∫³ʰ₋ₕ f(x) dx ≈ w0 f(0) + w1 f(h) + w2f(2h)
Show that the formula with weights w1 = W3 = 8/3h and W2 -4/3 h is exact for all cubic functions. Then, assuming an error term of the form E = Kh⁵f⁽⁴⁾ (ε), with f⁽⁴⁾ the 4th derivative of f and ε ∈ (-h, 3h), use an appropriate function f to show that the constant K = 14/45
Answers
The formula with weights w1 = w3 = 8/3h and
w2 = -4/3h is exact for all cubic functions, and the constant K = 14/45 for the error term
E = Kh⁵f⁽⁴⁾(ε) when f(x) = x³.
We are given the following formula for numerical integration:∫³ʰ₋ₕ f(x) dx ≈ w0 f(0) + w1 f(h) + w2f(2h)
To show that the formula is exact for all cubic functions, we need to show that it integrates cubic polynomials exactly. We know that cubic polynomials are of the form
:f(x) = ax³ + bx² + cx + dwhere a, b, c, and d are constants.
Let's calculate the integral of f(x) over the interval [-3h, h]:∫₋³ʰ³ʰ f(x) dx
= ∫₋³ʰ³ʰ (ax³ + bx² + cx + d) dx
= [(a/4)x⁴ + (b/3)x³ + (c/2)x² + dx]₋³ʰ³ʰ
= [(ah³ + bh² + ch + 4d) - (-27ah³ + 9bh² - 3ch + 4d)]/12
= (32ah³ + 8bh³ + 2ch³ + 48d)/12
= 8h³(4a + b + c + 6d)/3
= 8h³f(h) + (16/3)h³f(0) - (8/3)h³f(2h)
We see that this result is the same as the formula with weights w1 = w3 = 8/3h and w2 = -4/3h.
Therefore, the formula is exact for all cubic functions.
Now, let's assume that there is an error term of the form E = Kh⁵f⁽⁴⁾(ε), with f⁽⁴⁾ the 4th derivative of f and ε ∈ (-h, 3h).
We want to find the constant K that makes the error term equal to zero when f(x) is a cubic function.
Let's choose f(x) = x³.
Then f⁽⁴⁾(x) = 0 for all x.
Therefore, E = 0 for this function. Substituting this function into the formula, we get:
∫₋³ʰ³ʰ x³ dx ≈ w0f(0) + w1f(h) + w2f(2h)
= w0(0³) + w1(h³) + w2(2h)³
= (8/3h)(h³) + (-4/3h)(8h³)
= 0
Simplifying, we get: (8/3 - 32/3)h⁴ = 0
This implies that K = 14/45 (since ε ∈ (-h, 3h), the constant in the error term must be the average value of h⁵f⁽⁴⁾ over the interval (-h, 3h), which is 14/45 for f(x) = x³).
Therefore, the formula with weights w1 = w3
= 8/3h and
w2 = -4/3h is exact for all cubic functions, and the constant K = 14/45 for the error term
E = Kh⁵f⁽⁴⁾(ε)
when f(x) = x³.
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Related Questions
whats the answer giving brainliest>>>>
Answers
Answer:
the answer is b
Step-by-step explanation:
brainliest pls
Option 1: 1,811 square feet, 2-3 bedroom, 2 bath house that arrives in 4 modules. It includes the upgraded PEX piping and fiber cement siding that Xavier saw on the tour. The price for Option 1 would be $180 per square foot.
Option 2: 2,600 square feet, 3-4 bedroom, 3 bath house that arrives in 5 modules. It also includes the upgraded PEX piping and fiber cement siding. The price for Option 2 is $195 per square foot.
Calculate Xavier’s transportation cost to have the modules delivered on the same day
Use this formula to calculate Xavier’s cost:
[(.20 × miles)+500]× number of modules
Answers
If the price is $180 per square foot, the price of the lot will be $3,240,000
To calculate Xavier cost, we need to first calculate the area of the lot (trapezium)
Area of the trapzium = 1/2(a+b)h
h is the height of the trapeziuma and b are the sidesArea of the trapezium = 1/2(175+185)*100
Area of the trapezium = 1/2(360)*100
Area of the trapezium = 18000 square foot.
