3. Use the definition of the Riemann Sum to evaluate | 12(x3 – 2x)dx.
Answers
The value of the definite integral of 12(\(x^3\) - 2x)dx on the interval [0, 2] is -32/5.
To evaluate the definite integral ∫(1,2) 12(\(x^3\) - 2x) dx using the definition of Riemann Sum, we need to partition the interval [1, 2] into n subintervals of equal width.
Let Δx be the width of each subinterval, then:
Δx = (2 - 1)/n = 1/n
Let xi be the left endpoint of the ith subinterval, then:
xi = 1 + iΔx
The Riemann sum for this function is then:
Rn = ∑[i=1 to n] f(xi)Δx
Substituting the function f(x) = 12(\(x^3\) - 2x) and the expressions for xi and Δx, we get:
Rn = ∑[i=1 to n] 12(1 + iΔx)3 - 2(1 + iΔx)
Simplifying, we get:
Rn = 12/n ∑[i=1 to n] [(1 + iΔx)3 - 2(1 + iΔx)]
Expanding the cube, we get:
Rn = 12/n ∑[i=1 to n] (1 + 3iΔx + 3i2Δx2 + i3Δx3 - 2 - 2iΔx)
Simplifying, we get:
Rn = 12/n ∑[i=1 to n] (3i2Δx2 + 3iΔx + Δx3 - 1)
Now we can evaluate the summation:
∑[i=1 to n] 3i2Δx2 = 3Δx2 ∑[i=1 to n] i2 = 3Δx2 [(n(n+1)(2n+1))/6]
∑[i=1 to n] 3iΔx = 3Δx ∑[i=1 to n] i = 3Δx [(n(n+1))/2]
∑[i=1 to n] Δx3 = nΔx3
∑[i=1 to n] 1 = n
Substituting these back into the Riemann sum expression, we get:
Rn = 12/n [3Δx2(n(n+1)(2n+1))/6 + 3Δx(n(n+1))/2 + nΔx3 - n]
Simplifying and factoring out Δx, we get:
Rn = 12[(n+1)(2n+1)/2n2 - 3/2n + 1/2n3]Δx
Taking the limit as n approaches infinity, we get:
∫(1,2) 12(\(x^3\) - 2x) dx = lim(n→∞) Rn = 12(2+1)(2(2)+1)/(2(2)2) - 3/(2(2)) + 1/(2(2))
Simplifying, we get:
∫(1,2) 12(\(x^3\) - 2x) dx = 12(7/8 - 3/4 + 1/16) = 9
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Related Questions
Sandra’s rectangular garden is 24. 5 feet long, and the ratio of the length to the width is 7 to 4. What is the width of sandra’s garden?.
Answers
Answer:
width = 14 feet
Step-by-step explanation:
the 7 part of the ratio relates to the length of the garden , then
24.5 feet ÷ 7 = 3.5 feet ← value of one part of the ratio , so
width = 4 × 3.5 feet = 14 feet
the maintenance supervisor of an assembly line has two tool cabinets, one at each end of the assembly line. each morning, she walks from one end of the line to the other, and she is equally likely to begin the walk at either end. in the two tool cabinets are a total of six flashlights. at the beginning of her walk, the supervisor takes a flashlight (if one is available) from the tool cabinet at that location, and at the end of her walk, she leaves a flashlight (if she possesses one) from the tool cabinet at that location. model the movement of flashlights using a discrete-time markov chain.
Answers
A Markov chain models flashlight movement in an assembly line with seven states and transition probabilities. Steady-state probabilities are calculated to determine long-run proportions in each state.
Let's denote the state of the system by the number of flashlights in the tool cabinet at the starting end of the assembly line. Since there are two tool cabinets and a total of six flashlights, the state space consists of seven possible states: 0, 1, 2, 3, 4, 5, or 6 flashlights in the tool cabinet at the starting end.
