Assume that the polynomial P_9(x) interpolates the function f (x) = e^-2x at the 10 evenly-spaced points x = 0, 1/9, 2/9, 3/9, ....., 8/9, 1. (a) Find an upper bound for the error |f (1/2) - P_9(1/2)|. (b) How many decimal places can you guarantee to be correct if P_9(1/2) is used to approximate e^-1?
Answers
a) In = 9 because P_9(x) interpolates the function f(x) using 10 evenly-spaced points.
b) The error bound is approximately 0.0028, we can guarantee that the approximation P_9(1/2) of e^(-1) is accurate to at least three decimal places.
(a) To find an upper bound for the error |f(1/2) - P_9(1/2)|, we use the error formula for Lagrange interpolation:
|f(x) - P_n(x)| <= M/((n+1)!)|ω(x)|,
where M is an upper bound for the (n+1)-th derivative of f(x) on the interval [a, b], ω(x) is the Vandermonde determinant, and n is the degree of the polynomial interpolation.
In this case, n = 9 because P_9(x) interpolates the function f(x) using 10 evenly-spaced points.
(a) To find an upper bound for the error at x = 1/2, we need to determine an upper bound for the (n+1)-th derivative of f(x) = e^(-2x). Since f(x) is an exponential function, its (n+1)-th derivative is itself with a negative sign and a coefficient of 2^(n+1). Therefore, we have:
d^10/dx^10 f(x) = -2^10e^(-2x),
and an upper bound for this derivative on the interval [0, 1] is M = 2^10.
Now we can calculate the Vandermonde determinant ω(x) for the given evenly-spaced points:
ω(x) = (x - x_0)(x - x_1)...(x - x_9),
where x_0 = 0, x_1 = 1/9, x_2 = 2/9, ..., x_9 = 1.
Using x = 1/2 in the Vandermonde determinant, we get:
ω(1/2) = (1/2 - 0)(1/2 - 1/9)(1/2 - 2/9)...(1/2 - 1) = 9!/10! = 1/10.
Substituting these values into the error formula, we have:
|f(1/2) - P_9(1/2)| <= (2^10)/(10!)|1/10|.
Simplifying further:
|f(1/2) - P_9(1/2)| <= (2^10)/(10! * 10).
(b) To determine the number of decimal places guaranteed to be correct when using P_9(1/2) to approximate e^(-1), we need to consider the error term in terms of significant figures.
Using the error bound calculated in part (a), we can rewrite it as:
|f(1/2) - P_9(1/2)| <= (2^10)/(10! * 10) ≈ 0.0028.
Since the error bound is approximately 0.0028, we can guarantee that the approximation P_9(1/2) of e^(-1) is accurate to at least three decimal places.
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Related Questions
Jocelyn has apples and apricots in a ratio of 250:15. How many apples does she have if she has 3 apricots?
HELP ASAPP
Answers
Answer:50
Step-by-step explanation:
15 : 5 = 3
250 : 5 = 3
You are interested in the average reading achievement test score of the currently enrolled students in Edison Elementary School. The average score of just Ms. Grady's class would be referred to as a:
Answers
The average score of just Ms. Grady's class would be referred to as a "class average" or "average score of Ms. Grady's class."
What is the term used to refer to the average score of just Ms. Grady's class?
The given paragraph discusses the concept of average reading achievement test scores at Edison Elementary School. In this context, the average score specifically pertaining to Ms. Grady's class would be referred to as a "class average" or "average score of Ms. Grady's class."
To calculate the average score of Ms. Grady's class, the individual reading achievement test scores of all the students in her class would be added together and divided by the total number of students in the class.
This provides a measure of central tendency that represents the typical performance of students in her class on the reading achievement test.
The class average is a useful metric for assessing the overall performance and progress of students in a specific class, as it provides a consolidated representation of the collective achievement levels within that particular group.
By comparing the class average to school-wide or district-wide averages, educators can gain insights into the relative performance of Ms. Grady's class and make informed decisions regarding instructional strategies and interventions to support students' reading abilities.
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What are the 3 circle formulas?
Answers
Area of a circle: A = πr² , Circumference of a circle: C = 2πr , Diameter of a circle: D = 2r
1. The area of a circle formula is A = πr², where A is the area of the circle, π is a constant equal to 3.14, and r is the radius of the circle. To calculate the area, simply multiply π by the radius of the circle squared.