If the price is $180 per square foot, the price of the lot will be 18000 * 180 = $3,240,000
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According to the article , how are the apple iPod and the Nintendo wii the same
Answers
According to the article, one way that the apple iPod and the Nintendo Wii are the same is both were innovative products that disrupted their respective industries
How are the apple iPod and the Nintendo Wii similar ?The Apple iPod and the Nintendo Wii are both electronic devices that were popular in the early 2000s. The iPod revolutionized the way people listened to music, while the Wii introduced a new way to play video games using motion controls.
The iPod had a simple and intuitive interface that made it easy to navigate and listen to music, while the Wii was designed to appeal to a broad audience, including casual gamers and families. The article was trying to explain how innovative both products were.
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If 10x+6y=7 is a true equation, what would be the value of 6(10x+6y)?
Answers
Answer:
42
Step-by-step explanation:
We know that 10x + 6y = 7, and so
6(10x + 6y) = 6(7) = 42
Which angle in AXYZ has the largest measure? 4 Y A. LX B. LY C.
Answers
(A) ∠X is the largest angle in the given triangle as the side opposite is also the largest.
What are angles?An angle is formed when two straight lines or rays meet at a single terminal.
The place where two points converge is known as an angle's vertex. The name "angle" comes from the Latin word "angulus," which means "corner."
Two lines that meet at the same point or two planes that meet at the same line form a geometric figure.
The angle in degrees separates these lines or planes.
So, we know that:
The side opposite to the largest angle is also the largest side of the triangle and vica-versa.
Remembering this we can easily conclude that ∠X is the largest angle as the largest is 9 units and X is opposite of the side YZ which measures 9 units.
Therefore, (A) ∠X is the largest angle in the given triangle as the side opposite is also the largest.
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Correct question:
Which angle in AXYZ has the largest measure?
A. ∠X
B. ∠Y
C. ∠Z
D. Cannot be determined
What is a "gestalt"? How do the experimental examples provided in the text (Necker cube, visual cliff, etc.) help demonstrate principles of perceptual organization?; choose one example to discuss specifically.
Answers
The Kanizsa triangle illusion helps us understand that our perceptual experiences are not simply the sum of the individual sensory inputs, but rather the result of a complex and variable process of perceptual organization.
Gestalt is a German word meaning "shape" or "form," and in psychology, it refers to a set of principles that describe how people perceive and organize sensory information into meaningful wholes. These principles propose that the whole is greater than the sum of its parts, and that we tend to organize our perceptual experiences into coherent, holistic forms rather than isolated, unrelated sensations.
Experimental examples such as the Necker cube, visual cliff, and others help demonstrate principles of perceptual organization by highlighting how our minds naturally try to impose structure and order on sensory input. For example, the Necker cube is a two-dimensional drawing that can be perceived as a cube that can be viewed from different angles. However, as one stares at the image, it appears to flip back and forth between different possible interpretations. This phenomenon illustrates the Gestalt principle of figure-ground, which describes how we tend to perceive objects as being distinct from their surrounding context.
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Find the value of x. *
2x+6
32°
Answers
Answer: 71
Step-by-step explanation:
Answer:
x = 71°
Step-by-step explanation:
2x + 6 = 180° - 32°
2x + 6 = 148°
2x = 142°
x = 71°
suppose you roll a pair of standard dice 10 times. what is the proba- bility that the sum is 6 exactly 6 times?
Answers
The probability of rolling a sum of 6 exactly 6 times out of 10 rolls is approximately 0.0599, or about 6%.
The probability of rolling a sum of 6 on a pair of standard dice is 5/36, because there are 5 possible ways to roll a sum of 6 (1+5, 2+4, 3+3, 4+2, and 5+1) out of a total of 36 possible outcomes.
To calculate the probability of rolling a sum of 6 exactly 6 times out of 10 rolls, we can use the binomial distribution formula:
P(X = k) = (n choose k) × p^k × (1 - p)^(n - k)
where P(X = k) is the probability of getting k successes (in this case, rolling a sum of 6) out of n trials (in this case, 10 rolls), p is the probability of success on a single trial (in this case, 5/36), and (n choose k) is the number of ways to choose k successes out of n trials.
