Use this definition with right endpoints to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. f(x)=x x 3
+6
,1≤x≤4 A=lim n→[infinity]
∑ i=1
n
Answers
\(A = lim(n→∞) ∑[i=1 to n] A(i) = lim(n→∞) ∑[i=1 to n] Δx * f(xi)\). is the limit for the given question based on endpoints.
We are given the function f(x) = \(x^3 + 6\)and the interval [1, 4]. To find the area under the graph of this function, we can use right endpoints. We divide the interval into n subintervals of equal width, which can be calculated as (4 - 1) / n. Let's denote this width as Δx.
For each subinterval, we take the right endpoint as our x-value. Thus, the x-values for the subintervals can be expressed as xi = 1 + iΔx, where i ranges from 0 to n-1.
Next, we calculate the height of each rectangle by evaluating the function at the right endpoint. So, the height of the rectangle corresponding to the i-th subinterval is \(f(xi) = f(1 + iΔx) = (1 + iΔx)^3 + 6\).
The width and height of each rectangle allow us to calculate the area of each rectangle as A(i) = Δx * f(xi).
To find the total area under the graph, we sum up the areas of all the rectangles using sigma notation:
We are given the function f(x) = x^3 + 6 and the interval [1, 4]. To find the area under the graph of this function, we can use right endpoints. We divide the interval into n subintervals of equal width, which can be calculated as (4 - 1) / n. Let's denote this width as Δx.
For each subinterval, we take the right endpoint as our x-value. Thus, the x-values for the subintervals can be expressed as xi = 1 + iΔx, where i ranges from 0 to n-1.
Next, we calculate the height of each rectangle by evaluating the function at the right endpoint. So, the height of the rectangle corresponding to the i-th subinterval is \(f(xi) = f(1 + iΔx) = (1 + iΔx)^3 + 6\).
The width and height of each rectangle allow us to calculate the area of each rectangle as A(i) = Δx * f(xi).
To find the total area under the graph, we sum up the areas of all the rectangles using sigma notation:
\(A = lim(n→∞) ∑[i=1 to n] A(i) = lim(n→∞) ∑[i=1 to n] Δx * f(xi).\)
Taking the limit as n approaches infinity allows us to express the area under the graph of f(x) as a limit of a sum. However, the evaluation of this limit requires further calculations, which are not included in the given prompt.
Taking the limit as n approaches infinity allows us to express the area under the graph of f(x) as a limit of a sum. However, the evaluation of this limit requires further calculations, which are not included in the given prompt.
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Related Questions
Penelope and miranda found four consecutive odd integers such that 5 times the sum of the first two was 5 less than 19 times the fourth. What were the integers
Answers
The four consecutive odd integers are -11, -9, -7, and -5.
Let's define the consecutive odd integers and set up an equation using the given conditions:
Let the four consecutive odd integers be:
x, x+2, x+4, x+6
The problem states that 5 times the sum of the first two is 5 less than 19 times the fourth:
5(x + (x+2)) = 19(x+6) - 5
Now, let's solve for x step-by-step:
Distribute the 5 and 19:
5(2x + 2) = 19x + 114 - 5
Simplify further:
10x + 10 = 19x + 109
Move all terms with x to one side by subtracting 10x from both sides:
10 = 9x + 109
Subtract 109 from both sides:
-99 = 9x
Divide by 9:
x = -11
Now that we have x, we can find the consecutive odd integers:
-11, -9, -7, -5.
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Which expression is equivalent to n2 26n 88 for all values of n?
Answers
The value of the expression n² + 26n + 88 is equivalent to negative 4 and negative 22.
What is a factorization?It is the method to separate the polynomial into parts and the parts will be in multiplication. And the value of the polynomial at this point will be zero.
The expression is n² + 26n + 88
Put that expression to get the solution
n² + 26n + 88 = 0
n² + 22n + 4n + 88 = 0
n(n + 22) + 4(n + 22) = 0
(n + 4)(n + 22) = 0
The value of n is negative 4 and negative 22.
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1 .Find the Product of Prime factors 1352.
2. Find the L.C.M of 10 and 12.
Answers
The Prime Factors of 1352 are: 2, 2, 2, 13, 13.
