What is the least common multiple of two numbers that have no common factors greater than 1

Let f be defined by the function f(x) = 1/(x^2+9)

riadik2000 [5.3K]

(a)

$\displaystyle\int_3^\infty \frac{\mathrm dx}{x^2+9}=\lim_{b\to\infty}\int_{x=3}^{x=b}\frac{\mathrm dx}{x^2+9}$

Substitute x = 3 tan(t ) and dx = 3 sec²(t ) dt :

$\displaystyle\lim_{b\to\infty}\int_{t=\arctan(1)}^{t=\arctan\left(\frac b3\right)}\frac{3\sec^2(t)}{(3\tan(t))^2+9}\,\mathrm dt=\frac13\lim_{b\to\infty}\int_{t=\arctan(1)}^{t=\arctan\left(\frac b3\right)}\mathrm dt$

$=\displaystyle \frac13 \lim_{b\to\infty}\left(\arctan\left(\frac b3\right)-\arctan(1)\right)=\boxed{\dfrac\pi{12}}$

(b) The series

$\displaystyle \sum_{n=3}^\infty \frac1{n^2+9}$

converges by comparison to the convergent p-series,

$\displaystyle\sum_{n=3}^\infty\frac1{n^2}$

(c) The series

$\displaystyle \sum_{n=1}^\infty \frac{(-1)^n (n^2+9)}{e^n}$

converges absolutely, since

$\displaystyle \sum_{n=1}^\infty \left|\frac{(-1)^n (n^2+9)}{e^n}\right|=\sum_{n=1}^\infty \frac{n^2+9}{e^n} < \sum_{n=1}^\infty \frac{n^2}{e^n} < \sum_{n=1}^\infty \frac1{e^n}=\frac1{e-1}$

That is, ∑ (-1)ⁿ (n ² + 9)/eⁿ converges absolutely because ∑ |(-1)ⁿ (n ² + 9)/eⁿ| = ∑ (n ² + 9)/eⁿ in turn converges by comparison to a geometric series.

5 0

3 years ago

Antonio is a biologist studying life in a pond. He needs to know how deep the water is. He notices a water lily sticking straigh

quester [9]

For this problem we can represent the situation as a rectangle triangle.
x: depth of water.
40: Base. "Antonio pulls the lily to one side, keeping the stem straight, until the blossom touches the water at a spot"
x + 8: Hypotenuse. "He notices water lily sticking straight up from the water, whose blossom is 8 cm above the water's surface." Antonio pulls the lily to one side, keeping the stem straight, until the blossom touches the water at a spot".
By the Pythagorean theorem we have:
x ^ 2 + 40 ^ 2 = (x + 8) ^ 2
Clearing x:
x ^ 2 + 1600 = x ^ 2 + 16x + 64
x ^ 2 - x ^ 2 = 16x + 64 - 1600
0 = 16x -1536
1536 = 16x
1536/16 = x
x = 96
answer:
1) x ^ 2 + 40 ^ 2 = (x + 8) ^ 2
2) the depth of the water is
x = 96

6 0

3 years ago

The first transformation for this composition is [________], and the second transformation is a translation down and to the righ

Marysya12 [62]

Answer:

Step-by-step explanation:

B just took the until test

5 0

2 years ago

Read 2 more answers

Is ab perpendicular to cd ? Explain.

Afina-wow [57]

We know that
if two lines are perpendicular
then
the slopes
m1*m2=-1

step 1
find the slope AB
A (0,2)
B (-3,-3)
m=(y2-y1)/(x2-x1)-----> m=(-3-2)/(-3-0)-----> m=-5/-3----> m1=5/3

step 2
find the slope CD
C (-4,1)
D (0,-2)
m=(y2-y1)/(x2-x1)-----> m=(-2-1)/(0+4)-----> m=--3/4----> m2=-3/4

step 3
multiply mi*m2
(5/3)*(-3/4)-----> -15/12
so
15/12 is not -1
therefore
AB is not perpendicular to CD

4 0

3 years ago

Write the ratio as a percent.

natali 33 [55]

Answer:

62%

Step-by-step explanation:

This one's easy to solve as a percentage because the total number was 100 to begin with. Simply divide the number of employed students by the total amount of students, 62/100, which gets 0.62, or 62%.

5 0

2 years ago