3. Express the given integral as the limit of a Riemann sum but do not evaluate:
Answers
Expression of the integral [tex]\int\limits^3_0 {(x^{3}-6x) } \, dx[/tex] as the limit of a Riemann sum without any evaluation will be
Lim(n → ∞) ∑(n = 1 → ∞) [{(27i³/n³) - (18i/n)} * (3i/n)]
As per the question statement, we are provided with an integral [tex]\int\limits^3_0 {(x^{3}-6x) } \, dx[/tex] ,
And we are required to determine the expression of the above mentioned integral as the limit of a Riemann sum without any evaluation.
To start with, we need to know the formula [Δx = {(b - a)/n}]
And here, from our given integral [tex]\int\limits^3_0 {(x^{3}-6x) } \, dx[/tex], we get that, (a = 0) and
(b = 3). Therefore substituting the values of "a" and "b" in the formula to calculate Δx, we get,
[Δx = {(3 - 0)/n} = (3/n)]
Also, [(x[tex]_{i}[/tex]) = {a + (Δx)i} = {0 + (3/n)i) = (3i/n)],
Given, [ρ(x) = (x³ - 6x)], and thus, [ρ(x[tex]_{i}[/tex]) = ρ(3i/n)]
Or, [ρ(x[tex]_{i}[/tex]) = {(3i/n)³ - 6(3i/n)}]
Or, [ρ(x[tex]_{i}[/tex]) = {(27i³/n³) - (18i/n)}]
Then, Lim(n → ∞) ∑(n = 1 → ∞) ρ(x[tex]_{i}[/tex])Δx
= Lim(n → ∞) ∑(n = 1 → ∞) [{(27i³/n³) - (18i/n)} * (3i/n)]
Reimann Sum: In Mathematics, a Riemann sum is a certain kind of approximation method for an integral by a finite sum. Named after renowned German mathematician Bernhard Riemann, one very common application of the Reimann Sum is in approximating the area of functions or lines on a graph, and also the length of curve.To learn more about Integrals and Reimann Sum, click on the link below.
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Related Questions
A machine can fill 5,400 bottles in 3 hours. How many bottles can it fill in 8 hours?
Answers
Answer:
14400
Step-by-step explanation:
The function h is defiend by the following rule. h(x)=5x+4.
Answers
we have the function
h(x)=5x+4.
Create a table
For x=-4
substitute the value of x in the function to obtain h(x)
so
h(-4)=5(-4)+4
h(-4)=-20+4
h(-4)=-16
For x=-3
h(-3)=5(-3)+4
h(-3)=-15+4
h(-3)=-11
For x=1
h(1)=5(1)+4
h(1)=9
For x=2
h(2)=5(2)+4
h(2)=14
For x=5
h(5)=5(5)+4
h(5)=29
The graph of F(x), shown below, resembles the graph of G(X) = x2, but it hasbeen changed somewhat. Which of the following could be the equation ofF(x)?600 = x2FUO = ?O. A. F(x) = 0.272 - 3B. F(x) = -x2 - 3C. F(x) = 2x2 - 3D. F(X) = 32 - 3
Answers
You have G(x) = x².
take into account that G(x) can be considered as an streched of the F(X), moreover, F(x) is a translation of G(x) downward 3 units. If G(x) is an strech of F(x), then, G(x) is multipled by a constant lower than 1.
Then, based on the previous considerations, you have that the form of F(x) is:
F(x) = 0.2x² - 3
The solution process is shown for any equation. Justify each step in the process with the appropriate property. Select the correct answer from each drop down menu.
