4n%7D%7D%7B%282n%29%21%7D%3Df%27%28x%29)
c. This series also converges everywhere. By the ratio test, the series converges if
![\displaystyle\lim_{n\to\infty}\left|\frac{(-1)^{n+1}\frac{(n+1)x^{4(n+1)}}{(2(n+1))!}}{(-1)^n\frac{nx^{4n}}{(2n)!}}\right|=|x|\lim_{n\to\infty}\frac{\frac{n+1}{(2n+2)!}}{\frac n{(2n)!}}=|x|\lim_{n\to\infty}\frac{n+1}{n(2n+2)(2n+1)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%7C%5Cfrac%7B%28-1%29%5E%7Bn%2B1%7D%5Cfrac%7B%28n%2B1%29x%5E%7B4%28n%2B1%29%7D%7D%7B%282%28n%2B1%29%29%21%7D%7D%7B%28-1%29%5En%5Cfrac%7Bnx%5E%7B4n%7D%7D%7B%282n%29%21%7D%7D%5Cright%7C%3D%7Cx%7C%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7B%5Cfrac%7Bn%2B1%7D%7B%282n%2B2%29%21%7D%7D%7B%5Cfrac%20n%7B%282n%29%21%7D%7D%3D%7Cx%7C%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7Bn%2B1%7D%7Bn%282n%2B2%29%282n%2B1%29%7D%3C1)
The limit is 0, so any choice of
satisfies the convergence condition.