Answers
You should recognize the [tex]\dfrac3n=\dfrac{b-a}n[/tex] term on the right to represent the width of each equally-spaced subinterval of the integration interval [tex][a,b][/tex]. We don't know the endpoints' exact values just yet.
[tex]n[/tex] represents the total number of subintervals. If we fix it to some manageably small number, we can get a better picture of what the [tex]\left(\dfrac{3k}n-1\right)^4[/tex] term represents.
If [tex]n=3[/tex], the partial sum reduces to
[tex]\displaystyle\sum_{k=1}^3\left(\dfrac{3k}3-1\right)^4\dfrac33=\sum_{k=1}^3(k-1)^4=0^4+1^4+2^4[/tex]
This suggests the sum represents the left-endpoint Riemann summation to find the integral of [tex]x^4[/tex], and in particular, the integral seems to be taken over the interval [0, 3]. Taking the limit, we get
[tex]\displaystyle\lim_{n\to\infty}\sum_{k=1}^n\left(\dfrac{3k}n-1\right)^4\frac3n=\int_0^3x^4\,\mathrm dx[/tex]
Final answer:
In calculus, summation involves adding discrete values, while integration is akin to adding an infinite number of infinitesimally small products to find areas, volumes, etc. Integral calculus extends these concepts to continuous functions and is used, for example, to find position from acceleration by integrating twice.
Explanation:
The transition from summation to integral in calculus is an important concept that bridges the discrete world with the continuous. When we summate, we're adding together values to find a total amount. In a similar vein, an integral can be thought of as a continuous sum, where we're adding up infinitely small products to calculate an area, volume, or other quantities.
When performing an integral, the dimensions of what we're integrating must be consistent. For example, if we're integrating velocity (v) with respect to time (t), we obtain a dimension that is the product of the two, such as distance. This principle is rooted in the requirement of dimensional consistency, analogously understood in the phrase "You can't add apples and oranges."
Understanding the relationship between discrete sums and continuous integrals is crucial for applications in various fields, such as physics where integrals can represent cumulative effects over time or space. For example, integrating an acceleration function gives us velocity, and further integration yields position. This process allows us to work backward from acceleration to position, utilizing the power of integral calculus.
Related Questions
QM Q2.) Find the multiplicities of the greatest zero, second greatest zero, and the smallest zero.
Answers
First check out the zeros (first box).
Step One.
Put brackets around terms 1 and 2 and another set around terms 3 and 4.
f(x) = (x^3 + 8x^2) - (4x + 32) Watch out that you change the sign on the 32. Minus signs can be devastating. The idea is that when the brackets are removed, the question will be exactly as it was before you put the brackets there.
Step Two
Pull out any common factors.
f(x) = x^2(x + 8) - 4(x + 8)
Step Three.
If there is a common binomial pull it out as well. This is the hardest part of the problem. The binomials have to be exactly alike (in this case they are).
f(x) = (x + 8) (x^2 - 4)
Step Four
Factor (x^2 - 4). This is an exact difference of squares. It factors into
x^2 - 4 = (x + 2)(x - 2)
Step five
Find the zeros.
f(x) = (x + 8)(x - 2)(x + 2)
x + 8 = 0
x = - 8
x + 2 = 0
x = - 2
x - 2 = 0
x = 2
I know you didn't ask for all of this, but when someone asks a question, it's always a good idea to check the work done. In your case it was all correct. That's a very good thing.
Definition
Multiplicity is the number of times a root appears in the solution of a polynomial.
For example, g(x) = (x - 5)^2 (x - 4)
The five would have a multiplicity of 2 because the equation is really
g(x) = (x - 5)(x - 5) ( x - 4). There are 2 roots that are the same (both are 5).
4 has a multiplicity of 1 in this example.
Answer
The largest root is 2. It has a multiplicity of 1
The second largest root is -2. It has a multiplicity of 1
The tiniest root is - 8 . It has a multiplicity of 1
Which of the following is the correct simplification of 7•4²÷ 8 -12
Answers
This can be found by solving with PEMDAS. First, you do the 4^2 (4*4), which equals 16. Next, you multiply 16 by 7, to get 112. Then, you'd divide 112 by 8. This gives you 14. Finally, subtract 12 from 14 to get your final answer of 2.
