Find the measure of angle x in the figure below:

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

My grandson thinks he can cook better than any other

Molodets [167]

The answer is (B). Employing a comma to separate two independent phrases connected by the conjunction "and," avoids the mistake of the original.

<h3>What are jumbled words?</h3>

The reason was given for Wrong Answer A: Incorrect coordination is present in Option (A). Where a comma is required, a semicolon is used instead.

The reason was given for the Wrong Answer C: The sentence produced by option (C) is illogical. It illogically implies that the grandson's success in winning blue ribbons is due to his belief that he is the fair's top chef.

Justification for Wrong Answer D: In the case of option (D), the sentence is irrational. The initial portion of the line, "To think... fair," and the latter section of the sentence has no connection to one another ("my grandson . . . to prove it.").

Justification for Wrong Answer E: The sentence produced by option (E) is irrational. It is illogical to claim that the grandson's blue ribbons demonstrate that he was "thinking he can cook better... fair."

To learn more about it jumbled words, visit:

brainly.com/question/16029795

#SPJ4

4 0

1 year ago

For the following questions answer independent, dependent, or both. a) This probability causes you to subtract one from the tota

Elden [556K]

Answer:

(a)Dependent

(b)Both

(c)Independent

Step-by-step explanation:

a) When selections are made without replacement, the second(next) outcome is dependent of the first(previous) outcome. Therefore, we subtract one from the draw on the total.

Dependent

(b)The probability of Event A and Event B is the multiplication of the probability of event A and the probability of event B. Events A and B can either be dependent or independent.

Both

(c)When selections are made with replacement, the next outcome is independent of the previous outcome. Therefore, we say that the two events are independent.

Independent

7 0

3 years ago

Write the coordinates of the vertices after a reflection over the line y = x

Tomtit [17]

Answer:

what ?? I couldn't understand

7 0

3 years ago

Does anyone know the answer plz ?

Natalija [7]

Answer:

x = 5; y = 5

Step-by-step explanation:

One angle is a right angle, so its measure is 90 degrees.

One angle measures 45 degrees, so the unknown angle must also measure 45 degrees.

x and y are opposite two 45-degree angles, so x = y.

In a 45-45-90 triangle, the ratio of the lengths of the sides is

1 : 1 : sqrt(2)

We have

x : y : 5sqrt(2)

x/1 = 5sqrt(2)/sqrt(2)

x = 5

y = 5

5 0

3 years ago