At each step of the Markov chain, the supervisor takes a flashlight from the tool cabinet at the starting end (if one is available), and leaves a flashlight (if she possesses one) in the tool cabinet at the ending end. This means that the Markov chain is time-homogeneous, since the transition probabilities depend only on the current state and not on the time at which the transition occurs.
Let's calculate the transition probabilities between the states. If the supervisor starts at a state with k flashlights in the tool cabinet at the starting end, then there are 6 - k flashlights in the tool cabinet at the ending end. Therefore, the probability of moving to a state with j flashlights in the tool cabinet at the starting end is equal to the probability of taking a flashlight from the starting end (which is k/6 if k > 0) multiplied by the probability of leaving a flashlight at the ending end (which is (6 - k)/6 if j > 0) multiplied by the probability of starting at the ending end (which is 1/2 since the supervisor is equally likely to start at either end). Formally, we have:
P(k, j) = (k/6) * ((6 - k)/6) * (1/2) if j > 0
P(k, 0) = (6 - k)/6 * (1/2) if j = 0
Note that since the supervisor always takes a flashlight from the tool cabinet at the starting end, it is impossible to transition to a state with more flashlights at the starting end than the current state (i.e., P(k, j) = 0 if j > k).
We can represent the transition probabilities between the states using a transition probability matrix, which is a 7x7 matrix where element (i,j) is the probability of transitioning from state i to state j:
| P(0,0) P(0,1) P(0,2) P(0,3) P(0,4) P(0,5) P(0,6) |
| P(1,0) P(1,1) P(1,2) P(1,3) P(1,4) P(1,5) P(1,6) |
| P(2,0) P(2,1) P(2,2) P(2,3) P(2,4) P(2,5) P(2,6) |
| P(3,0) P(3,1) P(3,2) P(3,3) P(3,4) P(3,5) P(3,6) |
| P(4,0) P(4,1) P(4,2) P(4,3) P(4,4) P(4,5) P(4,6) |
| P(5,0) P(5,1) P(5,2) P(5,3) P(5,4) P(5,5) P(5,6) |
| P(6,0) P(6,1) P(6,2) P(6,3) P(6,4) P(6,5) P(6,6) |
We can fill in the entries of this matrix using the transition probabilities we calculated above.
For example, to find P(2,3), we use the formula we derived above, with k=2 and j=3:
P(2,3) = (2/6) * ((6 - 2)/6) * (1/2) = 1/12
Similarly, we can find all the other entries of the matrix.
Once we have the transition probability matrix, we can use it to calculate the steady-state probabilities of each state. These are the probabilities that the system will be in each state in the long run, assuming that the Markov chain has reached a steady state. The can be found by solving the equation:
πP = π
where π is a row vector of the steady-state probabilities and P is the transition probability matrix. Since the sum of the probabilities in any row of P is 1, we also have the normalization condition that the sum of the probabilities in π is 1.
We can solve for π using various methods, such as row reduction or matrix inversion. The steady-state probabilities tell us the long-run proportion of time that the system will spend in each state.
In summary, we can model the movement of flashlights using a discrete-time Markov chain with a state space of seven possible states (corresponding to the number of flashlights in the tool cabinet at the starting end), and transition probabilities that depend on the probabilities of taking and leaving flashlights at each end of the assembly line. We can calculate the steady-state probabilities of each state, which tell us the long-run proportion of time that the system will spend in each state.
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Reasoning with similarity geometry please help. 40 points
Answers
Answer:
ABC and CEF are right triangles | definition of right triangle
ABC and CEF are 30-60-90 triangles | definition of 30-60-90 triangle
ABC is similar to CEF | Angle-Angle-Angle similarity theorem
Solve each system by substitution. Check your answers.
x + 12y = 68 , x=8y-12
Answers
By substitution, the solution of the system of equations, x + 12y = 68 and x = 8y - 12, is (20 , 4).
A system of linear equations is a set of two or more equations which includes common variables. To solve system of equations, we must find the value of the unknown variables used in the equations that must satisfy both equations.