2. The circumference of a circle formula is C = 2πr, where C is the circumference of the circle, π is a constant equal to 3.14, and r is the radius of the circle. To calculate the circumference, simply multiply 2π by the radius of the circle.
3. The diameter of a circle formula is D = 2r, where D is the diameter of the circle and r is the radius of the circle. To calculate the diameter of a circle, simply multiply 2 by the radius of the circle.
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12 miles below sea level
Answers
Answer:
12 miles below sea level
Step-by-step explanation:
Peter bought three Note 20 Android phones costing 1050 dollars each. If he had 4000 dollars in his checking account, write a numerical expression and then find his balance after he paid for the phones. Solution
Answers
Answer:
4000 - 3(1050)
= 850
Step-by-step explanation:
Start with the 4000 that's in the account. Then subtract 3(1050) which represents 3×1050, the cost of three phones.
4000 - 3(1050)
= 4000 - 3150
= 850
He has 850 left in his account.
|x+5| = 3
|x-5| = 3
Please help
NO LINKS
Answers
Answer:
the answer for |x+5|=3=x=-2and x=-8
and for the other one is x=8 and x=2
Which equations can be used to solve for y, the length of the room? Select three options.
y(y + 5) = 750
y2 – 5y = 750
750 – y(y – 5) = 0
y(y – 5) + 750 = 0
(y + 25)(y – 30) = 0
Answers
Ava pours 3 gallons of paint equally into 5 cans. How many gallons of paint will there be in each can?
Answers
Answer:
0.6
Step-by-step explanation:
first you well have a 3 and then dived it by 5 and after you dived them to you well get the answer 0.6
Each month Edna spends all her income on rent, on travel and on other living expenses She spends 1/3 of her income on rent. she spends 1/5 of her income on travel. She spends $420 of her income on other living expenses. Work out her income each month.
Answers
Answer:I think $1064
Step-by-step explanation:
First you multiply 1/5 by 420 and then add, so 1/5 x 420 = 84 & 420 + 84 = 504. Then you multiply 1/3 by 420 and also add, so 1/3 x 420 = 140 & 420 + 140 = 560. After you add both totals to get your final answer, so 504 + 560 = 1064. Her income each month is $1064.
The first approximation of 537 can be written a b' where the greatest common divisor of a and bis 1, with a= type your answer... b = = type your answer...
Answers
The first approximation of 537 can be written as a = 536 and b = 1, where the greatest common divisor of a and b is 1.
In number theory, the greatest common divisor (GCD) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. In this case, since a = 536 and b = 1, the GCD of a and b is 1. This means that 536 and 1 have no common factors other than 1.
The first approximation of 537 can be expressed as a product of these two numbers, a and b, where a represents the larger part of the approximation and b represents the smaller part. In this case, a is equal to 536, and b is equal to 1. Since the GCD of a and b is 1, it indicates that 536 and 1 have no common factors other than 1. This approximation may not be the most accurate, but it satisfies the condition of having a GCD of 1.
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Write an equivalent expression for -2/5x + (3e + 7)
Answers
Answer:
(3e+7) - (2/5)x
Step-by-step explanation:
Instead of the - 2/5x we can remove the - from it and convert the addition problem into a subtraction problem and they will be equivalent.
|x+3| if x>5 pls hepl fasttt
Answers
8 if you mean x=5
but if you meant greater than well x could be 6 or higher
Algebraic Identity Formulas!
Answers
Answer:
The algebraic equations is
-b±√b²-4ac/2a
Monica hiked west from her campsite for 24 miles and then hiked north for 7 miles, as shown in the diagram.
What is the shortest distance Monica can hike to return to the campsite?
A. 25 miles
A. 25 miles,
B. 17 miles
B. , , 17 miles, , ,
C. 31 miles
C. , , 31 miles, , , ,
D. 41 miles
Answers
Answer: A. 25 miles
Step-by-step explanation: I used Pythagorean’s Theorem.
In the ale the original price are reduced by 15%
a. Calculate the ale price of a book that ha an original price of $12
b. Calculate the original price of a jacket that ha a ale price of $38. 25
Answers
Answer:b
Step-by-step explanation:
True or False? The point at which a tangent line meets a circle is called the point of tangency. O A. True O B. False.
Answers
True, The point at which a tangent line meets a circle is called the point of tangency.
What is Line segment?
