A sprinkler sprays water in a circle. the distance from the sprinkler to the outer edge of the circle is 2.5 m

Aliun [14]

(28) 2x+10+x-4=21
(29) 3x-16+4x-8=60
(30) 2x-8+3x-10=17

6 0

3 years ago

Consider the function f(x)equalscosine left parenthesis x squared right parenthesis. a. Differentiate the Taylor series about 0

dybincka [34]

I suppose you mean

$f(x)=\cos(x^2)$

Recall that

$\cos x=\displaystyle\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}$

which converges everywhere. Then by substitution,

$\cos(x^2)=\displaystyle\sum_{n=0}^\infty(-1)^n\frac{(x^2)^{2n}}{(2n)!}=\sum_{n=0}^\infty(-1)^n\frac{x^{4n}}{(2n)!}$

which also converges everywhere (and we can confirm this via the ratio test, for instance).

a. Differentiating the Taylor series gives

$f'(x)=\displaystyle4\sum_{n=1}^\infty(-1)^n\frac{nx^{4n-1}}{(2n)!}$

(starting at $n=1$ because the summand is 0 when $n=0$ )

b. Naturally, the differentiated series represents

$f'(x)=-2x\sin(x^2)$

To see this, recalling the series for $\sin x$ , we know

$\sin(x^2)=\displaystyle\sum_{n=0}^\infty(-1)^{n-1}\frac{x^{4n+2}}{(2n+1)!}$

Multiplying by $-2x$ gives

$-x\sin(x^2)=\displaystyle2x\sum_{n=0}^\infty(-1)^n\frac{x^{4n}}{(2n+1)!}$

and from here,

$-2x\sin(x^2)=\displaystyle 2x\sum_{n=0}^\infty(-1)^n\frac{2nx^{4n}}{(2n)(2n+1)!}$

$-2x\sin(x^2)=\displaystyle 4x\sum_{n=0}^\infty(-1)^n\frac{nx^{4n}}{(2n)!}=f'(x)$

c. This series also converges everywhere. By the ratio test, the series converges if

$\displaystyle\lim_{n\to\infty}\left|\frac{(-1)^{n+1}\frac{(n+1)x^{4(n+1)}}{(2(n+1))!}}{(-1)^n\frac{nx^{4n}}{(2n)!}}\right|=|x|\lim_{n\to\infty}\frac{\frac{n+1}{(2n+2)!}}{\frac n{(2n)!}}=|x|\lim_{n\to\infty}\frac{n+1}{n(2n+2)(2n+1)}$

The limit is 0, so any choice of $x$ satisfies the convergence condition.

3 0

3 years ago

Which is a simplified form of the expression -6a + 2(2a + 2)?

abruzzese [7]

-2a+4 because you have to distribute

5 0

3 years ago

Solve for<br> m2 = x+69<br> 2<br> 60"<br> Xe

Doss [256]

Answer:

8.8

x

+

1.2

=

6

y

−

2

(

3

y

+

4

)

=

4

y

+

20

−

2

x

=

3

Step-by-step explanation:

5 0

2 years ago

On Tuesday the Soto salad restaurant served 6 1/2 cups of Italian salad dressing. If the restaurant serves 1/4 cup of dressing w

Black_prince [1.1K]

Answer:

26 salads were ordered with Italian dressing

Step-by-step explanation:

Here, we want to know the number of salads ordered with Italian dressing;

To find the number of salads with Italian dressing, we shall need to divide the number of cups of italian salad dressing served by the number of cups of dressing served with each salad

Mathematically, that would be;

6 1/2 divided by 1/4

= 13/2 divided by 1/4

= 13/2 * 4/1 = 26

5 0

3 years ago