There are three methods that can be used to solve system of linear equations.
1. Elimination
2. Substitution
3. Graphing
Using substitution method, given two linear equations in x and y,
x + 12y = 68 (equation 1)
x = 8y - 12 (equation 2)
Since the second equation is already expressed in x in terms of y, substitute the value of x to the first equation.
x + 12y = 68 (equation 1)
(8y - 12) + 12y = 68
8y - 12 + 12y = 68
Combining all terms containing the variable y on one side and the constants on the other side of the equality, and solving for y.
8y - 12 + 12y = 68
8y + 12y = 68 + 12
20y = 80
y = 4
Substitute the value of y in the second equation and solve for x.
x = 8y - 12 (equation 2)
x = 8(4) - 12
x = 32 - 12
x = 20
Hence, the solution of the given system of equations is (20 , 4).
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fociaral and state governmenta? A. \( \$ 268,10 \) B. \( \$ 14155 \) c. \( \$ 255.54 \) b. \( \$ 11400 \)
Answers
Answer:
Step-by-step explanation:
A. $268,10 B. $14155 c. $255.54 b. $11400. fociaral and state governmenta? ... record payroll tax expense and pay to the federal and state governments :.
1 answer
·
Top answer:
Amount of state and federal unemployment tax that the employer must record payroll tax expense and pay to the federal and state governments : According ...
Missing:255.54
Solve for n: 3+4n+n=2n+15
what is N??
Answers
Answer:
N = 4
Step-by-step explanation:
Please mark brainliest.
Answer:
n=4
Step-by-step explanation:
3+4n+n=2n+15
5n=2n+12
3n=12
n=4
A spinner is spun twice with 4 equal sections colored red, orange, green, and blue. What is the P(spinning two Reds)?
1 over 2
1 over 4
1 over 8
1 over 16
Answers
The probability of the spinner landing on 2 reds is P ( 2 Reds ) = 1/8
Given data ,
To find the probability of spinning two reds, we need to calculate the probability of spinning a red on the first spin and then multiply it by the probability of spinning a red on the second spin.
The probability of spinning a red on the first spin is 1/4 since there is one red section out of four equal sections.
Now , the value of P ( R ) = 1/4
when the spinner is spun twice ,
P ( A ) = P ( R ) P ( R )
P ( A ) = 1/4 ( 1/4 )
On simplifying , we get
P ( A ) = 1/8
Hence , the probability is P ( A ) = 1/8.
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What is the distance from point N to LM in the figure below?
Answers
Answer:
It is 7.5
Step-by-step explanation:
A point is reflected across the x-axis. The new point is located at (4.75, -2.25) Where was the original point located.
Answers
Answer:
(4.75, 2.25)
Step-by-step explanation:
Given the coordinate (x,y). If this coordinate is reflected over the x axis, the resulting coordinate will be (x, -y)
Note that the y coordinate was negated.
Let the original point needed be (x, y)
If the new point is located at (4.75, -2.25)
Since the y coordinate was negated, then;
-y = -2.25
y = 2.25
x = 4,75 (x coordinate remains the same)
Hence the original point is (4.75, 2.25)
5. What is the largest value of x that is not a
solution to -(9x - 4) + 12 + 18x > 79?
Answers
Answer:
7
Step-by-step explanation:
- 9x + 4 + 12 + 18x > 79
9x+16>79
9x>63
x>7
All numbers greater than7 are solution to -(9x - 4) + 12 + 18x > 79.
So, the largest value of x that is not a
solution to -(9x - 4) + 12 + 18x > 79 is 7.
if the probability of a dancing lady accepting an invitation to dance is 0.18, find the expected number of ladies you would have to ask before one accepts.
Answers
If the probability of a dancing lady accepting an invitation to dance is 0.18.The accepted number of ladies asked before is 7.14 .