Line segment is a part of the line which have two endpoints and bounded by two distinct end points and contain every point on the line which is between its endpoint.
Given that;
To check the point at which a tangent line meets a circle is called the point of tangency.
Now, We know that;
A line which touches the circle at a single point is known as a tangent to a circle.
And, The point where tangent meets the circle is called point of tangency.
Thus, The given statement is true.
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Please answer this question now in two minutes
Answers
Answer:
m∠C = 102°
Step-by-step explanation:
This diagram is a Quadrilateral inscribed in a circle
The first step is to determine what m∠B
is
The sum of opposite angles in an inscribed quadrilateral is equal to 180°
m∠D + m∠B = 180°
m∠B = 180° - m∠D
m∠B = 180° - 80°
m∠B = 100°
Second step is we proceed to determine the exterior angles of the circle
m∠ADC = 2 × m∠B
m∠ADC = 2 × 100°
m∠ADC = 200°
m∠ADC = m∠CD + m∠AD
m∠AD = m∠ADC - m∠CD
m∠AD = 200° - 116°
m∠AD = 84°
The third step is to determine m∠BAD
m∠BAD = m∠AD + m∠AB
m∠BAD = 84° + 120°
m∠BAD = 204°
The final step Is to determine what m∠C is
It is important to note that:
m∠BAD is Opposite m∠C
Hence
m∠C = 1/2 × m∠BAD
m∠C = 1/2 × 204
m∠C = 102°
ANSWER ALL FOR BRAINLIEST !!!
Answers
B. X<8 due to multiplication of 5 fraction needs to be less than 1(7,6,5,4,3,2,1,0,-1...)
C. X=2 since 2/x needs to equal to 1 you need the denominator to be 2 as well when mutiplying
Answer:
A. 4 (any number over 3)
B. 1
C. 2
Step-by-step explanation:
A. It needs to equal more than 11, so the fraction would have to be greater than 1.
B. 5*1/8 = 0.625
C. Fraction has to equal 1, 2/2 is equal to 1.
What is the equation of the line that passes through the point (4, -8) and has
a slope of -3/4
Answers
Answer:
y = (-3/4)x - 5
Step-by-step explanation:
m = (-3/4); (4, -8)
(x₁, y₁)
y - y₁ = m(x - x₁)
y - (-8) = (-3/4)(x - 4)
y + 8 = (-3/4)x + 3
-8 -8
-------------------------
y = (-3/4)x - 5
I hope this helps!
Answer: y + 8 = -3/4(x - 4)
Explanation:
Point slope form formula: Y – y1 = m (x – x1)
m = the slope
y1 equals the y of the point
x1 equals the x of the point
Plug in the numbers and you have your equation
Y – y1 = m (x – x1) -----> y - (-8) = -3/4(x - 4)
Simplify:
y - (-8) = -3/4(x - 4) ------> y + 8 = -3/4(x - 4)
Your final answer: y + 8 = -3/4(x - 4)
-----------------------------------
If your answer requires slope intercept, here is your answer:
y = -3/4x - 5
Charlotte must have at least 320 points in her science class to get a B or better. She needs at least a B so she can play on the softball team. Charlotte currently has 168 points from four tests and 79 points from her quizzes. Charlotte will be taking her final exam worth 100
points. What must Charlotte score on the final exam to reach her goal? (This is a inequality connection btw)
Answers
Answer:
73
Step-by-step explanation:
First, you want to add 168 +79 = 247
Then, to see how many she needs, you subtract 320 - 247 = 73
What is the slope of the equation? y=2/3x+4
Answers
Answer:
Slope m = 2/3
General Formulas and Concepts:
Algebra I
Slope-Intercept Form: y = mx + b
m - slope b - y-interceptStep-by-step explanation:
Step 1: Define
y = 2/3x + 4
Step 2: Break Function
Identify Parts.
Slope m = 2/3
y-intercept b = 4
Malik wants to make a cubic frame using a piece of wire with a length of 150cm.If the volume of the cube is 2197 cm cube , is the length of the wire sufficient? Give a reason for your answer
Answers
Answer:
Yes, the wire is sufficient for making a cube of volume 2197 cm³
Step-by-step explanation:
Volume of cube = L³
Volume of the cube = 2197 cm³
Length of the wire = 150 cm
Let's see if the wire is sufficient for making a cube.
Volume of the cube that could be made from the wire is:
V = (L)³
V = (150)³
V = 3,375,000 cm³
So,
The volume of the cube using a 150 cm length wire is much more grater than 2197 cm³.
That is why, the wire is sufficient for making a cube of volume 2197 cm³.
PLEASE HELP!! Identify the vertex, axis of symmetry, and min/max value of each and show your work.
f(x)=x^2 - 18x + 86
Answers
The quadratic function in this problem is defined as follows:
f(x) = x² - 18x + 86.
The coefficients are given as follows:
a = 1, b = -18 and c = 86.
Hence the axis of symmetry, representing the x-coordinate of the vertex, is given as follows:
x = -b/2a
x = 18/2
x = 9.
The coefficient a is positive, hence the function has a minimum value, which is given as follows:
f(9) = 9² - 18(9) + 86
f(9) = 5.
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PLZZZ help 30 points
Answers
Answer:
20*3 1/5
3 1/5= 15+1=16
16/3*20
320/3=
106 1/3
Answer:
64 is ur answer
Step-by-step explanation:
3 1/5 is equal to 16/5
20/1 x 16/5
320/5 = 64
What is the solution set for 5x−5=15, given the replacement set {0, 1, 2, 3, 4}?
x = 4
x = 3
x = 2
x = 1
x = 0
Answers
Answer:
x=4
Step-by-step explanation:
Answer:
x=4
Step-by-step explanation:
Replace X with the numbers given and see which ones work
Twenty seven percent of the employees in a company have managerial positions, and 5 percent of the employees in the company have MBA degrees. Also, 59 percent of the managers have MBA degrees. Using the probability formulas, find the proportion of employees who either have MBAs or are managers.
Answers
Answer:
Step-by-step explanation:
Given that :
27% employees managerial position
5% Employee have MBA
59% of managers have MBA
Proportion of employees who either have MBA or are managers :
P(either manager or MBA) = P(Manager U MBA)
= P(manager) + P(MBA) - [P(manager n MBA) * P(Manger)]
P(manager ) = 0.27
P(MBA) = 0.05
P(manager n MBA) * P(Manger) = 0.59 * 0.27 = 0.1593
P(manager) + P(MBA) - [P(manager n MBA) * P(Manger)]
= 0.27 + 0.05 - 0.1593
= 0.1607
pls help if you can asap!!!!
Answers
Answer: x= 6
Step-by-step explanation:
Since the shape is a parallelogram, the angles will either be equal to each other or add up to 180.
You can see they do not look the same so they add up to equal 180
12x + 3 +105 = 180
12x + 108 = 180
12x = 72
x = 6
) how many third-order partial derivatives are there for a function f(x,y)? what are they? why do we have that many?
Answers
There are 8 third-order partial derivatives for a function f(x, y). They are:
1. ∂³f/∂x³ - third derivative with respect to x
2. ∂³f/∂x²∂y - mixed derivative with two x's and one y
3. ∂³f/∂x∂y² - mixed derivative with one x and two y's
4. ∂³f/∂y³ - third derivative with respect to y
5. ∂³f/∂x∂y∂x - mixed derivative with x, y, and x
6. ∂³f/∂y∂x∂y - mixed derivative with y, x, and y
7. ∂³f/∂x∂y∂y - mixed derivative with x, y, and y (equivalent to 3)
8. ∂³f/∂y∂x∂x - mixed derivative with y, x, and x (equivalent to 2)
We have 8 third-order partial derivatives because there are two variables (x and y), and we can take the derivative up to three times with respect to either variable or a combination of both. This results in different ways to distribute the three derivative operations among the two variables (x and y), leading to the 8 combinations mentioned above. Note that some of these derivatives are equivalent due to the symmetry of mixed derivatives when the function f(x, y) has continuous second-order partial derivatives.
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who envented 0 name the person
Answers
Answer:
Brahmagupta invented the number 0
Step-by-step explanation:
Answer:
Brahmagupta
Step-by-step explanation:
Can you please help me?
Answers
Answer:
-3, -4, -4, 0, 16, 64, 192.
Step-by-step explanation:
I don't know for sure if this is correct, but when i did it theses are the answers i got. You do y=8*2^x. Meaning you substitute the x for the x values such as -3, -2, and -1.
Also if you have one there's a setting you can go to on an actual calculator and type in this on a Y/X chart. I didn't use one this time because i don't have one, but thought you'd find that information useful.