What is Prime factors?The breakdown of a composite number into a sum of smaller integers is known as integer factorization in number theory. By repeatedly dividing a number by prime factors until the remainder equals one, the simplest method for determining a number's prime factors is to keep going till we have the answer. For instance, when we factor a number by its prime factors, we obtain 30/2 = 15, 15/3 = 5, and 5/5 = 1. It cannot be further factored because we received the leftover money.The Prime Factor Tree's Prime Factor Tree only uses prime numbers that are prime factors of 1352. The math is illustrated below:352 ÷ 2 = 676
676 ÷ 2 = 338
338 ÷ 2 = 169
169 ÷ 13 = 13
13 ÷ 13 = 1
Again, all the prime numbers you used to divide above are the Prime Factors of 1352. Thus, the Prime Factors of 1352 are: 2, 2, 2, 13, 13.
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The cylinder has a diameter of 4cm and a height of 14cm
i) Find the circumference of the base
ii)find the area of the base
iii)what is the volume of the cylinder
take pi=22\7
Answers
The circumference and area of the base, and the volume of the cylinder are 88/7 cm, 88/7 cm², and 176 cm³ respectively.
What is the circumference of the base, the area of the base, and the volume of the cylinder?A cylinder is simply a 3-dimensional shape having two parallel circular bases joined by a curved surface.
The circumference of the base of a cylinder is expressed as:
C = 2πr
The area is expressed as:
A = πr²
The volume of a cylinder is expressed as;
V = π × r² × h
Where r is the radius of the circular base, h is height and π is constant pi ( π = 22/7 )
Given that:
Diameter d = 4cm
Radius d/2 = 4/2 = 2cm
Height h = 14cm
i) Circumference of the base:
C = 2πr
C = 2 × 22/7 × 2cm
C = 88/7 cm
ii) Area of the base:
A = π × r²
A = 22/7 × 2²
A = 88/7 cm²
iii) Volume of the cylinder:
V = π × r² × h
V = 22/7 × 2² × 14
V = 176 cm³
Therefore, the volume is 176 cubic centimeters.
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-DEFINITIONS and APPLICATION: Respond to only four (4) from the list of eight (8) concepts below. Each explanation is worth six (6 Doints. The DEFINITION and APPLICATION questions therefore in total a
Answers
DEFINITIONS and APPLICATION:
1A short answer is a concise response that directly addresses the question or prompt. It typically consists of a few sentences or a paragraph and is focused on providing a specific answer without unnecessary details or elaboration.
Example: When asked "What is the capital of France?", a short answer would be "Paris."
A 100-word response refers to providing a written explanation or description within a specific word limit of 100 words. This limitation requires the writer to be concise and selective with their information while still conveying a comprehensive understanding of the topic.
Example: If asked to describe the water cycle in 100 words, a response could include information about evaporation, condensation, precipitation, and how water moves between the Earth's surface, atmosphere, and bodies of water.
A conclusion is the final part of a written piece that summarizes the main points and provides a closing statement. It serves to wrap up the information discussed in the body of the text and leave a lasting impression on the reader.
Example: In an essay about the benefits of exercise, the conclusion may restate the key advantages, such as improved physical and mental health, increased energy levels, and reduced risk of chronic diseases, while also encouraging the reader to prioritize regular physical activity.
4. Application: In the context of these concepts, application refers to the practical use or implementation of a particular idea, theory, or skill in real-life situations. It involves taking what has been learned and using it to solve problems, make decisions, or accomplish specific tasks.
Example: If learning about the Pythagorean theorem, applying it would involve using the formula to find the length of the hypotenuse in a right triangle when given the lengths of the other two sides.
These are four of the eight concepts related to definitions and applications. Remember to provide clear explanations, relevant examples, and use a step-by-step approach when addressing each concept.
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A carton of grapefruit juice displays the nutritional information shown below. How many grams of sugar are there in a 200 ml glass of juice? Grapefruit juice 250 ml contains Carbohydrate Sugar Protein 19.5 g | 16.5 g | 1.5 g
Answers
Answer:
13.2 g
Step-by-step explanation:
let x = grams sugar in a 200 ml glass
16.5 g sugar / 250 ml = x g sugar / 200 ml
x(250) = (16.5)(200)
x = (16.5)(200) / (250) = 3300 / 250 = 13.2
Answer: there are 13.2 g sugar in a 200 ml glass of juice
Maricella solves for x in the equation 4 x minus 2 (3 x minus 4) + 4 = negative x + 3 (x + 1) + 1. She begins by adding –4 + 4 on the left side of the equation and 1 + 1 on the right side of the equation. Which best explains why Maricella’s strategy is incorrect?