Answers
Answer:
Explanation:
Here, we want to get the values in the segments
a) Here we would have to open up the brackets
Now, to do this, we are going to use the distributive property
By using the distributive property, we will be able to open up the brackets
Doing this, we can get the values in the brackets
So the answer here is distributive property
b) Here, we have
14 = -y after combining the like terms
The correct answer here is the subtraction property of equality
We simply subtract 3y from both sides of the equation to arrive at this answer
c) -14 = y
We have the multiplication property of equality
The reason for this is that we multiplied both sides by -1 to arrive at this answer
d) y = -14
This is the symmetric property of equality
We have this here because if two values are equal on both sides, we can switch each to the opposite sides and still retain the same equality values
Use Vocabulary in Writing 9. Explain how you can find the product 4 X 2 and the product 8 X 2 Use at least 3 terms from the Word List in your explanation.
Answers
Okay, here we have this:
4 students from a class of 15 are going to be chosen to be on the dance committee. Findthe number of different 4-person committees that can be made.
Answers
Answer:
[tex]C(15,4)=1365\text{ different committees}[/tex]Step-by-step explanation:
This situation can be approached using the formula for combinations:
[tex]\begin{gathered} C(n,r)=\frac{n!}{r!(n-r)!} \\ \text{where,} \\ n=\text{ number of possible items that can be }selected \\ r=\text{ number of items that were selected} \end{gathered}[/tex]Therefore, solve for n=15 and r=4.
[tex]\begin{gathered} C(15,4)=\frac{15!}{4!(15-4)!} \\ C(15,4)=1365\text{ different committees} \end{gathered}[/tex]Math help!!! Only a tutor that can give me answers to 9 and 10!!
Answers
9.
Alternative interior angles are congruent
Therefore;
5x + 42 = 18x -12
collect like term
5x - 18x = -12 - 42
-13 x = -54
Divide both-side of the equation by -13
x=4.15
True or False? The end behaviors of each end of any quadratic function are always inthe same direction.
Answers
In general, given a quadratic function,
[tex]\begin{gathered} f(x)=ax^2+bx+c \\ a,b,c\rightarrow\text{ constants} \end{gathered}[/tex]The end behaviors of each end of the function are given by the limits of f(x) when x approaches +/-infinite.
Therefore,
[tex]\lim_{x\to\infty}f(x)=\lim_{x\to\infty}ax^2=a\lim_{x\to\infty}x^2=a*\infty[/tex]and
[tex]\lim_{x\to-\infty}f(x)=\lim_{x\to-\infty}ax^2=a\lim_{x\to-\infty}x^2=a*\infty[/tex]Thus, the two limits are the same and depend on the sign of a.
Hence, the answer is True, the statement is True.how long will it take for $2700 to grow to $24500 at an interest rate of 2.2% if the interest is compounded quarterly? Round to the nearest hundredth.
Answers
Let n be the number of quarterlies.
Then
[tex]\begin{gathered} 24500=2700(1+0.022)^n \\ \Rightarrow1.022^n=\frac{245}{27} \\ \Rightarrow n=\frac{\log _{10}\frac{245}{27}}{\log _{10}1.022} \end{gathered}[/tex]Hence the number of months = 3n = 304.04 months
and the number of years = n / 4 = 25.34 years
Which fraction is represented by point A on the numb 0 -1 O- 1 4. 4 1 1
Answers
The fraction that represent the point A is -1/4.
Since there are 4 divition in between -1 and 0.
Thus the length of one division is,
[tex]undefined[/tex]The equation 3x + 2y = 120 models the number of passengers who can sit in a train car, where isthe number of adults and y is the number of children. Explain what the 2- and y-intercepts mean.
Answers
Explanation:
Given the equation that models the number of passengers who can sit in a car expressed as 3x + 2y = 120
x is the number of adults
y is the number of children
The x-intercept is the point where y is zero i.e. the number of adults when there is no number of children.
when y = 0
3x + 2(0) = 120
3x = 120
x = 120/3
x = 40
This means that there will be 40 adults if there are no children
The y-intercept is the point where x is zero i.e. the number of children when there is no number of adults.
when x = 0
3(0) + 2y = 120
2y = 120
y = 120/2
y = 60
This means that there will be 60children if there are no adults
write the linear equation that passes through the two given points (2,-2) and (0,-1)
Answers
Given the points:
(x1, y1) ==> (2, -2)
(x2, y2) ==> (0, -1)
To find the linear equation, use the form:
y = mx + b
where m is the slope.