I hope this helps!
What is the similarity ratio of the smaller to the larger prisms? Enter your answer as a:b
Answers
Because the lengths are 6 and 9 (2:3), the widths are 8 and 12 (2:3), and the heights are 6 and 4 (2:3).
Answer:
6:9>2:3
Step-by-step explanation:
In order to do this, you just have to equalize the ratio you just need to search for the same sides, so the 6 side is similar with the 9 side on the larger, and then the 8 to the 12 and the 4 to the 6, so you just equalize any of those equal we will use the 8 and 12:
8:12
Now we just reduce it with the same:
4:6
2:3
So the ratio would be 2:3.
Instructions:Select the correct answer from the drop-down menu. In the figure, sin ∠MQP = cos N and sin R sin R and sin N cos N and sin M cos R and sin N .
Answers
sin ∠MQP = sin(56) = 0.829cos ∠N = cos(56) = 0.559sin ∠R = sin(34) = 0.559sin ∠N = sin(56) = 0.829cos ∠N = cos(56) = 0.559sin ∠M = sin(90) = 1cos ∠R = cos(34) = 0.828
Based on these findings, it is obvious that sin ∠MQP is equal to cos ∠R and sin ∠N.Thus, the correct choice is "cos R and sin N".
Answer:
Cos R and sin N
Step-by-step explanation:
how much tax you need to pay on a taxable income of 40,000
Answers
2017 RETURNS FILED IN 2018:
Single earning in this tax bracket $37,951 to $91,900, the tax rate is 25%.
Married/Joint earning in this tax bracket $18,651 to $75,900, the tax rate is 15%.
2018 RETURNS FILED IN 2018:
Single earning in this tax bracket $38,701 to $82,500, the tax rate is 22%.
Married/Joint earning in this tax bracket $19,051 to $77,400, the tax rate is 12%.
Hope this helps! :)
The average weekly income of teachers in one school district is $750 with a standard deviation of $85. Find the percentage of teachers earing more than $825 a week?
Answers
Standard deviation = $85
P(X>$825) = ?
Z = (X-mean)/Standard deviation = (825-750)/85 ≈ 0.88
From Z-table, P(X=825) = 0.8106
Therefore,
P(X>$825) = 1 - P(X=$825) = 1-0.8106 = 0.1894 = 18.94%
Percentage of teachers earning more than $825 is 18.94%.
A dog kennel uses 90 lb of dog food per day. How many pounds are used in a year?
Answers
Lindsay's watering can holds 12 quarts of water. She uses 1 pint of water on each of her flowers.
How many flowers can she water?
Answers
12 quarts x 2 = 24 pints
1 flower = 1 pint
she can water 24 flowers
She can water 24 flowers using 12 quarts.
1 quart consists of how many pints?1 quart consists of 2 pints
So, 12 quarts = 12*2 =24 pints
It is given that she uses 1 pint of water on each flower.
So, 24 pints can be used to water =24*1 =24 flowers.
Therefore, She can water 24 flowers using 12 quarts.
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Solve for the variable 10/8=n/10
Answers
Answer: 50/4 or 12.5
Step-by-step explanation:
We can cross-multiply in this situation.
Cross-multiplying...
[tex]\frac{10}{8} = \frac{n}{10}[/tex]
8n = 10(10) = 100
In other words, we multiply the denominator of 10/8, which is 8, by the numerator of n/10, which is n, and vice-versa.
Dividing 8 on both sides....
n = 100/8 = 50/4 = 12.5
A test is worth 50 points. Multiple-choice questions are worth 1 point, and short-answer questions are worth 3 points. If the test has 20 questions, how many multiple-choice questions are there?
A. t for test, q for questions
B. m for multiple choice, s for short answer
C. s for short answer, t for test
D. p for points, m for multiple choice
apex mcr
Answers
S * Q + M * Q = P
Representations:
S = The amount of points for short-answer questions.