Answers
Answer:
It is wrong because she should simplify both sides of the equation,distribute, combine like terms,subtract and divide to get the final answer.
4x−2(3x−4)+4=−x+3(x+1)+1
Step 1: Simplify both sides of the equation.
4x−2(3x−4)+4=−x+3(x+1)+1
4x+(−2)(3x)+(−2)(−4)+4=−x+(3)(x)+(3)(1)+1(Distribute)
4x+−6x+8+4=−x+3x+3+1
(4x+−6x)+(8+4)=(−x+3x)+(3+1)(Combine Like Terms)
−2x+12=2x+4
−2x+12=2x+4
Step 2: Subtract 2x from both sides.
−2x+12−2x=2x+4−2x
−4x+12=4
Step 3: Subtract 12 from both sides.
−4x+12−12=4−12
−4x=−8
Step 4: Divide both sides by -4.
−4x−4=−8−4
x=2
Answer:
x=2
Answer:
it is in order to combine like terms on one side of the equation, the inverse operations must be used
Step-by-step explanation:
trust me I took it on edge
What is an interval of increase ?
Answers
Given f(x)=x*-x³-6x², for what values of x will f(x) > 0?
Answers
The values of x will f(x) > 0 for x < 0, and f(x) < 0 for -6 < x < 0 and x > -6.
To determine the values of x for which f(x) > 0, we need to find the intervals where the function is positive. Let's analyze the function f(x) = x*-x³-6x².
First, let's factor out an x from the expression to simplify it: f(x) = x(-x² - 6x).
Now, we can observe that if x = 0, the entire expression becomes 0, so f(x) = 0.
Next, we analyze the signs of the factors:
1. For x < 0, both x and (-x² - 6x) are negative, resulting in a positive product. Hence, f(x) > 0 in this range.
2. For -6 < x < 0, x is negative, but (-x² - 6x) is positive, resulting in a negative product. Therefore, f(x) < 0 in this range.
3. For x > -6, both x and (-x² - 6x) are positive, resulting in a negative product. Thus, f(x) < 0 in this range.
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Write an inequality for the word phrase "x increased by 5 is at least 13"
Answers
Answer:
x+5≥13
Step-by-step explanation:
how do you find area of a shape?
Answers
Answer:
The area of a rectangle is the length times width.
Step-by-step explanation:
Josea went out to dinner at an Indian restaurant. The
total bill was $38. She wanted to leave a 15% tip.
If you use the idea of scaling to find the tip amount,
what would she need to multiply by?
Answers
H
determine the standard form of an equation of the parabola subject to the given conditions. vertex: (−1,−3); directrix: x=−5
Answers
To determine the standard form of an equation of the parabola with the given vertex and directrix, we need to use the following formula:
y = (1/4a) x^2 + (1/2)ap + k
where (h,k) is the vertex and a is the distance between the vertex and the focus (which is the same as the distance between the vertex and the directrix). In this case, the vertex is (-1,-3) and the directrix is x=-5.
First, let's find the value of a. Since the directrix is a vertical line, we know that the parabola is opening horizontally. The distance between the vertex and the directrix is 4 units (since the vertex is 4 units to the right of the directrix), so we have:
a = 1/2 * 4 = 2
Now we can substitute the values of a, h, and k into the formula:
y = (1/4*2) x^2 + (1/2)2(-1) - 3
Simplifying this equation, we get:
y = (1/8) x^2 - x - 3
So the standard form of the equation of the parabola with vertex (-1,-3) and directrix x=-5 is:
y = (1/8) x^2 - x - 3
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An elevator has a placard stating that the maximum capacity is 1884 lb-12 passengers. So, 12 adult male passengers can have a mean weight of up to 1884/12=157 pounds. If the elevator is loaded with 12 adult male passengers, find the probability that it is overloaded because they have a mean weight greater than 157 lb. (Assume that weights of males are normally distributed with a mean of 165 lb and a standard deviation of 32 lb.) Does this elevator appear to be safe? BICICIE The probability the elevator is overloaded is (Round to four decimal places as needed) Does this elevator appear to be safe? OA. No, there is a good chance that 12 randomly selected adult male passengers will exceed the elevator capacity OB. No, 12 randomly selected people will never be under the weight limit. OC. Yes, there is a good chance that 12 randomly selected people will not exceed the elevator capacity OD. Yes, 12 randomly selected adult male passengers will always be under the weight limit. A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, find the probability of a rating that is between 200 and 275. Round to four decimal places. www OA. 0.4332 OB. 0.9332 OC. 0.5000 OD. 0.0668 Find the value of the linear correlation coefficient r. The paired data below consist of the costs of advertising (in thousands of dollars) and the number of products sold (in thousands). Cost 9 2 3 5 9 10-> 4 2 68 67 Number 52 55 85 A. 0.235 OB. 0.708 OC. 0.246 OD. -0.071 86 83 73
Answers
The answer is option A. No, there is a good chance that 12 randomly selected adult male passengers will exceed the elevator capacity.
Probability that it is overloaded if 12 adult male passengers have a mean weight greater than 157 lb is 0.0229.Round to four decimal places as needed.Based on the calculations the elevator does not appear to be safe.The solution for the given problem is as follows:
Given that, the maximum capacity of the elevator is 1884 lb - 12 passengers.
We can write as below:
Maximum capacity per person=1884/12=157lb.
And, weights of males are normally distributed with a mean of 165 lb and a standard deviation of 32 lb.Thus, Z = (157-165) / (32 / √12) = -1.7321Then, P(Z > -1.7321) = 0.9586
Hence, the probability that it is overloaded if 12 adult male passengers have a mean weight greater than 157 lb is:P(Z > -1.7321) = 1 - P(Z < -1.7321) = 1 - 0.0229 = 0.9771 (rounded off to 4 decimal places).This probability is greater than 5% and therefore, the elevator does not appear to be safe.
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Somebody please help me I need to send this tomorrow
4. Mr. Wilson has two cats, named Ramses and Kilala. Kilala weighs 6 pounds. Ramses weighs 3 times as much as Kilala. How much does Ramses weigh? Show how you figured this out.
Answers
Answer:
18
Step-by-step explanation:
6lb
*3
6lb*3=18
PLEASE HELP!!!
The lines shown below are parallel.
A. True
O
O
B. False
Answers
To find the quotient of 726.02 and 1,000, move the decimal point in 762.02___Places to the ____
Choices for blank 1.
A. 2
B. 3
C. 4
D. 5
Choices for blank 2.
A. Left
B. Right
NO LINKS!!
Answers
from blank one five
from blank two is right
The sum of two numbers is 39 and the difference is 15. What are the numbers?
Smaller number:
Larger number:
Answers
In order to calculate the numbers, we must first create two equations with the given information so that we have variables to solve for.
Given:
The sum of the two numbers is 39 The difference is 15Let S = smaller number and L = larger number
If the sum of the two numbers is 39,
Then S + L = 39
If the difference is 15
Then L - S = 15
We know that the smaller number is being subtracted from the larger number as the outcome is positive.
Solving for S and LNow that we have created two equations we can solve them as a system of equations. We can use the substitution method as we can easily rearrange either equation to define one of the variables.
L - S = 15
==> rearrange equation to define L
L = 15 + S
==> we now plug in L = 15 + S into the other equation
S + (15 + S) = 39
==> remove parenthesis and combine like terms
2S + 15 = 39
==> subtract 15 from both sides
2S = 24
==> divide both sides by 2
S = 12
We now plug this into one of the equations to find the L
L - S = 15
==> plug in s = 12
L - 12 = 15
==> add 12 to both sides
L = 27
12 + 27 = 39 and 27 - 12 = 15 ✓
The smaller number is 12 and the larger number is 27
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3) A lottery ticket says that the chances of winning are 1 in 8. Suppose you buy 10 of these lottery tickets. Find the probability that at least one of them will be a winner
Answers
The probability of at least one of your 10 lottery tickets being a winner is approximately 0.638 or 63.8%.
The probability of winning on a single lottery ticket is 1/8, which means that the probability of not winning is 7/8. If you buy 10 of these lottery tickets, the probability of not winning on any of them is:
(7/8)^10 = 0.362
This means that there is a 36.2% chance that none of your tickets will be a winner. To find the probability that at least one of your tickets will be a winner, we can use the complementary probability:
P(at least one winner) = 1 - P(no winners)
P(at least one winner) = 1 - 0.362
P(at least one winner) = 0.638
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Consider the following geometric sequence 2,6,18,54
Answers
Answer:
Multiply the number by 3 to get the next number.
Step-by-step explanation:
PLLLLLLLLZZZZZ Solve |2x - 5| = 4.