To find the slope, use the formula below:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]Thus, we have the slope as:
[tex]m=\frac{-1-(-2)}{0-2}=\frac{-1+2}{0-2}=\frac{1}{-2}=-\frac{1}{2}[/tex]Input 2 for x, -2 for y, and -1/2 for b to find b.
[tex]\begin{gathered} -2=-\frac{1}{2}(2)+b \\ \\ -2=-1+b \\ \\ -2+1=b \\ \\ -1=b \end{gathered}[/tex]Therefore, the linear equation is:
[tex]y=-\frac{1}{2}x-1[/tex]ANSWER:
[tex]y=-\frac{1}{2}x-1[/tex]please show me how to solve this triangle, thank you!
Answers
Statement Problem: Solve for the missing sides of the triangle;
Solution:
The sum of angles in a triangle is 180degrees. Thus,
[tex]\begin{gathered} \angle A+\angle B+\angle C=180^o \\ \angle B=180^o-\angle A-\angle C \\ \angle B=180^o-42^o-96^o \\ \angle B=42^o \end{gathered}[/tex]Since measure angle A and measure angle B are equal. Thus, the triangle is isosceles and the two sides are equal.
[tex]a=b[/tex]We would apply sine rule to find the missing side a.
[tex]\begin{gathered} \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} \\ \frac{\sin A}{a}=\frac{\sin C}{c} \end{gathered}[/tex][tex]\begin{gathered} \frac{\sin42^o}{a}=\frac{\sin96^o}{12} \\ a=\frac{12\sin42^o}{\sin96^o} \\ a=8.07 \\ a\approx8.1 \end{gathered}[/tex]Thus,
[tex]a=b=8.1[/tex]CORRECT ANSWERS:
[tex]\begin{gathered} a=8.1 \\ b=8.1 \\ m\angle B=42^o \end{gathered}[/tex]name a 2 digit odd number that is composite
Answers
We should know that:
All the odd integers which are not prime are odd composite numbers. Examples of composite odd numbers are 9, 15, 21, 25
Solve the given equation:x = -8y + 9
Answers
We have to solve the equation.
[tex]x=-8y+9[/tex]We have 2 unknowns and one equation, so we can only express one in function of the other.
We already have x in function of y, so we will now express y in function of x:
[tex]\begin{gathered} x=-8y+9 \\ x-9=-8y+9-9 \\ \frac{x-9}{-8}=\frac{-8y}{-8} \\ \\ -\frac{x}{8}+\frac{9}{8}=y \\ \\ y=-\frac{x}{8}+\frac{9}{8} \end{gathered}[/tex]Answer:
y = -x/8 + 9/8
The _________ is a point that is equidistant from all points on the perimeter of the circle.
Answers
The center is a point that is equidistant from all points on the perimeter of the circle, where this distance is the radius.
help me please. using the axis of symmetry find the vertex for the follow quadratic function. f (x)=3x^2-6x+8
Answers
Answer:
[tex]P(1,5)[/tex]
Explanation: Axis of symmetry is a vertical line that makes function symmetrical along either side:
In case of parabla function or:
[tex]y(x)=3x^2-6x+8[/tex]We get axial symmetry where the first derivate is zero, and in fact, that is the x value for vertex:
Therefore:
[tex]\begin{gathered} f^{\prime}(x)=(3x^2-6x+8)^{\prime}=6x-6=0 \\ \therefore\rightarrow \\ x=\frac{6}{6}=1 \end{gathered}[/tex]And the corresponding y-value is:
[tex]f(1)=3(1)^2-6(1)+8=5[/tex]Therefore vertex is at the point:
[tex]P(1,5)[/tex]A company produces 11 times as many rings on shift 1+ shift to if I total of 12,000 rings were produced how many were produced on each shift
Answers
Shift 1 produced 11000 rings and Shift 2 produced 1000 rings.