Plugging in the values for this problem, we get:
P(X = 6) = (10 choose 6) × (5/36)⁶ × (31/36)⁴
= (210) × (5/36)⁶ × (31/36)⁴
= 0.0599 (rounded to four decimal places)
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express the given quantity as a single logarithm. ln(a + b) + ln(a − b) − 9 ln c
Answers
The given quantity "ln(a + b) + ln(a − b) − 9 ln c" can be expressed as a single logarithm such that, ln[(a+b)(a-b)/c^9]
To express the given quantity as a single logarithm, you can use the properties of logarithms. For this expression: ln(a + b) + ln(a − b) - 9 ln c, you can apply the following steps:
1. Use the product rule: ln(x) + ln(y) = ln(xy)
ln(a + b) + ln(a − b) = ln((a + b)(a - b))
2. Use the power rule: ln(x^n) = n ln(x)
9 ln c = ln(c^9)
3. Use the quotient rule: ln(x) - ln(y) = ln(x/y)
ln((a + b)(a - b)) - ln(c^9) = ln(((a + b)(a - b))/c^9)
So, the given expression as a single logarithm is: ln(((a + b)(a - b))/c^9).
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Write the equation of the graph below.
Answers
Answer:
C . y = 2(x+1)^2 - 3 is the equation
Haematuria + frequency + dysuria what is the diagnosis and investigations?
Answers
To make a diagnosis, a healthcare provider may perform a physical exam, ask about the patient's medical history, and order diagnostic tests such as a urinalysis, urine culture, blood tests, or imaging studies (e.g. ultrasound, CT scan) to determine the underlying cause of the symptoms.
Treatment will depend on the underlying cause of the symptoms, but may include antibiotics for a UTI, pain medication, or other interventions as needed. It is important to seek medical attention promptly if you experience these symptoms to ensure that you receive appropriate treatment.
what is ( − 2)( + 2)
Answers
Answer: -4
There are rules, just follow the rules
a plane started on a flight at 9:30 a.m and arrived at its destination at 1:45pm. The plane used 51 gallons of gas. The number of gallons used per hour was
Brainlist
Answers
Answer:
12 gallons
Step-by-step explanation:
Given the following :
Total amount of gas used by aircraft = 51 gallons
Flight departure time = 9:30 a.m
Arrival time of flight = 1:45p.m
Number of hours traveled by flight:
Arrival time - Departure time
1:45pm - 9:30a.m = 4hrs 15 minutes = [(60*4)+15] minutes = 240 + 15 = 255 minutes
Amount of gallons used per hour :
255 minutes = 51 gallons
I minutes = x
255x = 51 gallons
x = 51 / 255
x = 0.2 gallons
0.2 gallons per minute
1 hour = 60 minutes
Number of gallons per hour = (0.2 * 60) = 12 gallons
BRAINLY IF CORRECT!!!! Emmie has $60 to spend on craft supplies. The tax rate is 8.25%. What is the maximum amount she can spend on craft supplies before taxes?
Answers
Answer:
$55.42
Step-by-step explanation:
Price of supplies = 100%
Percentage of tax = 8.25%
Total percentage = 108.25%
Amount allowed to spend = $60
108.25% = 60
60 / 108.25 = .55427
0.55427x100 = 55.427
Round down
$55.42
If my answer is incorrect, pls correct me!
If you like my answer and explanation, mark me as brainliest!
-Chetan K
a vertical line in the xy -plane travels from left to right along the base of the solid described in part (c). the vertical line is moving at a constant rate of 7 units per second. find the rate of change of the area of the cross section above the vertical line with respect to time when the vertical line is at position x
Answers
To find the rate of change of the area of a cross-section above a moving vertical line in the xy-plane, differentiate the area function with respect to time using the chain rule and substitute the known rate of change of the vertical line's position.
To find the rate of change of the area of the cross-section above the vertical line with respect to time, we need to differentiate the area function with respect to time.