Answers
Answer:
2x-5=4
2x=9
x=9/2=4.5
or
2x-5=-4
2x=1
x=1/2
Step-by-step explanation:
Answer:
x=1/2 and x=9/2
Step-by-step explanation:
7/10 × 4/9 as a fraction
Answers
Answer: 14/45
Step-by-step explanation:
.2. Determine whether the feasible set for each of the following systems of constraints is convex, and if not, indicate points x^1 and x² that violate definition. a) (x1)² + (x2)² > 9
x1 + x2 ,10
x1, x2 > 0
Answers
The feasible set for this system of constraints is not convex, and the points (5, 5) and (3, 7) violate the convexity definition.
To determine whether the feasible set for each system of constraints is convex, we need to analyze the constraints individually and examine their intersection.
a) (x1)² + (x2)² > 9
This constraint represents points outside the circle with a radius of √9 = 3. The feasible set includes all points outside this circle.
b) x1 + x2 ≤ 10
This constraint represents points that lie on or below the line x1 + x2 = 10. The feasible set includes all points on or below this line.
c) x1, x2 > 0
This constraint represents points in the positive quadrant, where both x1 and x2 are greater than zero.
Now, let's analyze the intersection of these constraints:
Considering the first two constraints (a and b), we can see that the feasible set consists of all points outside the circle (constraint a) and below or on the line x1 + x2 = 10 (constraint b).
To determine whether the feasible set is convex, we need to check if any two points within the set create a line segment that lies entirely within the set.
If we consider the points (5, 5) and (3, 7), both points satisfy the individual constraints (a) and (b). However, the line segment connecting these two points, which is the line segment between (5, 5) and (3, 7), exits the feasible set since it passes through the circle (constraint a) and above the line x1 + x2 = 10 (constraint b).
Therefore, the feasible set for this system of constraints is not convex, and the points (5, 5) and (3, 7) violate the convexity definition.
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what rounds 9,823 to the nearest hundred
Answers
write three equations of lines with a positive slope
Answers
Based on this data, for every inch in height, the arm span increases by 1.1 inches.
Here, we have,
I put the height of people on the x-axis because this is what I am trying to correlate the arm span to. I put the arm span on the y-axis because it is dependent on height.
I used point (60,61) and (70,72) to draw the line of best fit and to get the equation. First, to get the slope I would use the slope formula of m = 72-61/70-60 or 11/10 to get a slope of approximately 1.1. Then I used (60,61) to get the rest of the equation by plugging it in: y-61 = 1.1(x -60). This becomes y-61 = 1.1x - 66. Add 61 to both sides and the final equation is y = 1.1x – 5.
The slope represents the rate of change arm span based on height. The y intercept is the arm span.
Using points (66,68) and points (70,72) I plug each into the equation y=1.1x-5 to figure each out based on the difference from the actual points and get .4 for (66,68) and 0 for point (70,72) so I would say, based on .4 and 0 for these two points that this is a good line of best fit.
Based on this data, for every inch in height, the arm span increases by 1.1 inches.
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complete question:
There are many measurements of the human body that are positively correlated. For example, the length of one's forearm (measured from elbow to wrist) is approximately the same length as the foot (measured from heel to toe). They are positively correlated because, as one measurement increases, so does the other measurement.
Use Euler's method with step size h = 0.2 to approximate the solution to the initial value problem at the points x = 4.2, 4.4, 4.6, and 4.8. points x -4.2,4.4,4.6, and 4.8.
y=3/x(y^2+y), y(4)=3
Answers
Using Euler's method with step size h = 0.2, the approximate solution to the initial value problem at x = 4.2, 4.4, 4.6, and 4.8 is 4.8, 10.344, 21.719650882754717, and 65.86222981762188, respectively.
To approximate the solution to the initial value problem using Euler's method with a step size of h = 0.2, we can follow these steps:
Step 1: Define the differential equation:
Given initial value problem:
dy/dx = 3/x(y^2 + y), y(4) = 3
Step 2: Define the step size:
h = 0.2
Step 3: Define the points at which we want to approximate the solution:
x = 4.2, 4.4, 4.6, and 4.8
Step 4: Implement Euler's method:
We'll use the iterative formula:
y[i+1] = y[i] + h * f(x[i], y[i])
where x[i+1] = x[i] + h, and f(x[i], y[i]) represents the value of dy/dx at x[i] and y[i].
Using the given initial value y(4) = 3, we'll start with x = 4 and y = 3.