What does "parent company" mean and how Do Parent Companies Work?
A single firm that owns a majority stake in another company or groups of companies is known as a parent company.
Parent corporations are created through acquisition, merger, spin-off, or carving out of subsidiaries.
A parent company is a business that controls a significant portion of another business, giving it operational authority over that business.
Given :-
Production of Shift 1 = 11 times of Production of Shift 2
total production = 12,000 rings
production of ( Shift 1 + Shift 2 ) = Total production
on solving we get,
Production of Shift 1 = 11,000 rings
Production of Shift 2 = 1,000 rings
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please help me work through this, thank you very much!
Answers
Given
[tex]plane-height=650m[/tex]To Determine: The angle function
Solution
The information can be represented as shown below
From the diagram below
[tex]\begin{gathered} tan\theta=\frac{650}{x} \\ \theta(x)=tan^{-1}(\frac{650}{x}) \end{gathered}[/tex]10. A recipe for banana bread calls for 3 bananas for every 6 cups of
What is the ratio of bananas to sugar?
Answers
1:2
Explanation:
If we divide both terms in the ratio 3:6 by the number 3, it equals 1:2
Have an AWESOME day! :)
please help ………………. …………. ………… i already have the answer for part A but im having trouble with Parts B and C
Answers
In part B we must perform the following operation:
[tex](5a^3+4a^2-3a+2)+(a^3-3a^2+3a-9)[/tex]The key here is to group the terms according to the power of a they have:
[tex](5a^3+4a^2-3a+2)+(a^3-3a^2+3a-9)=(5a^3+a^3)+(4a^2-3a^2)+(-3a+3a)+(2-9)[/tex]Then, we can use the distributive property of the multiplication but in reverse:
[tex]b\cdot a+c\cdot a=(b+c)\cdot a[/tex]If we do this in each of the terms between parenthesis we get:
[tex]\begin{gathered} (5a^3+a^3)+(4a^2-3a^2)+(-3a+3a)+(2-9)= \\ =(5+1)a^3+(4-3)a^2+(-3+3)a-7 \\ (5+1)a^3+(4-3)a^2+(-3+3)a-7=6a^3+a^2-7 \end{gathered}[/tex]Then the answer for part B is:
[tex]6a^3+a^2-7[/tex]In part C we must simplify:
[tex](4y^3-2y+9)-(2y^3-3y^2+4y+7)[/tex]Here is important to remember that a negative sign before a parenthesis means that you have to change the sign of all the terms inside it. Then we have:
[tex](4y^3-2y+9)-(2y^3-3y^2+4y+7)=4y^3-2y+9-2y^3+3y^2-4y-7[/tex]Now we can do the same thing we did in part B, we group the terms according to the powers of y:
[tex]4y^3-2y+9-2y^3+3y^2-4y-7=(4y^3-2y^3)+3y^2+(-2y-4y)+(9-7)[/tex]Then we apply the distributive property in reverse:
[tex]\begin{gathered} (4y^3-2y^3)+3y^2+(-2y-4y)+(9-7)=(4-2)y^3+3y^2+(-2-4)y+2 \\ (4-2)y^3+3y^2+(-2-4)y+2=2y^3+3y^2-6y+2 \end{gathered}[/tex]Then the answer for part C is:
[tex]2y^3+3y^2-6y+2[/tex]There are 4 options on the dessert menu at a restaurant. Bill and Laura like all of the choices equallyeach choose a dessert at random from the menu. What is the probability that Bill will choose apple pLaura will choose strawberry cheesecake for dessert? Express your answer as a decimal. If necessalyour answer to the nearest thousandth.0 0.938O 0.063O 0.25O 0.083
Answers
Solution
If we have 4 options and we want to find that Bill select one option and then Laura a different second option is:
1/2 * 1/2= 1/4= 0.25
Then the best answer is:
0.25
If licorice costs $6.59 a pound, how much would it cost to buy a quarter-pound of licorice?