M = The amount of points for multiple-choice questions.
Q = Number of questions for each of the type of questions that add up to the total number of questions on the test (20 questions).
P = Total points on the test.
Since there is a total of 50 points on the 20 question test, you would need to divide up the amount of questions there are for each of the type there are on the test, short-answer, and multiple-choice.
3 * Q + 1 * Q = P
Now find how many of the type of questions there are out of 20 questions on the test that would add up to the total number of points on the test.
3 * Q + 1 * Q = 50
(15 + 5 = 20 questions on the test, 15 and 5 can be used for the number of questions for each type of question on the test).
3 * 15 + 1 * 5 = 50
45 + 5 = 50
50 = 50
So, your total number of multiple choice questions are: 5.
And, your total number of short answer questions are: 15.
I would go with D. for your answer. I may be wrong.
I hope this helps.
~ Notorious Sovereign
Find the area of a regular hexagon whose side measure is 5 cm.
Answers
Area of the Hexagon
[tex] = ( \frac{3 \sqrt{3} \times 5 {}^{2} }{2} ) \: sq \: cm[/tex]
[tex] = \frac{75}{2} \sqrt{3 \:} sq \: cm[/tex]
Exercise 3.8 asked you to write a method called quadratic that solves quadratic equations and prints their roots. recall that a quadratic equation is a polynomial equation in terms of a variable x of the form a x2 + b x + c = 0. the formula for solving a quadratic equation is:
Answers
x=[-b+/-√(b²-4ac)]/2a
where a,b and c are the respective coefficients as indicated in the quadratic form.
What is the degree measure of angle A?_____ degrees
Answers
Vertical angles are congruent
Angle A is vertical angles to the angle that measures 80
The answer is 80
Hope this helps!
in a survey of a town 56% of residents own a car, 21% of residesnts own a truck
Answers
Cost per mile is calculated by dividing total cost by total miles. Using the notation P = cost per mile, C = total cost, and M = total miles, show this relationship in algebraic equation form.
Answers
Substituting the variables defined in the problem statement, this is ...
P = C/M
Answer:
(cost per mile) = (total cost)/(total miles)
Substituting the variables defined in the problem statement, this is ...
P = C/M
Step-by-step explanation:
I do have a time limit. I appreciate any help
If YB = ZA find the value of X and the length of YB
Answers
Also, we can see that YZ and AB are perpendicular to both ZA and YB, thus
YZ = AB
16x - 4 = 4 - 4x
16x + 4x = 4 + 4
20x = 8
x= 8/20 = 2/5
YB = ZA = 20x - 5 = 20(2/5) - 5 = 3
Thus, the value of x is 2/5 and length of YB is 3 units.
So, the correct answer is option A
Refer to the above data. in which year is there a balanced budget? year 1 year 2 year 3 year 4
Answers
Which describes a cost that fluctuates depending on the number of units produced A) Liability B) Revenue C) fixed cost D) Variable cost
Answers
Answer: D: Variable cost
Explanation: There are two aspects of cost in the process of production. One is fixed and the other is variable
The variable component is called as variable cost which fluctuates based on the number of units produced. The examples of variable costs include commissions, direct labor costs, manufacturing costs etc which depend on the number of units produced and sold
A cost that fluctuates depending on the number of units produced is called a variable cost.
Option D is the correct answer.
What is a variable cost?A variable cost is a cost that changes proportionally to the level of activity or production volume.
We have,
The cost per unit varies depending on how many units are produced.
As more units are produced, the variable cost increases, and as fewer units are produced, the variable cost decreases.
This is in contrast to a fixed cost, which remains constant regardless of the level of production.
Examples of variable costs include direct labor, raw materials, and production supplies.
Thus,
A cost that fluctuates depending on the number of units produced is called a variable cost.
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at 6am the temperature is -5°f the temperature rises 6°f by noon what is the temperature at noon?
Answers
(-5)+6=1
If you use a number line, and start from -5 and "count right 6 times", you should get 1 degree Fahrenheit.