Probability is a branch of mathematics that quantifies the likelihood of an event occurring or the likelihood of a statement being true. Probability is a number between 0 and 1, with 0 generally indicating impossibility and 1 indicating certainty that an event will occur. A simple example is tossing a fair (unbiased) coin. Since the coin is fair, the two outcomes (heads and tails) are equally likely. The probability of heads is the same as the probability of tails. Since no other outcome is possible, the probability of heads or tails is 1/2 (also written as 0.5 or 50%).
According to the above discussion:
P{x(bar) (0.18)} = 0.07142371
= 7.14%
Therefore, the expected number is 7.14%.
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wsdbfmnsedbznvmsdbn jkcmendz jvmdsbn xzjkv,dmzb cnxrfjklserjdzcfilksdjcfiovk,dsjnifkvsjdxioglmhsrndxjkflgbnesjkdxhbnk
Answers
Answer:
0.149> 0.128
Step-by-step explanation:
Hope this helps!
Translate the triangle by
(-2
-4)
Answers
Translating the triangle using the vector \((^{-2}_{-4})\) means translating 2 units left and 4 units down.
What is transformation?Transformation is the movement of a point from its initial location to a new location. Types of transformations are reflection, translation, rotation and dilation.
Rigid transformation preserves the shape and size of the figure. Reflection, translation, rotation are rigid transformations.
Translating the triangle using the vector \((^{-2}_{-4})\) means translating 2 units left and 4 units down.
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The population of your town is about 30,000. This is about 1/10 the population of your friend’s town about what is the population of your friend’s town?
Answers
The population of your friend's town is 300,000 if the population of your town is about 30,000.
To solve this problem, we can use the fact that if one quantity is a fraction of another quantity, we can multiply or divide the given quantity by that fraction to find the other quantity. In this case, we know that the population of your town is 1/10th of your friend's town's population, so we can multiply your town's population by 10 to get your friend's town's population.
If the population of your town is about 30,000, and it's about 1/10 the population of your friend's town, we can calculate your friend's town's population by multiplying your town's population by 10.
So, the population of your friend's town is
30,000 x 10 = 300,000
Therefore, your friend's town has a population of about 300,000.
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PLEASE help me with this question!!! REALLY URGENT!
Answers
Answer:
The third table is the correct answer
Step-by-step explanation:
Here in this question, we are concerned with determine which of the tables correctly represents what an exponential function is.
An exponential function is a function of the form;
y = x^n
where the independent variable x in this case is raised to a certain exponent so as to give the results on the dependent variable axis (y-axis)
In the table, we can see that we have 2 segments, one that contains digits 1,2 and so on while the other contains purely the powers of 10.
Now, let’s set up an exponential outlook;
y = 10^x
So we have;
1 = 10^0
10 = 10^1
1/10 = 10^-1
1000 = 10^3
1/100 = 10^-2
We can clearly see here that we have an increase in the value of y, depending on the value of the exponent.
However it is only this table that responds to this successive correctness as the other tables in the answer do have a point where they fail.
For example;
10^-2 is not 10 which makes the fourth table wrong
10^4 is not 100 which makes the first table wrong
we have same error on second table too
Can someone help me? Please Thank you.
Answers
Answer:
He can fit 2 more DVD's on the shelf.
And he'll have about 12 mm of extra space left.
Step-by-step explanation:
40×1.4 = 56cm This is the length of space the 40 DVD's will take up.
60-56 = 4cm This is the amount of space left.
4/1.4 = 2.857 the number of DVD's that will fit.
Of course, the decimal represents a partial DVD which is worthless.
Just to be sure, 2×1.4 = 2.8cm OK. But 3×1.4= 4.2cm which is 2mm too much
A manufacturer produces cell phone she estimates that by selling them for x dollars she will be able to sell them for 285 minus 2X each week determine what the maximum revenue will be round your answer to two decimal places
Answers
Answer:
38.30
Step-by-step explanation:
because 230 -6x=0
Chris's family plans to drive 220 miles to their vacation spot. They would like to complete their drive in 4 hours. Find the average speed in miles per hour needed in order to make the trip in the required time.
Answers
Answer:
24.59 m/s
Step-by-step explanation:
Average speed(m/s) = distance travelled(m) / time taken(s)
Distance = 220 miles
220 miles = 354,056 meters
Time taken = 4 hours
4 hours = 14,400 seconds
Average speed = distance travelled / time taken
= 354,056 / 14,400
= 24.59 meter/seconds
= 24.59 m/s
Kevin is 3 years older than Daniel. Two years ago, Kevin was 4 times as old as Daniel. Let k be Kevin's age and let d be Daniel's age. Which system of equations represents this situation?
Answers
Step-by-step explanation:
Let k be Kevin’s age and let d be Daniel’s age.
Given : Kevin is 3 years older than Daniel.
i.e. k=d+3
Kevin's age two years ago= k-2
Two years ago, Kevin was 4 times as old as Daniel.
i.e. k-2=4(d-2)
Hence, the system of equations represents this situation :
k=d+3
k-2=4(d-2)
Write an equation in slope-intercept form of the line that passes through (6, −1) and (3, −7)?
Answers
Answer:
\(y=2x-13\)
Step-by-step explanation:
(6, −1) and (3, −7)
First, you find the slope of the two points.
\(\frac{-7-(-1)}{3-6} = \frac{2}{1} =2\)
Then, you substitute slope and a point into the y = mx + b equation.
\(-1 = 2(6) + b\)
Next, you solve the equation to find b.
\(-1 = 2(6) + b\\-1 = 12 + b\\-13 = b\)
Finally, you have your slope and y-intercept. Now you just write your equation.
\(y=2x-13\)
evaluate det ka if a is an n × n matrix and k is a scalar. justify your answer.'
Answers
Evaluate det(ka) by raising k to the power of n and multiplying the result by det(a).
How to evaluate det(ka)?If we multiply any row (or column) of a matrix by a scalar k, the determinant of the resulting matrix is also multiplied by k.
Specifically, if we denote the determinant of a by det(a), then we have:
\(det(k a) = k^n det(a)\)
where n is the size of the matrix (i.e., n = number of rows = number of columns).
To see why this is true, note that the determinant is a linear function of each row (or column) of the matrix.If we multiply a row (or column) of a by k, then the corresponding entry in the matrix of cofactors (which is used to compute the determinant) is also multiplied by k.So the overall effect on the determinant is to multiply it by k.Now, in the given problem, we are asked to evaluate det(ka) for a given n × n matrix a and scalar k. Using the above formula, we have:\(det(ka) = k^n det(a)\)
Therefore, we can evaluate det(ka) by raising k to the power of n and multiplying the result by det(a).
Note that if k = 0, then det(ka) = 0 for any nonzero matrix a, since any matrix with a row (or column) of zeros has determinant zero.
If k = 0 and a is the zero matrix, then det(ka) = 0 as well.
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just answer 15 I need help on math
Answers
Answer:
15.3
Step-by-step explanation:
Can someone please explain how to do this ?
Answers
Answer:
1) 26
2) 5
3) 20
Step-by-step explanation:
1. 7 x 3 + 5 Do Multiplication before addition
21 + 5
26
2. 8÷4 + 3 Do division before addition
2 + 3
5
3. 2(12-4) + 4 Do what is in the parentheses first
2(8) + 4 Multiply before you add
16 + 4
20
a man of mass 50 kg climbs up stairs each of height 0.2 m in 20 seconds .calculate the power of the man
Answers
Answer:
Given that ,
A man whose mass is 50kg climbs up 30 steps of a stair in 30s
each step is 20 cm high
Height at 30 steps , h=30×0.2=6 m
Change in potential energy , =mgh=50×10×6=3000 J
So, Work done by the man , W=3000 J
Power used , P=
t
W
=
30
3000
=100 W
Step-by-step explanation:
Simplify 4 sqrt 2 + 7 sqrt 2 - 3 sqrt 2
Answers
Hello
4/2 + 7/2 - 3/2
= 11/2 - 3/2
= 8/2
= 4
The correlation coefficient for poor hearing and loud music in a group of people is 0. 67. Analyze the following statement: Poor hearing is caused by listening to loud music. Is this a reasonable conclusion? Yes; everyone who listens to loud music has hearing trouble. Yes; the correlation coefficient is above 0. 5, so that implies causation. No; the data does not suggest causation, and many people who listen to loud music can hear well. No; poor hearing and listening to loud music are completely unrelated.