If a 10-foot piece of electrical tape has 0.037 feet cut from it. What is the new length of tape?
A director replayed 231 of the 1000 scenes filmed for a movie. Write a decimal to represent the part of the movie the director replayed.
If you had half a dollar, three quarters, eight dimes, six nickels, and nine pennies, how much money would you have altogether?
What is the combined thickness of these shims: 0.008, 0.125, 0.15, 0.185, and 0.005 cm?
All the people of a neighborhood pooled together and won the lottery. They won $10,000,000 and each person got a 0.02 share. How much money did each person receive?
Answers
Answer:
1. 1.6475.
2. 9.963.
3. 0.231
4. $1.69
5. 0.473 cm.
6. x = $200,000
Step-by-step explanation:
1.$6.59 ÷ 4 = 1.6475.
2. 10-0.037 = 9.963
3. 231 divided by 1000.
4. $0.50 + $0.80 + $0.30 + $0.09 = $1.69
5. 0.008 + 0.125 + 0.15 + 0.185 + 0.005
6. x =$10,000,000(0.02) where x is the amount of money each person will receive. x=$200,000 (multiply)
5-74.The number of girls at Middle SchoolCyber Summer Camp was six morethan twice the number of boys. Therewere a total of 156 middle schoolstudents at the camp. Use the 5-DProcess to find the number of boysand the number of girls at camp.
Answers
We have the following:
Describe/Draw
The statement tells us that the number of girls was 6 plus twice the number of boys and that in total there are 156 students.
Define
Number of girls: y, y = 6 + 2x
Number of boys: x
Number of students: 154
Do
[tex]\begin{gathered} x+y=156 \\ x+6+2x=156 \\ 3x=156-6 \\ x=\frac{150}{3} \\ x=50 \end{gathered}[/tex]for y:
[tex]y=6+2\cdot50=6+100=106[/tex]Decide
The answer is correct because the sum of both is equal to 156 students
Declare
In total there are 50 boys and 106 girls
Given the function defined in the table below, find the average rate of change, in simplest form, of the function over the interval ≤ x ≤ 9.
Answers
The average rate of change is given by the rate of change of both variables.
"Rate" refers to a division. We want to divide the change of y, Δy, by the change of x, Δx:
Δy/Δx
("Δ" means "change").
We want to analyze the change over the interval 3 ≤ x ≤ 9.
Step 1: change of x (Δx)The change from x = 3 and x = 9 is
Δx = 9 - 3 = 6
Step 2: change of y (Δy)We observe the right column of the table. When x = 3, y = 28 and when x = 9, y = 4.
The change from y = 28 to y = 4 is
Δy = 4 - 28 = -24
Step 3: rate of changeThen, the average rate of change is:
Δy/Δx = -24/6 = -4
Answer: -4
Answer the questions below about the quadratic function.g(x) = 3x² + 12x+8Does the function have a minimum or maximum value?MinimumMaximumWhere does the minimum or maximum value occur?x =0What is the function's minimum or maximum value?
Answers
Plot the function on the graph.
From the graph it can be observed that graph of function opening upwards and it has minimum value at x = -2.
Thus function has minimum value.
The minimum value of the function occurs at x = -2. So mimimum value of function occurs at x = -2.
The value of the function at x = -2 is -4. So function's minimum value is -4.