Hope this helped!!! c:
Which events are independent? Check all that apply. rolling two number cubes at the same time choosing a card, and then flipping a coin choosing a first, second, and third place winner drawing a name from a hat, keeping it, then drawing another name drawing a marble from a bag, replacing it, then drawing another marble flipping the same coin three times
Answers
Answer:
1.rolling two number cubes at the same time
2.choosing a card, and then flipping a coin
5.drawing a marble from a bag, replacing it, then drawing another marble
6.flipping the same coin three times
Compute the simple interest earned on a deposit of $5000 at 8.5% for 120 days
Answers
[tex]\bf ~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\to& \$5000\\ r=rate\to 8.5\%\to \frac{8.5}{100}\to &0.085\\ t=years\to \frac{120}{365}\to &\frac{24}{73} \end{cases} \\\\\\ I=(5000)(0.085)\left( \frac{24}{73} \right)\implies I \approx 139.72602739726[/tex]
please show me how to answer this
Answers
Choose the equation that represents the line passing through the point (-3,-1) with a slope of 4
Answers
y +1 = 4(x +3) . . . . . . point-slope form
y = 4(x +3) -1 . . . . . .. another point-slope form
y = 4x +11 . . . . . . . . . slope-intercept form
4x -y +11 = 0 . . . . . . . general form
4x -y = -11 . . . . . . . . . standard form
Use part 1 of the fundamental theorem of calculus to find the derivative of the function. y = cos x (3 + v5)8 dv sin x
Answers
[tex]y(x)=\displaystyle\int_{\cos x}^{\sin x}(3+v^5)^8\,\mathrm dv[/tex]
(If the limits are in the wrong order, just multiply the result by -1)
Split the integral at an arbitrary value between [tex]\cos x[/tex] and [tex]\sin x[/tex], and write [tex]y(x)[/tex] as
[tex]y(x)=\displaystyle\left\{\int_{\cos x}^c+\int_c^{\sin x}\right\}(3+v^5)^8\,\mathrm dv[/tex]
[tex]y(x)=\displaystyle\int_c^{\sin x}(3+v^5)^8\,\mathrm dv-\int_c^{\cos x}(3+v^5)^8\,\mathrm dv[/tex]
Then by the FTC,
[tex]\dfrac{\mathrm dy}{\mathrm dx}=(3+\sin^5x)^8\cdot\dfrac{\mathrm d\sin x}{\mathrm dx}-(3+\cos^5x)^8\cdot\dfrac{\mathrm d\cos x}{\mathrm dx}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\cos x(3+\sin^5x)^8+\sin x(3+\cos^5x)^8[/tex]
The first part of the Fundamental Theorem of Calculus provides a method to find the derivative of a function. However, clarity is needed in the question as it has extra elements. If we consider the derivative of y = cos x (3 + √5)8 sin x, it would be f'(x) = -sin x (3 + √5)8 sin x + cos x (3 + √5)8 cos x.
Explanation:Since the question asks for the derivative of the function y = cos x (3 + √5)8 dv sin x, we'll need to use the First Part of the Fundamental Theorem of Calculus. This theorem states that if a function f is continuous over the interval [a, b] and F is an antiderivative of f on [a, b], then the definite integral from a to b of f(x) dx equals F(b) - F(a). In this case, we have y = cos x (3 + √5)8 dv sin x which sadly is not clear due the presence of apparently extra and possibly typographical elements.
Assuming, the question intends to find derivative of y = cos x (3 + √5)8 sin x, we can proceed as follows:
Let f(x) = cos x (3 + √5)8 sin x. Then by applying the chain rule, we get: f'(x) = -sin x (3 + √5)8 sin x + cos x (3 + √5)8 cos x.
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If it takes 2 nurses 2 minutes to measure the blood pressure of 2 patients, how long would it take 200 nurses to measure the blood pressure of 200 patients (in minutes)?
Answers
Final answer:
One nurse takes 2 minutes to measure the blood pressure of one patient. With 200 nurses, each assigned to one patient, it will take all nurses 2 minutes to measure the blood pressure of 200 patients.
Explanation:
The question is asking for the time it would take a group of nurses to measure the blood pressure of an equal number of patients given a scenario. It works on the principle that if the number of nurses is directly proportional to the number of patients, the time it takes for one nurse to measure one patient's blood pressure will be the same, regardless of how many nurses or patients there are, as long as the ratio remains 1:1.
If it takes 2 nurses 2 minutes to measure the blood pressure of 2 patients, that means one nurse takes 2 minutes to measure the blood pressure of one patient. Therefore, if you have 200 nurses each measuring the blood pressure of one patient, it would still take 2 minutes for all 200 patients to have their blood pressure measured since each nurse is working with one patient individualy.
SCIENCE HELP 10 POINTS!!!!!
The greenhouse effect traps heat near earth's surface. Greenhouse gases in the atmosphere—carbon dioxide, water vapor, methane, and nitrous oxide—allow the shorter waves of ultraviolet radiation from the sun to enter earth's lower atmosphere. However, these gases do not allow the longer wavelengths of infrared radiation to pass back out to the outer atmosphere. The result is that the lower atmosphere heats up.This result happens naturally, but it can be accelerated by human activities. Since the late 1800s, human activity has greatly altered the amount of greenhouse gases that enter the atmosphere. Burning fossil fuels, farming, and clearing forests, as well as industry, have all contributed to increases of carbon dioxide (CO2), methane, nitrous oxide, and other greenhouse gases in the atmosphere.The result is that temperatures in the troposphere have gone up. Many scientists identify this increase in greenhouse gas emissions as contributing directly to global warming. This suggests that human activity has had an impact on the global climate.
The author argues that in modern times, the greenhouse effect *
A.helps keep the Earth warm
B.has increased because of human activity
C.is in a natural cycle of changing
D.has changed local weather patterns
Answers
C.is in a natural cycle of changing
Which figure shows how a shape can be rotated about an axis to form the given object?
Answers
The answer is as shown in the attachment which is the first diagram (on the left) on the second row which is highlighted in the attachment
That is the only figure which when rotated about the axis will form a cylindrical form as required.
The top two will form into a solid sphere and the last would form into a solid cylinder and not the one as required.
Answer:
Option C.
Step-by-step explanation:
In this question we will go by every option.
Option A). When a circle showing in the figure is rotated then a hollow ring will be formed.
Option B). In this option when a circle like structure with two layers is rotated about an axis it will forms a ring with two layers. Inner part of the ring will be empty.
Option C). In this option when a vertical strip is rotated by the axis, it will form the cylinder similar to the object shown in the figure.
Option D). In this option when a strip attached to the axis is rotated, a sold cylinder is formed.
Therefore Option C. is the correct option.
If the graph of the function h is defined by = h(x)+x^2+9 is translated vertically downward by 9 units, it becomes the graph of a function g. Find the expression for
gx
Answers
Solution:
The graph of h(x) = x²+9 translated vertically downward by 9 units means that each point (x, h(x)) is shifted onto the point (x, h(x) - 9), that is,
(x, h(x)) → (x, h(x) - 9)
The translated graph that represents the function is defined by the expression for g(x):
g(x) = h(x) - 9 = x² + 9 - 9 = x²
h(x) = x²+9 → g(x) = x² shows that the graph of the equation g(x) = x² moves the graph of h(x) = x²+9 down nine units.
Bernard weighs 7.5 pounds at birth. assuming his development proceeds normally, he should weigh about _____ pounds by his first birthday. 22.5 24.5 20.5 18.5
Answers
Given Bernard weighs 7.5 pounds at birth.The question is asking us to assume that Bernard development proceeds normally.We need to find Bernard birth after a year.
It is observed that birth of a child triples or is thrice (3 times ) the weight at birth.As Bernard is weighing 7.5 pounds so he will triple or be 3times of this weight.
Bernard weight after a year= 3x7.5=22.5 pounds.
help how woudl i do this ive attached a picture
Answers
m angle b° = arc/2
m angle b° = 60°/2
m angle b°=30°
b=30
Answer: Fourth option: 30