For x = 4.2:
x[1] = 4 + 0.2 = 4.2
y[1] = y[0] + h * f(x[0], y[0])
= 3 + 0.2 * (3/4 * (3^2 + 3))
= 3 + 0.2 * (3/4 * 12)
= 3 + 0.2 * 9
= 3 + 1.8
= 4.8
For x = 4.4:
x[2] = 4.2 + 0.2 = 4.4
y[2] = y[1] + h * f(x[1], y[1])
= 4.8 + 0.2 * (4.2/4.8 * (4.8^2 + 4.8))
= 4.8 + 0.2 * (4.2/4.8 * 31.68)
= 4.8 + 0.2 * 27.72
= 4.8 + 5.544
= 10.344
For x = 4.6:
x[3] = 4.4 + 0.2 = 4.6
y[3] = y[2] + h * f(x[2], y[2])
= 10.344 + 0.2 * (4.6/10.344 * (10.344^2 + 10.344))
= 10.344 + 0.2 * (4.6/10.344 * 127.051344)
= 10.344 + 0.2 * 56.87825441377359
= 10.344 + 11.375650882754718
= 21.719650882754717
For x = 4.8:
x[4] = 4.6 + 0.2 = 4.8
y[4] = y[3] + h * f(x[3], y[3])
= 21.719650882754717 + 0.2 * (4.8/21.719650882754717 * (21.719650882754717^2 + 21.719650882754717))
= 21.719650882754717 + 0.2 * (4.8/21.719650882754717 * 1007.8317707837554)
= 21.719650882754717 + 0.2 * 220.7128946743358
= 21.719650882754717 + 44.14257893486716
= 65.86222981762188
Therefore, the approximate solution to the initial value problem at the points x = 4.2, 4.4, 4.6, and 4.8 is:
For x = 4.2, y ≈ 4.8
For x = 4.4, y ≈ 10.344
For x = 4.6, y ≈ 21.719650882754717
For x = 4.8, y ≈ 65.86222981762188
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How many miles can a car travel on 15 gallons of gas if it travels 28 miles on 1 gallon of gas?
A. 43 miles
B. 270 miles
C. 405 miles
D. 420 miles
Answers
a measure of variability of the sample means xbar in repeated samples of n observations randomly selected from a given population is
Answers
The correct answer is B i.e. Standard Error of the Mean (SE) = σ / √(n)
The measure of the variability of the sample means, also known as the standard error of the mean, can be calculated using the formula:
Standard Error of the Mean (SE) = σ / √(n)
sigma is the population standard deviation,
n is the sample size.
The standard error of the mean represents the variability or dispersion of the sample means obtained from repeated samples of size n from a population. It indicates how much the sample means are likely to differ from the true population mean.
The smaller the standard error of the mean, the less variability is expected among the sample means, indicating a more precise estimate of the population mean.
Therefore, the correct answer is B i.e. Standard Error of the Mean (SE) = σ / √(n)
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Given question is incomplete, the complete question is below
A measure of variability of the sample means xbar in repeated samples of n observations randomly selected from a given population is
a. s
b. σ/√(n)
c. s/√(n)
d. σ
The other one was cut this is the ful picture
Answers
Answer:H?????
Step-by-step explanation:
Determine the equation of the tangent plane and normal line of
the curve f(x,y,z)=x2+y2-2xy-x+3y-z-4 at p(2,
-3, 18)
Answers
To determine the equation of the tangent plane and normal line of the given curve at the point P(2, -3, 18), we need to find the partial derivatives of the function f(x, y, z) = x^2 + y^2 - 2xy - x + 3y - z - 4.
Taking the partial derivatives with respect to x, y, and z, we have:
fx = 2x - 2y - 1
fy = -2x + 2y + 3
fz = -1
Evaluating these partial derivatives at the point P(2, -3, 18), we find:
fx(2, -3, 18) = 2(2) - 2(-3) - 1 = 9
fy(2, -3, 18) = -2(2) + 2(-3) + 3 = -7
fz(2, -3, 18) = -1
The equation of the tangent plane at P is given by:
9(x - 2) - 7(y + 3) - 1(z - 18) = 0
Simplifying the equation, we get:
9x - 7y - z - 3 = 0
To find the equation of the normal line, we use the direction ratios from the coefficients of x, y, and z in the tangent plane equation. The direction ratios are (9, -7, -1).Therefore, the equation of the normal line passing through P(2, -3, 18) is:
x = 2 + 9t
y = -3 - 7t
z = 18 - t
where t is a parameter representing the distance along the normal line from the point P.
To learn more about tangent plane click here : brainly.com/question/33052311
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