Solve system of equations using the method of substitution. Identify wether the system represents parallel, coincident, or parallel lines.5x+2y=167.5x+3y=24
Answers
Given
5x+2y=16 ---(1)
7.5x+3y=24 ----(2)
Find
1) value of x and y
2) Type of system
Explanation
From equation (1)
[tex]\begin{gathered} 5x+2y=16 \\ 5x=16-2y \\ x=\frac{16-2y}{5} \end{gathered}[/tex]Putting this value of x in equation 2
[tex]\begin{gathered} 7.5x+3y=24 \\ 7.5(\frac{16-2y}{5})+3y=24 \\ 1.5(16-2y)+3y=24 \\ 24-3y+3y=24 \end{gathered}[/tex]From here we cannot find the values of x and y as 3y and -3y will cancel each other. Hence there is not a particular solution
Checking the type of system
From these equations we get
[tex]\frac{a1}{a2}=\frac{b1}{b2}=\frac{c1}{c2}[/tex]Therefore the lines are coincident to each other
Therefore the lines have infinte solutions
Final Answer
Therefore the lines have infinte solutions
The lines are coincident to each other
Which graph represents the solution of −2x≤4(x−6)?
Answers
Answer:
See attachments.
Step-by-step explanation:
Given inequality:
[tex]-2x\leq 4(x-6)[/tex]
Solve the inequality by first expanding the brackets:
[tex]\implies -2x\leq 4x-24[/tex]
Subtract 4x from both sides:
[tex]\implies -2x-4x\leq 4x-24-4x[/tex]
[tex]\implies -6x\leq -24[/tex]
Divide both sides by -6 (remembering to reverse the inequality sign as we are dividing by a negative number).
[tex]\implies \dfrac{-6x}{-6}\leq \dfrac{-24}{-6}[/tex]
[tex]\implies x\geq 4[/tex]
When graphing inequalities on a coordinate plane:
< or > : dashed line.≤ or ≥ : solid line.< or ≤ : shade under the line.> or ≥ : shade above the line.Therefore, to graph the given inequality on a coordinate plane:
Draw a solid line at x = 4.Shade above the line (i.e. shade to the right of the line).(See attachment 1).
When graphing inequalities on a number line:
< or > : open circle.≤ or ≥ : closed circle.< or ≤ : shade to the left of the circle.> or ≥ : shade to the right of the circle..Therefore, to graph the given inequality on a number line:
Place a closed circle at 4.Shade to the right of the circle.(See attachment 2).
Which is the equivalent of 6 14’ 48’’ written in decimal form Round to the nearest thousandth of a degree A. 6.145 B. 6.367 C. 6.247 D. 6.313
Answers
Answer
Step-by-step explanation
First, we need to convert the 48'' into minutes. Using the conversion factor: 1' = 60'', we get:
[tex]\begin{gathered} 48^{\prime}^{\prime}=48^{\prime}^{\prime}\cdot\frac{1^{\prime}}{60^{\prime}^{\prime}} \\ 48^{\prime\prime}=\frac{48}{60}^{\prime} \\ 48^{\prime}^{\prime}=0.8^{\prime} \end{gathered}[/tex]Then, 14 minutes and 48 seconds are equivalent to 14 + 0.8 = 14.8 minutes. To convert this amount of minutes into degrees we need to use the conversion factor 1° = 60', as follows:
[tex]\begin{gathered} 14.8^{\prime}=14.8^{\prime}\cdot\frac{1\degree}{60^{\prime}^{\prime}} \\ 14.8^{\prime}=\frac{14.8}{60}\degree \\ 14.8^{\prime}=0.247\operatorname{\degree} \end{gathered}[/tex]In consequence, 6° 14’ 48’’ is equivalent to 6 + 0.247 = 6.247°
Melina made a scale drawing of a building.She used a scale in which 0.5 inch represents 1 foot. Which graph represents this relationship?
Answers
From the graph, the y - axis 10 uints while the x - axis is 5 units
The x - axis is labeled inches and its half of the feet
For every half inch on x - axis you have 1 feet
The graph that displays the scale is graph D
The answer is OPTION D