Answers
The correlation coefficient for poor hearing and loud music in a group of people, No, the statement that poor hearing is caused by listening to loud music is not a reasonable conclusion.
A correlation coefficient measures the strength and direction of the linear relationship between two variables. In this case, the correlation coefficient of 0.67 suggests a moderately positive correlation between poor hearing and loud music in the group of people. However, correlation does not imply causation.
The statement that poor hearing is caused by listening to loud music assumes a causal relationship, implying that everyone who listens to loud music will have hearing trouble. This is an overgeneralization and ignores other factors that may contribute to poor hearing. It is possible for people who listen to loud music to have good hearing, and for people who do not listen to loud music to have poor hearing.
Therefore, the reasonable conclusion is that the data does not suggest causation, and many people who listen to loud music can hear well.
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Write the log equation as an exponential equation. You do not need to solve for x.
Answers
Given a log function log₅ₓ (x) = 7/2. Its exponential equation is (5x)^(7/2) = x or x^(5/2) = 1/125√5
Logarithmic function is an inverse of exponential function. The relation between logarithm and exponent is given by:
exponent form: y = aˣ
logarithmic form: x = logₐ y
Those two equations are equivalent. The constant a is called "base".
Example:
y = 2³ (exponent form)
x = log₂8 (logarithm form)
In the problem, the logarithm function is:
log₅ₓ (x) = 7/2
Hence, the exponential form is:
(5x)^(7/2) = x
5^(7/2) · x^(7/2) = x
x^(5/2) = 1/125√5
Your question is incomplete. Most likely it was:
Write the log equation as an exponential equation. You do not need to solve for x.
log₅ₓ (x) = 7/2
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What is the equation of the perpendicular to 5y-8x=19 passing through the point (-1,-7).
Answers
The equation of perpendicular to the equation 5y-8x=19 passing through the point (-1,-7) is 5x+8y=27.
What is perpendicular?Perpendicular lines are two separate lines that cross each other at a right angle or a 90° angle. Two geometric objects are perpendicular in simple geometry if they intersect at a right angle. The perpendicular symbol, ⟂, can be used to graphically represent the condition of perpendicularity.It can be defined between two planes, two lines, or two planes and another line.
So, calculate as follows:
5x - 8y =19-8y = - 5x+19y = (-5/8)x + 19/8Then,
y = MX + bm = -8/5Now,
-7 = -8/5(-1) + b7 = 8/5 + bb = 7-8/5b=27/5Finally, we obtain:
y = (-5/8)x + 27/525x+40y=1355x+8y=27Therefore, the equation of perpendicular to the equation 5y-8x=19 passing through the point (-1,-7) is 5x+8y=27.
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Can someone help me on this plzzz??
Answers
Answer:
add to elimanate y.
Step-by-step explanation:
Yesterday 170 guests at a hotel called for room service , and 255 guests did not call for room service . What percentage of the guests at this hotel called for room service yesterday ?
Answers
Answer:
40%
Step-by-step explanation:
add both numbers together to get the total amount of guests
170 + 255 = 425
divide the number of guests who called for room service by the total guests
170 ÷ 425 = 0.4
convert the decimal to a percentage
0.4 = 40%
Part : 170
Whole : 255
Percent :
170/255 • x/100
(• is the multiplication sign, And you have to cross multiply 170/255 and x/100.)
17,000/255x is what you’ll get, So now you divide, So your answer is 66.66666667% , BUT write 66.67% (: