For a certain continuous function f, the right Riemann sum approximation of\int_{0}^{2} f(x) dx with n subintervals of equal length is\frac{2(n+1)(3n+2)}{n^2}for all n. What is the value of\int_{0}^{2} f(x) dx?
A. 2
B. 6
C. 12
D. 20
Answers
The value of $\int_{0}^{2} f(x) dx$ is $\boxed{\textbf{(B) }6}$.
Let's use the formula for the right Riemann sum approximation:
\begin{align*}
\int_{0}^{2} f(x) dx &= \lim_{n\to\infty} \frac{2}{n} \sum_{i=1}^n f\left(\frac{2i}{n}\right) \\
&= \lim_{n\to\infty} \frac{2}{n} \sum_{i=1}^n f\left(2 - \frac{2(n-i)}{n}\right) \quad \text{(re-indexing the sum)} \\
&= \lim_{n\to\infty} \frac{2}{n} \sum_{i=1}^n f\left(2 - 2\cdot\frac{n-i}{n}\right) \\
&= \lim_{n\to\infty} \frac{2}{n} \sum_{j=0}^{n-1} f\left(2 - 2\cdot\frac{j}{n}\right) \quad \text{(substitute }j=n-i\text{)}\\
&= \int_{0}^{2} f(2-x) dx. \quad \text{(re-write as definite integral)}
\end{align*}
Now we have an expression for the integral in terms of a reverse function $f(2-x)$. Notice that the given formula for the right Riemann sum approximation is also in terms of $n$ and $f\left(\frac{2i}{n}\right)$, which suggests that we might be able to relate this to the integral we want to find. In particular, we can compare the two expressions:
$$\frac{2(n+1)(3n+2)}{n^2} = \frac{2}{n} \sum_{i=1}^n f\left(\frac{2i}{n}\right) = \int_{0}^{2} f(2-x) dx.$$
Taking the limit as $n\to\infty$, we have
$$\lim_{n\to\infty} \frac{2(n+1)(3n+2)}{n^2} = \int_{0}^{2} f(2-x) dx.$$
The left side simplifies as follows:
\begin{align*}
\lim_{n\to\infty} \frac{2(n+1)(3n+2)}{n^2} &= \lim_{n\to\infty} \frac{6+\frac{4}{n}}{1+\frac{1}{n}} \\
&= 6.
\end{align*}
Therefore, the value of $\int_{0}^{2} f(x) dx$ is $\boxed{\textbf{(B) }6}$.
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Related Questions
which of the following is a duplex fiber optic connection
Answers
The LC connector is a duplex fiber-optic connection.
The duplex fiber-optic connection among the options provided is the LC connector. The LC connector is a small form-factor connector commonly used in fiber-optic networks. It features a push-pull design and is used for both single-mode and multimode fibers. The LC connector allows for duplex connections, meaning it supports bidirectional communication over a single fiber by utilizing separate transmit and receive channels. The other options mentioned, RJ-45, ST, and SC connectors, are not specifically designed for duplex fiber-optic connections. RJ-45 is commonly used for Ethernet connections, while ST and SC connectors are more commonly used for simplex (single-direction) fiber-optic connections.
Complete Question:
Which of the following is a duplex fiber-optic connection?
RJ-45
LC
ST
SC
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A floor plan of a house is given at the right. What is the area of the floor if each unit is one foot?
Answers
The coordinates of the vertices of the floor plan are given in the image provided in the question.
Before we can calculate the area, we need to get the lengths of the sides. This we can do by subtracting the coordinates of consecutive vertices. Note that the vertices are in the form (x, y). The x-coordinates are for horizontal lengths, while the y-coordinates are for vertical lengths.
For instance, the length first vertical line to the extreme left of the image can be gotten by subtracting as shown below:
\(13-0=13ft\)and the base of the image can be gotten to be:
\(9-0=9ft\)The image of the shape of the floor plan with the floor lengths is shown below:
To find the area, we can divide the shape into manageable chunks. These are shown below:
Area of A
Using the formula to find the area of a rectangle given to be:
\(A=l\times b\)the area of A is calculated to be:
\(A_A=13\times5=65ft^2\)Area of B
Using the formula for the area of a rectangle as well, we have:
\(A_B=16\times4=64ft^2\)Area of C
This is calculated to be:
\(A_C=8\times5=40ft^2\)Combined Area of the Floor
This is calculated as the sum of all the areas. This is gotten to be:
\(\begin{gathered} A=A_A+A_B+A_C \\ A=65+64+40=169ft^2 \end{gathered}\)The area of the floor is 169 square feet.
Greta mows lawns in the summer to earn money. She can mow 3 lawns in 5 hours. Which graph has a slope that beat represents the number of lawns Greta can now per hour
Answers
The graph that best represents the number of lawns Greta can mow per hour will have slope of 0.6
What is the general equation of a straight line?
The general equation of a straight line is -
y = mx + c
Where -
[m] is the slope of the line.
[c] is the y - intercept of the line.
Given is Greta who can mow 3 lawns in 5 hours.
The slope [m] of a straight line represents the rate of change of variable along the y - axis with respect to the variable along the x - axis. Mathematically -
[m] = dy/dx
or
[m] = Δy/Δx
In easy words, it gives us the unit rate of change of variable [y]. Now,
Greta can mow 3 lawns in 5 hours. We can write -
3 lawns will be mowed in 5 hours
In 1 hour, the number of lawns mowed will be (3/5) or 0.6.
Now, label the x - axis representing the the time taken in hours and the y - axis representing lawns mowed . The slope of this line will be 0.6 . Refer to the graph attached for graphical interpretation.
Therefore, the graph that best represents the number of lawns Greta can mow per hour will have slope of 0.6
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When Zak first planted his flower garden, there were only 5 weeds. Every week, the number of weeds doubles. Assuming he never removes the weeds, when will there first
be 3000 weeds in his garden? (Round your answer to the nearest week)
There will be 3,000 weeds in Zak's garden after
***
A weeks.
Answers
I need help please and thank you
(show work please)
Answers
Answer:
1) 22
2)25
3) 28
Step-by-step explanation:
the equation is y = 3x + 10
input x = 4 in the equation
y = 3(4) + 10 which means y = (3 × 4) + 10 = 22
input x = 5 in the equation
y = 3(5) + 10 which means y = (3 × 5) + 10 = 25
input x = 6 in the equation
y = 3(6) + 10 which means y = (3 × 6) + 10 = 28
Help me please I’ll give brainliest if your correct
Answers
To find the selling price that will yield the maximum profit, we need to find the vertex of the quadratic function given by the profit equation y = -5x² + 286x - 2275.The x-coordinate of the vertex can be found using the formula:
x = -b/2a
where a = -5 and b = 286.
x = -b/2a
x = -286/(2(-5))
x = 28.6
So, the selling price that will yield the maximum profit is $28.60 (rounded to the nearest cent).
Therefore, the widgets should be sold for $28.60 to maximize the company's profit.
Hope I helped ya...
Answer:
29 cents
Step-by-step explanation:
The amount of profit, y, made by the company selling widgets, is related to the selling price of each widget, x, by the given equation:
\(y=-5x^2+286x-2275\)
The maximum profit is the y-value of the vertex of the given quadratic equation. Therefore, to find the price of the widgets that maximises profit, we need to find the x-value of the vertex.
The formula to find the x-value of the vertex of a quadratic equation in the form y = ax² + bx + c is:
\(\boxed{x_{\sf vertex}=\dfrac{-b}{2a}}\)
For the given equation, a = -5 and b = 286.
Substitute these into the formula:
\(\implies x_{\sf vertex}=\dfrac{-286}{2(-5)}\)
\(\implies x_{\sf vertex}=\dfrac{-286}{-10}\)
\(\implies x_{\sf vertex}=\dfrac{286}{10}\)
\(\implies x_{\sf vertex}=28.6\)
Assuming the value of x is in cents, the widget should be sold for 29 cents (to the nearest cent) to maximise profit.
Note: The question does not stipulate if the value of x is in cents or dollars. If the value of x is in dollars, the price of the widget should be $28.60 to the nearest cent.
Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. (Enter your answers as comma-separated lists.)P(x) = x^3 − x^2 − x − 5number of positive zeros possible number of negative zeros possible number of real zeros possible
Answers
The number of positive and negative real roots of the function
x³ - x² - x - 5 are 1 and 2 respectively.