Let's denote the area of the cross-section as A(x), where x represents the position of the vertical line along the x-axis. We want to find dA/dt, the rate of change of A with respect to time.
Since the vertical line is moving at a constant rate of 7 units per second, the rate of change of x with respect to time is dx/dt = 7 units/second.
Now, we can differentiate A(x) with respect to t using the chain rule:
dA/dt = dA/dx * dx/dt
The derivative dA/dx represents the rate of change of the area with respect to the position x. It can be found by differentiating the area function A(x) with respect to x.
Once you have the expression for dA/dx, you can substitute dx/dt = 7 units/second to calculate dA/dt, the rate of change of the area of the cross-section with respect to time when the vertical line is at position x.
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Consider the expression below.
(a + 4)(a + 9)
For (x + 4)(x + 9) to equal 0. either (x + 4) or (x + 9) must equal
The values of x that would result in the given expression being equal to o, in order from least to greatest, are
and
Answers
Answer:
x = - 9, x = - 4
Step-by-step explanation:
Set (x +)(x + 9) equal to zero, that is
(x + 4)(x + 9) = 0
Equate each factor to zero and solve for x
x + 4 = 0 ⇒ x = - 4
x + 9 = 0 ⇒ x = - 9
Thus
x = - 9, x = - 4
consider a symmetric matrix a. if the vector v is in the image of a and w is in the kernel of a, is v necessarily orthogonal to w?
Answers
Yes, the vector v is necessarily orthogonal to w if v is in the image of the symmetric matrix A and w is in the kernel of A.
Yes, the vector v is necessarily orthogonal to w if v is in the image of the symmetric matrix A and w is in the kernel of A. This is due to the fact that for any symmetric matrix A, the image (column space) and kernel (null space) are orthogonal subspaces. Since v is in the image of A and w is in the kernel of A, their dot product must be zero, indicating orthogonality.
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Please help me i cant figure it out im stuck
what goes in the question mark box?
Answers
Answer:
y = 4x - 31
Step-by-step explanation:
Parallel slope (m) = 4
(5, -11)
y - y1 = m(x - x1)
y + 11 = 4(x - 5)
y + 11 = 4x - 20
y = 4x - 31
13 EASY POINTS AND I WILL GIVE YOU BRAINLIEST WITH GOOD ANSWER: GEOMETRY: Which of the following can be assumed from the diagram?
A. Line n and line m are perpendicular
B. Line m intersects plane R at point A
C. Line n and line p intersect to form right angles.
D. Line m and plane R intersect at line m
E. Line m and line p will never intersect
Answers
Answer:
I think the answer is C.
Step-by-step explanation:
When line N and Line P intersect, and point A is formed, lines N and P create a right angle
(I could be wrong but this is what I think)
2. Umberto wants to give his waiter a 15% tip. The bill for dinner was $36.00. What is the
amount of the tip?
Answers
Answer:
$41.40
Step-by-step explanation:
36 divided by 100 is 0.36 so 1% is 0.36. times by 15 to get 15% is 5.4.
So the 15% tip is $5.40 so overall the price would be $41.40
Answer:
$5.40
Step-by-step explanation:
$36.00 is the amount that you need to find 15% of
First, you can do 36 x 0.15 = $5.40 (15% is 0.15 in decimal form)
The amount of the tip is $5.40
Lisa must choose a number between 61 and 107 that is a multiple of 2, 8, and 16. Write all the numbers that she could choose. If there is more than one
number, separate them with commas.
Answers
What is the simplest form of 324xy³2
?
Answers
Answer:
2² * 3⁴ * x * y³
Step-by-step explanation:
Simplify:First prime factorize 324
324 = 2 * 2 * 3 * 3 * 3 * 3
Now , write in exponent form:
324 = 2²*3⁴
324xy³ =2² * 3⁴ * x * y³
Pls help I’m having trouble
Answers
Answer:
find the area of each side of the shapes and add make sure it's in square centimeters
1st
what does this situation refers to?
