Answers
=======================================================
Explanation:
f(x) is continuous when x >= 1. The only discontinuity for f(x) is when x = 0, but 0 is not part of this interval.
f(x) is positive for any valid x value in the domain since x^6 is always positive. In general, x^n is positive for all x when n is any even number.
f(x) is decreasing. You can see this through a table of values or through a graph. For anything in the form 1/(x^k), it will be a decreasing function because x^k gets larger, so 1/(x^k) gets smaller, when x goes to infinity.
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The conditions to use the integral test have been met. So we have to see if [tex]\displaystyle \int_1^{\infty}f(x)dx[/tex] converges or not.
Let's integrate and find out
[tex]\displaystyle \int \frac{1}{x^6} dx = \int x^{-6} dx\\\\\\ \displaystyle \int \frac{1}{x^6} dx = \frac{1}{1+(-6)}x^{-6+1}+C\\\\\\ \displaystyle \int \frac{1}{x^6} dx = \frac{1}{-5}x^{-5}+C\\\\\\ \displaystyle \int \frac{1}{x^6} dx = -\frac{1}{5}*\frac{1}{x^5}+C\\\\\\ \displaystyle \int \frac{1}{x^6} dx = -\frac{1}{5x^5}+C\\\\[/tex]
So we have
[tex]\displaystyle g(x) = \int f(x) dx\\\\\\\displaystyle g(x) = \int \frac{1}{x^6} dx\\\\\\\displaystyle g(x) = -\frac{1}{5x^5}+C\\\\\\[/tex]
Meaning that,
[tex]\displaystyle \int_{a}^{b} f(x) dx = g(b)-g(a)\\\\\\\displaystyle \int_{a}^{b} \frac{1}{x^6} dx = \left(-\frac{1}{5b^5}+C\right)-\left(-\frac{1}{5a^5}+C\right)\\\\\\\displaystyle \int_{a}^{b} \frac{1}{x^6} dx = -\frac{1}{5b^5}+\frac{1}{5a^5}\\\\\\[/tex]
If we plug in a = 1 and apply the limit as b approaches positive infinity, then the expression [tex]-\frac{1}{5b^5}+\frac{1}{5a^5}[/tex] will turn into [tex]\frac{1}{5}[/tex]
Therefore,
[tex]\displaystyle \int_{1}^{\infty} \frac{1}{x^6} dx = \frac{1}{5}\\\\\\[/tex]
Because this integral converges, this means the series [tex]\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^6}[/tex] also converges as well by the integral test.
Related Questions
Carson needs to lease out a music studio to record his new album. The studio charges an initial studio-use fee plus an hourly fee for each hour in the studio. The total amount of money Carson would have to pay to use the studio for tt hours can be determined using the equation P=50t+100.P=50t+100. What is the yy-intercept of the equation and what is its interpretation in the context of the problem?
Answers
Answer:
The y-intercept of the equation is 100 and represents the initial studio-use fee.
Step-by-step explanation:
In this equation, our t variable (time) is the equivalent of the x-variable on a graph. This is because it is the variable that we 'change' to see its impact on y. We see how the amount of hours affects the price. So our P variable (price) is the equivalent of y on a graph. The y-intercept is where the line crosses the y-axis on a graph. At this point, x=0.
Since P is our y, and t is our x, to find the y-intercept, we simply need to make t = 0.
P = 50(0) + 100
P = 100
Therefore the y-intercept is 100.
In this context, t represents time, so even though the studio has been used for 0 hours, the price is still 100. This is because the 100 represents the initial studio-use fee, and using it for certain amounts of time adds onto the initial fee of $100. The hourly fee is represented by 50t so it costs $50 more for each hour of use.
Hope this helped!
Billy wants to buy a golf cart that costs $12,000. He has already saved $8,500 and will continue to save $100 a week. Write and solve an equation to find the number of weeks, w, he must save until he has enough to purchase the golf cart.
Answers
Answer:
35 weeks
Step-by-step explanation:
12000=8500+100x
8500 from 12000 is 3500.
3500/100 is 35. 35 is x.
to check answer:
(35*100)+8500=
12000
find the slope of the line that passes through points (-18,2) and (-18,17)
Answers
Answer:
there is no slope
Step-by-step explanation:
cause both x variables are the same
Which of the following explains the relationship between angles A and D
Answers
Answer:
I believe they are vertical angles
Step-by-step explanation:
Vertical angles are angles that are across from each other. They are also equal to each other.
What are the complete factors of 2x6 –12x4?
Answers
Answer:
-36 is the correct answer .Step-by-step explanation:
2×6 - 12×4= 12 - 48= -36mark BRAINLIEST
Thank you ☺️☺️
Answer:48-12= 36
Step-by-step explanation:
So 2 x 6= 12 & 12 x4 = 48
Then you subtract 48-12=36
Which of the sums below can be expressed as 7(4 + 9)?
O 28 +9
O 28 +63
07 +63
O 11 + 16
Answers
Answer:
28+63
Step-by-step explanation:
Reason:
First, we need to solve 7(4+9) are also known as 7x(4+9). 4 plus 9 is 13 and multiply by 7 is 91. 28+63 also equals 91.
Hope this helps!
why were Europeans so frightened by the Vikings
Answers
Answer:
Europeans were so afraid of the Vikings because they raided/ took over parts of their land in Western Europe.
Because they raided in all of their continents and settled there, and also destroyed all of their belongings.
(don't put this part but also because they were swedish)
have a good day/night <3
good luck <3
-Dan
Which of these sentences is always true for a parallelogram?
ОА.
All sides are congruent.
O B.
All angles are congruent.
O C.
The diagonals are congruent.
D.
Opposite angles are congruent.
Answers
Answer:
d
Step-by-step explanation:
when you look at a parallelogram, it could be a rectangle, square, or rhombus. so apply each of these statements to all of these shapes to see if they are true.
a. on a rectangle, not all sides are congruent. cant be that one
b. on a rhombus, not all angles are congruent. cant be that one
c. on a rhombus, the diagonals are not the same length. cant be that one
d. on every parallelogram, the opposite angles are congruent because that is what makes opposite sides parallel
hope this helped :)
What fraction of the student finished in less than 20 minutes?
Answers
Answer:
that depends on the number of students
Answer:
how many kids were involved
Many attempts have been made to relate happiness with various factors. One such study relates happiness with age and finds that holding everything else constant, people are least happy when they are in their mid-40s (The Economist, December 16, 2010). Data are collected on a respondent’s age and his/her perception of well-being on a scale from 0 to 100; the data is presented below.
Age Happiness
49 62
51 66
41 67
65 71
84 87
41 60
83 86
18 78
36 59
61 63
15 77
86 90
73 70
32 62
84 93
23 72
52 58
72 73
63 63
30 66
72 78
47 60
88 95
69 72
Required:
a. Calculate the sample correlation coefficient between age and happiness.
b. Interpret the sample correlation coefficient between age and happiness.
c. Draw a scatterplot to evaluate your interpretation of the calculated sample correlation value.
Answers
Answer:
a) 0.572
b) There is moderate positive relationship between age and happiness.
c)
Scatter plot is attached
Step-by-step explanation:
a)
[tex]Sample correlation coefficient=r=\frac{nsumxy-(sumx)(sumy)}{[nsumx^{2}-(sumx)^2][ nsumy^{2}-(sumy)^2] }[/tex]
Age(x) Happiness(y) x² y² xy
49 62 2401 3844 3038
51 66 2601 4356 3366
41 67 1681 4489 2747
65 71 4225 5041 4615
84 87 7056 7569 7308
41 60 1681 3600 2460
83 86 6889 7396 7138
18 78 324 6084 1404
36 59 1296 3481 2124
61 63 3721 3969 3843
15 77 225 5929 1155
86 90 7396 8100 7740
73 70 5329 4900 5110
32 62 1024 3844 1984
84 93 7056 8649 7812
23 72 529 5184 1656
52 58 2704 3364 3016
72 73 5184 5329 5256
63 63 3969 3969 3969
30 66 900 4356 1980
72 78 5184 6084 5616
47 60 2209 3600 2820
88 95 7744 9025 8360
69 72 4761 5184 4968
sumx=1335
sumy=1728
sumx²=86089
sumy²=127346
sumxy=99485
n=24
[tex]Sample correlation coefficient=r=\frac{24(99485)-(1335)(178)}{[24(86089)-(1335)^2][ 24(127346)-(1728)^2] }[/tex]
Sample correlation coefficient=0.572
b)
Sample correlation coefficient=0.572 shows the moderate positive correlation between age and happiness. As the age increases happiness also moderately increases.
c)
The attached scatter plot shows that the as age increases happiness also moderately increases.
What is the domain of this function? (-6.7, 12)(15,5.2)(1,9)(12,9)
Answers
Answer:
9 is the answdr i think sorry if wronf
A square has a perimeter of 4 yd. What is the length of each side?
Answers
add them together to get the perimeter of four
The length of each side of the square will be 1 yard.
What is the perimeter of the square?A square has four equal sides, and four vertices, and is a closed, two-dimensional object. It has parallel sides on either side. A square is also equivalent to a rectangle with equal length and width.
The square's perimeter is four times as long as its side.
For a square with a side length of a unit, we get:
Perimeter of the square = 4a unit
A square has a perimeter of 4 yards. Then the length of each side of the square will be given as,
4 = 4a
a = 1 yard
The length of each side of the square will be 1 yard.
More about the perimeter of the square link is given below.
https://brainly.com/question/11495285
#SPJ2
Hi what is the slope of the line on the graph below 1/5 1/3 3 5
Answers
Answer: B (1/3)
Hope this helps
1. 30% of ___ = 90
2. 20% of ___ = 40
Answers
Answer:
1. 300
2. 200
Hope this helps!
Answer:
30% of 300 = 90
20% of 200 = 40
use a model to divide 5/7 divided by 3 Express the answer in simplest form
Answers
Answer:
5/21
You flip the number 3 which will be "1/3", then you multiply that by 5/7 and you get 5/21
Given expression;
We are to divide 5/7 by 3;
Now;
[tex]\frac{\frac{5}{7} }{3}[/tex]
So;
[tex]\frac{5}{7}[/tex] divided by 3
= [tex]\frac{5}{7} x \frac{1}{3}[/tex]
= [tex]\frac{5}{21}[/tex]
The solution is [tex]\frac{5}{21}[/tex]
50 PONITS This model represents an equation. (a) Write an equation that could be used to solve x. (b) Solve the equation for x. Show your work.
Answers
Answer:
A) [tex]5x-1=-6[/tex]
B) [tex]x=-1[/tex]
Step-by-step explanation:
On the left, we have five (positive) Xs and one -1. So, we can represent the left with the expression [tex]5x-1[/tex].
On the right, we have six -1s. So, we can represent the right with the expression [tex]-6[/tex].
A)
Since the balance is balanced, we know that the two expressions are equal to each other. So, we can write the equation:
[tex]5x-1=-6[/tex]
B)
Now, we can solve the equation for x. Add 1 to both sides:
[tex](5x-1)+1=(-6)+1[/tex]
The left will cancel. Add on the right:
[tex]5x=-5[/tex]
Finally, divide both sides by 5:
[tex](5x)/5=(-5)/5[/tex]
The left side will cancel. So, the value of x is:
[tex]x=-1[/tex]
Answer:
A) 5x-1=-6
B) x=-1
Step-by-step explanation:
Jim is going from town A to town B.town B is located at (-4,5).he makes a stop at his friends house, which is halfway along the trip.what are coordinated of his top?
Answers
Answer:
Well hm It would help a lot if you told us where town A is on the map or showed us a picture. Could you do that?
Step-by-step explanation:
On Monday, 255 students went on a trip to the zoo. All 6
buses were filled and 9 students had to travel in cars. How many students were in each bus ?
Answers
Answer:
41
Step-by-step explanation:
255-9=246
246/6=41 students
In ΔUVW, u = 380 cm, ∠W=89° and ∠U=33°. Find the length of w, to the nearest 10th of a centimeter.
Answers
Answer:
697.6
Delta math
-10.12 + 51 + 3
212 + 4
Select the true statement
Answers
What is the y-intercept of the line shown?
0
1
3
3.5
Answers
Answer:
-1
Step-by-step explanation:
The line travels through -1, which is the y intercept.
Determine the square root for each perfect square.
1. √64
2. √100
3. √81
4. √16
5. \sqrt 4/9
6. \sqrt 1/4
Answers
What is 5(x-2)-14x when x=4
Answers
Answer:
-46
Step-by-step explanation:
5 (x - 2) - 14x. x=4
5 (4 - 2) - 14(4) (distribute)
20 - 10 - 14(4) (multiply)
20 - 10 - 56 (subtract and solve.)
10 - 56 = -46 - solution
hope this helps (:
Find the lengths of side BC.
Give your answer to 3 significant figures.
Answers
Step-by-step explanation:
The perimeter of a rectangle is the sum of all the sides and the width sides are equal, as well as the two length sides.
For this problem that would mean that two sides are equal to BC and two sides are equal to AB.
Since the perimeter (P) is equal to 60 , we can write:
P = 2 * BC + 2 * AB
We can substitute the expressions we are given for AB and BC and 60 for the perimeter:
60 = 2 (2z + 3 ) + 2 (3z + 2)
Using the distributive property and combining like terms, we can find the value of the variable , z:
60 = 4z + 6 + 6z + 4
60 = 10z + 10
Subtract 10 from both sides of the equation:
60 - 10 = 10z + 10 -110
50 = 10z
Divide both sides of the equation by 10:
50/10 = 10z/10
5 = z
Since z = 5, then BC is 2z + 3 or 2*5 + 3
BC = 13
Tell me if that's correct!
i need help in this qustion.each unit costs 14p.how much do 942 units costs?
Answers
Answer:
£131.88 or 13188 pence
Step-by-step explanation:
1 unit = 14p
942 units = 942 x 14
942 units = 13188p
£1 = 100p
13188 ÷ 100 = £131.88
Hope this helps!
An oil factory used 2/3 of a tin of
peanuts to prepare 3 drums of peanut
oil. How many tins of peanuts does the
factory use to fill in each drum?
Answers
Answer:16
Step-by-step explanation:
Classify each of the following attributes as either categorical or numerical. For those that are numerical, determine whether they are discrete or continuous.
a. Brand of computer purchased by a customer
b. State of birth for someone born in the United States
c. Price of a textbook
d. Concentration of a contaminant (micrograms per cubic centimeter) in a water sample
e. Zip code
f. Actual weight of coffee in a 1-pound can
Answers
Answer:
Explained below.
Step-by-step explanation:
Categorical variables are those variables that represents certain characteristic or attribute. Such as gender, eye color, age limit and so on.
Numerical variables are variables that assumes numerical values. Such as height, weight, time and so on.
Numerical variables are of two types: Discrete and Continuous.
Discrete variables are those variables that assumes finite whole number values. Such as number of girls in a house, number of children in the nursery and so on.
Continuous variables are those variables that assumes values from a fixed interval. Such as length of screw, amount of payment for donations and so on.
The attributes are classified as follows:
a. Brand of computer purchased by a customer : Categorical
b. State of birth for someone born in the United States : Categorical
c. Price of a textbook : Numerical and Discrete
d. Concentration of a contaminant (micrograms per cubic centimeter) in a water sample : Numerical and Continuous
e. Zip code : Numerical and Discrete
f. Actual weight of coffee in a 1-pound can: Numerical and Continuous
Jack decides to coast to a stop on his bicycle, so he quits pedaling. The table represents his velocity (in meters per second) as a
function of time (in seconds). Which statement best describes the significance of V(t) having a value of zero?
A)
Jack has coasted for 20 seconds
B)
Jack's bicycle will not be moving.
C)
Jack is pedaling his bicycle again
D)
Jack has traveled at least 200 meters
Answers
Answer:
B if your doing prep
Step-by-step explanation:
math
2/3x -4= -2
A.-3
B.-4
C.-9
D.3
Answers
Answer:
D
Step-by-step explanation:
mark me as brainliest?
Answer:
D. 3
Step-by-step explanation:
if you were trying to find the value of x, heres how i got the answer:
2/3x - 4 = -2
- add 4 to both sides
2/3x = 2
- divide both sides by 2/3
x = 3
A byciclist travels 1 mile in 5 minutes. If m represents minutes what does the expression m/5 represents
Answers
Answer:
It represents 5 miles in 5 minutes
Step-by-step explanation:
Because, if the bycicleist travels 1 mile in 1 minute, in 5 minutes he traveled 5 miles