What is Descartes' Rule?
Descartes' rule of sign is used to determine the number of real zeros of a polynomial function. It tells us that the number of positive real zeros in a polynomial function f(x) is the same or less than by an even numbers as the number of changes in the sign of the coefficients.
Number of positive real roots of f(x) ≤ Number of sign changes f(x)
Number of negative roots of f(x) ≤ Number of sign changes of f(-x)
According to the given questions:
The given polynomial function f(x) = x³ - x² - x - 5
We can observe the number of sign changes in the coefficients
The sign changes only once
hence there is one positive real root
Now, f(-x) = -x³ + x² - x + 5
Clearly sign changes 2 times evenly
Therefore the function has 2 negative real roots
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Which of the following are partial products for 43x17 choose all that apply
Answers
Answer:
10 multiply 40, 10 multiply 3, 7 multiply 40, and 7 multiply 3.
Step-by-step explanation:
hope it helps!
A line segment has more then one midpoint true or false
Answers
Answer:
false
Step-by-step explanation:
any line segment or distance has only one midpoint.
there is only one middle to everything.
how would that look like, if something had 2 middles ?
then they would not be a middle, and the middle between these 2 middles would then be the real middle ...
Answer:false
Step-by-step explanation: If a line segments has endpoints A and B, then the midpoint of that line segment is a point, M Such that M. Becuase A line segment can only IM SAYING ONLY 1 MIDPOINT!
Jane's curtain company is making curtains for a local hotel. She wants each curtain to be 24 inches long. At the fabric store, fabric is sold by the foot. If Jane's curtain company wants to make 48 curtains, how many feet of fabric will she need? 1 inch = 1 12 foot
Answers
Answer:
\(96\ \text{inches}\)
Step-by-step explanation:
Fabric used for one curtain = 24 inches = \(\dfrac{24}{12}=2\ \text{feet}\)
Number of curtains Jane's company wants to make is 48.
So, total feet of fabric used by Jane's company will be the product of the fabric used for one curtain and the number of curtains.
Total fabric needed = \(48\times 2=96\ \text{inches}\)
Hence, Jane will need \(96\ \text{inches}\) of fabric.
Determine the unique solution of the following differential equation by using
Laplace transforms: y ′′+4y=3H(t−4) The initial values of the equation are ( y(0)=1) and y′(0)=0
Answers
Since 3 is not equal to 0, there is no value for B(0) that satisfies the equation. Therefore, the partial fraction decomposition cannot be applied in this case.
The differential equation y'' + 4y = 3H(t-4) represents a forced harmonic oscillator. The general solution to the homogeneous equation y'' + 4y = 0
To find the unique solution of the given differential equation y'' + 4y = 3H(t-4) using Laplace transforms, we'll apply the Laplace transform to both sides of the equation. We'll also use the initial values y(0) = 1 and y'(0) = 0.
Let's denote the Laplace transform of y(t) as Y(s) and the Laplace transform of y'(t) as Y'(s).
Taking the Laplace transform of both sides of the equation, we have:
s^2Y(s) - sy(0) - y'(0) + 4Y(s) = 3e^(-4s)/s
Substituting the initial values y(0) = 1 and
y'(0) = 0, we get:
s^2Y(s) - s(1) - 0 + 4Y(s) = 3e^(-4s)/s
Simplifying, we have:
s^2Y(s) - s + 4Y(s) = 3e^(-4s)/s
Now, let's solve for Y(s):
(s^2 + 4)Y(s) = s + 3e^(-4s)/s
Dividing both sides by (s^2 + 4), we get:
Y(s) = (s + 3e^(-4s)/s) / (s^2 + 4)
Now, we need to find the inverse Laplace transform of Y(s) to obtain the solution y(t). To do this, we'll use partial fraction decomposition.
Let's decompose Y(s) into partial fractions:
Y(s) = A(s) + B(s)
(s + 3e^(-4s)/s) / (s^2 + 4) = A(s) + B(s)
Multiplying both sides by (s^2 + 4), we get:
s + 3e^(-4s)/s = A(s)(s^2 + 4) + B(s)
Now, we can find the values of A(s) and B(s) by comparing the coefficients of like terms on both sides.
From the equation, we have:
s = A(s)(s^2 + 4) + B(s)
Setting s = 0, we find:
0 = A(0)(0^2 + 4) + B(0)
0 = 4A(0)
Since A(0) = 0, we can disregard the first term A(s)(s^2 + 4) in the equation.
Now, we have:
s + 3e^(-4s)/s = B(s)
To find B(s), we can multiply both sides by s and take the limit as s approaches 0:
s^2 + 3e^(-4s) = B(s)s
Taking the limit as s approaches 0:
0 + 3e^(0) = B(0)(0)
3 = 0
Since 3 is not equal to 0, there is no value for B(0) that satisfies the equation.
Therefore, the partial fraction decomposition cannot be applied in this case.
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a color print 21 pages in 7 minutes How many minutes does it take per page?
Answers
Answer:
3
Step-by-step explanation:
Cause you would divide 21 by 7 and 7 can only go into 21 3 times
\(\huge \dag \sf{Answer \: it}\)
\(solve \: for \: x \\ \\ {2}^{ {x}^{2} } 2 \frac{3x \sqrt{3} }{2} = 8 \)
Don't spam and Don't copy.
Need answer with explanation
Don't answer if you don't know
Best of luck guys
Answers
Answer:
Answer: x = √3 and 8/(3√3)
Step-by-step explanation:
\({ \rm{({2}^{ {x}^{2} } )(2 \frac{3x \sqrt{3} }{2}) = 8}}\)
• For the first bracket:
\({ \rm{ {2}^{ {x}^{2} } = 8}} \\ \\ { \rm{ {2}^{ {x}^{2} } = {2}^{3} }} \\ \\ { \rm{ {x}^{2} = 3}} \\ \\ { \rm{x = \pm \sqrt{3} }}\)
• For the second bracket;
\({ \rm{ 2 \frac{3x \sqrt{3} }{2} = 8}} \\ \\ { \rm{3x \sqrt{3} = 8 }} \\ \\ { \rm{3x = \frac{8}{ \sqrt{3} } }} \\ \\ { \rm{x = \frac{8}{3 \sqrt{3} } }}\)
Solve the inequality.
9-7.25x ≤-20
Answers
The inequality is solved to give the value of x as x≤ 4
What are inequalities?
Inequalities are defined as those expressions with unequal comparison of numbers, expressions, variables or even terms.
It is important to note the different signs of inequalities. They are;
< represents less than> represents greater than≥ represents greater than or equal to≤ represents less than or equal toFrom the information given, we have that;
9-7.25x ≤-20
Now, collect the like terms, we have;
-7. 25 ≤ - 20- 9
subtract the values given, we get;
-7.25x ≤ -29
Divide both sides by the coefficient of x, we get;
x ≤ - 29/7.25
Divide the values, we get;
x ≤ 4
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The average price of 1 acre of land in Berrien County is currently $6236 and is increasing at an annual rate of 3.2%. Which function represents the estimated average price of 1 acre of land in Berrien County after x years?
Answers
The function that represents the estimated average price of 1 acre of land in Berrien County after x years can be written as:
\($$P(x) = 6236(1 + 0.032)^x$$\)
What is average?
In mathematics, the average, also known as the mean, is a measure of central tendency that represents the sum of a set of values divided by the number of values in the set.
The current price of 1 acre of land in Berrien County is $6236. After one year, the price of land will increase by 3.2%, which is equivalent to multiplying the current price by 1.032. After two years, the price will increase by another 3.2%, which is equivalent to multiplying the price after one year by 1.032, and so on.
Therefore, the function that represents the estimated average price of 1 acre of land in Berrien County after x years can be written as:
\($$P(x) = 6236(1 + 0.032)^x$$\)
where \($P(x)$\) is the estimated average price of 1 acre of land in Berrien County after \($x$\) years.
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What’s 50 to the nearest 10?!
Answers
50.
Rounding 50 to the nearest 10 is like doing nothing. 50 is already in the spot.
Hope this helps :)
Answer:
10
Step-by-step explanation:
i think so it may helps you
A large rectangular prism is 5 feet long, 3 feet wide, and 4 feet tall. A small rectangular prism is 2.5 feet long, 1.5 feet wide, and 2 feet tall.
How many small prisms would it take to fill the large prism?
Write your answer as a whole number or decimal. Do not round.
Answers
Answer:
2 small prism could fit.
Step-by-step explanation:
the large prism is 4 feet tall, 5 feet long, And 3 feet wide. So it would take to small prism to fit because the small prism is 2 feet tall, 2.5 feet long, And 1.5 feet long. Do you see it the small prism is exactly half the size of the large prism there the large prism could fit two small prism with in its self.
On a sine function y = Asin(Bx) +C , the maximum value is 19 and the minimum value is -5. Which of the following must be the value of |A| - C ?
Answers
Answer:
|A| - C = 5
Step-by-step explanation:
The maximum value that a sine function can have is 1, and the minimum value is -1.
With this information, to find the minimum value of y, we can use sin(Bx) = -1, and to find the maximum value, we can use sin(Bx) = 1. Then, we have that:
19 = A + C
-5 = -A + C
Summing both equations, we have:
14 = 2C
C = 7
Now, to find the value of A, we can use the first equation:
19 = A + C
19 = A + 7
A = 12
So the value of |A| - C is:
|12| - 7 = 12 - 7 = 5
Which multiplication problem is represented by the model?
Answers
Answer = 6 x 2
This is determined by the number of columns and rows
Rows = 2
Columns = 6
After multiplying both numbers that number of squares is 12
plz help will mark brainliest
Answers
Answer:
50.4
Step-by-step explanation:
You are driving your car to school and all of sudden the gauges go crazy and your car starts sputtering and smoking. You have to pull off on the side of the road.It will cost $48 to get a tow truck to pick you up. He then will charge you $0.51 per mile. If your total bill was $64.83, how many miles did you have to be towed?
Answers
What is the domain of the function need help please
Answers
An isosceles triangle in which the two equal sides, labeled a, are longer than the base, labeled b.
This isosceles triangle has two sides of equal length, a, that are longer than the length of the base, b. The perimeter of the triangle is 15.7 centimeters. The equation can be used to find the side lengths.
If one of the longer sides is 6.3 centimeters, what is the length of the base?
cm
Answers
If one of the longer sides of the Isosceles triangle is 6.3 centimeters, the length of the base is 3.1 centimeters.
Let's solve the problem step by step:
1. Identify the given information:
- The triangle is isosceles, meaning it has two equal sides.
- The two equal sides, labeled "a," are longer than the base, labeled "b."
- The perimeter of the triangle is 15.7 centimeters.
- One of the longer sides is 6.3 centimeters.
2. Set up the equation based on the given information:
Since the triangle is isosceles, the sum of the lengths of the two equal sides is twice the length of the base. Therefore, we can write the equation:
2a + b = 15.7
3. Substitute the known value into the equation:
One of the longer sides is given as 6.3 centimeters, so we can substitute it into the equation:
2(6.3) + b = 15.7
4. Simplify and solve the equation:
12.6 + b = 15.7
Subtract 12.6 from both sides:
b = 15.7 - 12.6
b = 3.1
5. Interpret the result:
The length of the base, labeled "b," is found to be 3.1 centimeters.
Therefore, if one of the longer sides of the isosceles triangle is 6.3 centimeters, the length of the base is 3.1 centimeters.
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a box contains six red light bulbs, nine blue light bulbs, and five green light bulbs. what is the probability of randomly selecting a red light bulb? (round to 2 decimal places) what is the probability of randomly selecting a blue light bulb? (round to 2 decimal places) what is the probability of randomly selecting a red light bulb or a blue light bulb? (round to 2 decimal places) what is the probability of randomly selecting a green light bulb? (round to 2 decimal places) what is the probability of randomly selecting a blue or a green light bulb? (round to 2 decimal places)
Answers
Using it's definition, the probabilities are given as follows:
Red: 0.3 = 30%.Blue: 0.45 = 45%.Red or Blue: 0.75 = 75%.Green: 0.25 = 25%.Blue or Green: 0.6 = 60%.What is a probability?The probability of an event in an experiment is calculated as the number of desired outcomes in the context of the experiment divided by the number of total outcomes in the context of the experiment.
The total number of bulbs in this problem is given as follows:
6 + 9 + 5 = 20.
Six of them are red, hence the probability of a red bulb is:
p = 6/20 = 0.3 = 30%.
Nine of them are blue, hence the probability of a blue bulb is:
p = 9/20 = 0.45 = 45%.
Five of them are green, hence the probability of a green bulb is:
p = 5/20 = 0.25 = 25%.
The probability of a red or blue bulb is:
p = 0.3 + 0.45 = 0.75.
The probability of a blue or green bulb is:
p = 0.45 + 0.25 = 0.6.
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If y varies jointly as x and z and y = 4 when x = 2 and z = 3, what is y when x = -6 and z = 2?
Answers
Answer:
Step-by-step explanation:
Y = kxz
Putting values of y x and z in the equation
4 = k(2)(3)
4 = k6
2/3 = k
Now finding y when x = - 6 and z = 2
Y = kxz
= 2/3(-6)(2)
= - 8
A 12-inch diameter pizza has 22 calories per square inch.
About how many calories are in the entire pizza?
Use 3.14 for pi.
Select from the drop-down menu to correctly complete the statement.
There are about
Choose...
calories in the pizza. optains : 264, 414, 2487, 9948
Answers
Answer:
2486.88
Step-by-step explanation:
Find the area of the pizza. A=πr²
the radius is half the diameter so here it's 6.
A= π6²
6²=36
36 × 3.14= 113.04
the pizza is 113.04 in². multiply that by 22.
Answer:
2487
Step-by-step explanation:
This is correct for k12 people
Sports science researchers determined that, of those people who skateboard, 22% have never had an injury, 45% have had one injury, 18% have had two injuries, and 15% have had three or more injuries. What is the probability that a randomly chosen skateboarder has not had three or more injuries?
0.15
0.25
0.50
0.85
Answers
Answer:
Step-by-step explanation:
0.63 hope this helps
The required value of probability can be found as 0.85. The correct option is (D).
What is probability?Probability is the branch of Mathematics that deals with the measurement of the chance of occurrence of a random event.
The probability of any event always lie in the close interval of 0 and 1 [0,1].
The percentage of people having no, one, two and three or more injuries are given as 22%, 45%, 18% and 15% respectively.
Then, the probability of an event is equivalent to the percent value.
Now, the probability of skateboarders not having 3 or more injuries is given as,
⇒ 1 - P(Three or more injury)
⇒ 1 - 15/100
⇒ 0.85
Hence, the probability can be obtained for not having 3 or more injuries as 0.85.
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Write the next thee numbers in each number pattern. Describe the pattern.
a) 2, 4, 3, 6, 5, 10, 9, 16, …
b) 7, 5, 8, 6, 9, 7, 10, …
Answers
Answer:
a) 2, 4, 3, 6, 5, 10, 9, 16, ...
Step-by-step explanation:
Find the indicated term of the arithmetic sequence with the given description. The first term is 3550 , and the common difference is −17. Which term of the sequence is 2734? n=
Answers
The 49th term of the given arithmetic sequence with the first term of 3550 and the common difference of -17 is equal to 2734.
Given the first term, a1 = 3550
The common difference, d = -17
The formula to find the nth term of an arithmetic sequence is given by,
an = a1 + (n - 1)d
Where, n - the required nth term
an - nth term of the sequence
a1 - first term of the sequence
d - common difference of the sequence
To find the nth term of the sequence that is equal to 2734, we have to plug in the given values in the above formula as follows;
2734 = 3550 + (n - 1) (-17)
2734 - 3550 = -17(n - 1)
-816 = -17(n - 1)
⇒ -816 / (-17) = n - 1
⇒ 48 = n - 1
⇒ n = 49
Therefore, the 49th term of the arithmetic sequence is equal to 2734.
The 49th term is 2734.
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Jordan is running 5 miles for cross-country practice (track). She ran the first 2 miles in 12.5 minutes. How long will it take her to run all 5 miles at this speed?
Answers
Answer:
31.25 minutes
Step-by-step explanation:
12.5 minutes divided by 2 (miles) = 1 mile per 6.25 minutes
6.25 x 5 = 31.25