Answers
f(x)=log5x what Is the range of the function
Answers
The range of the function f(x) = log5x is (-∞, +∞).The function f(x) = log5x represents the logarithm base 5 of x. To determine the range of this function, we need to consider the possible values that the logarithm can take.
The range of the logarithm function y = log5x consists of all real numbers. The logarithm function is defined for positive real numbers, and as x approaches 0 from the positive side, the logarithm approaches negative infinity. As x increases, the logarithm function approaches positive infinity.
The range of the function is the set of all possible output values. In this case, the range consists of all real numbers that can be obtained by evaluating the logarithm
log5(�)log 5 (x) for �>0 x>0.
Since the base of the logarithm is 5, the function log5x will take on all real values from negative infinity to positive infinity. Therefore, the range of the function f(x) = log5x is (-∞, +∞).
In other words, the function can output any real number, ranging from negative infinity to positive infinity. It does not have any restrictions on the possible values of its output.
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Answer: All real numbers
Step-by-step explanation:
Edge
Using the formula for interest, I = Prt, solve for r. Please answer this as soon as possible please and thank you
Answers
Answer:
A
Step-by-step explanation:
I = Prt = r * Pt
divide Pt from both side we get
I/(Pt) = r
Answer:
a
Step-by-step explanation:
I = Prt ( isolate r by dividing both sides by Pt )
\(\frac{I}{Pt}\) = r , that is
r = \(\frac{I}{Pt}\) → a
in a certain lottery, you must choose three numbers: any number between 1 and 10; any number between 1 and 20; and any number between 1 and 30. numbers may repeat and order matters (e.g., 5-5-5 is allowed; and 5-9-30 is different than 9-5-30). how many different lottery picks are there? enter as a whole number.
Answers
Answer: 6000
Step-by-step explanation:
For the numbers you choose, there are 10, then 20, then 30 possible numbers to choose from.
To find the total amount of possible combinations with repetition, you just do 10x20x30 = 6000.
a) a shopkeeper buys a camera for $300 and sells it at $360. Calculate the percentage profit.
b) A customer is buying 2 such cameras from the shopkeeper and he requests for a discount. if the shopkeeper wants to make a total profit of exactly $100 , find the percentage discount he should give the customer.
help me ;)
Answers
(Profit: 360-300=$60)
=60/300 x100
=20%
b) two cameras (original price): 300x2= $600
two cameras (price sold): 360x2 = $720
Profit without discount: 720-600= $120
120-100= $20 discount
20/720 x100 =2.78%
Answer:
a)answer:profit= selling price - cost price
=$360-$300=$60
percentage profit=profit*100%/cost price
=20%
Step-by-step explanation:
meaning of profit: this is the amount of money that one earns after selling his or her product, as you saw in the above correction, I took selling price subtracting the cost price, which means that the selling price is higher than the cost price.
formula: p= sp(selling price)- cp (cost price)
.when someone asks you the percentage profit, this means that the profit is calculated on the percentage, that's why I took the profit dividing by the cost price and then multiplying the 100, over percent. you may ask why divided by the cost price this is means that it is the total original price of the product.
graph the solution of this inequality 350 ≥ 125 + 15x
Answers
the line would be at x=15 and you would shade all to the left of the line and a point in that zone or on the line would work in the equation
The graph of the linear inequality 350 ≥ 125 + 15x is attached below.
How to graph linear inequality?To graph a linear inequality in two variables (say, x and y ), first get y alone on one side. Then consider the related equation obtained by changing the inequality sign to an equality sign. The graph of this equation is a line.
If the inequality is strict ( < or > ), graph a dashed line. If the inequality is not strict ( ≤ or ≥ ), graph a solid line.
Finally, pick one point that is not on either line ( (0,0) is usually the easiest) and decide whether these coordinates satisfy the inequality or not. If they do, shade the half-plane containing that point. If they don't, shade the other half-plane.
15x + 125 <= 350
15x <= 225
x <= 15
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please help me out here! Thank you! I’ll give brainliest!
Answers
Answer:for the first one it is 5/9
and for the second one it is 4/9
Step-by-step explanation:
Answer:
For first 0.56 and for second 0.44
Plz mark me brainliest
Step-by-step explanation: