Using Simpson's [tex]\frac{1}{3}[/tex] rule with 6 strips, the approximate value of the integral ∫[2.0, 2.8] f(x) dx is -3.8492.
To integrate the function [tex]\begin{equation}y = f(x) = \frac{ax^2}{b + x^2}[/tex] using Simpson's 1/3 rule, we need to divide the interval [2.0, 2.8] into an even number of strips (in this case, 6 strips). The formula for approximating the integral using Simpson's 1/3 rule is as follows:
[tex]\begin{equation}\int_a^b f(x) dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right][/tex]
Where:
h is the width of each strip ([tex]\begin{equation}h = \frac{b - a}{n}[/tex], where n is the number of strips)
[tex]x_0[/tex] is the lower limit (2.0)
[tex]x_n[/tex] is the upper limit (2.8)
f(xi) represents the function evaluated at each strip's midpoint
Given the values of a = 1.2 and b = -0.587, we can proceed with the calculations.
Step 1: Calculate the width of each strip (h):
[tex]\begin{equation}h = \frac{b - a}{n} = \frac{-0.587 - 1.2}{6} = \frac{-1.787}{6} \approx -0.2978[/tex]
Step 2: Calculate the function values at each strip's midpoint:
x₀ = 2.0
x₁ = x₀ + h = 2.0 + (-0.2978) = 1.7022
x₂ = x₁ + h = 1.7022 + (-0.2978) = 1.4044
x₃ = x₂ + h = 1.4044 + (-0.2978) = 1.1066
x₄ = x₃ + h = 1.1066 + (-0.2978) = 0.8088
x₅ = x₄ + h = 0.8088 + (-0.2978) = 0.511
x₆ = x₅ + h = 0.511 + (-0.2978) = 0.2132
xₙ = 2.8
Step 3: Evaluate the function at each midpoint:
[tex]f(x_0) = \frac{1.2 \times 2^2}{-0.587 + 2^2} = \frac{4.8}{3.413} \approx 1.406 \\\\f(x_1) = \frac{1.2 \times 1.7022^2}{-0.587 + 1.7022^2} \approx 2.445 \\\\f(x_2) = \frac{1.2 \times 1.4044^2}{-0.587 + 1.4044^2} \approx 2.784 \\\\f(x_3) = \frac{1.2 \times 1.1066^2}{-0.587 + 1.1066^2} \approx 2.853 \\\\[/tex]
[tex]f(x_4) = \frac{1.2 \times 0.8088^2}{-0.587 + 0.8088^2} \approx 2.455 \\f(x_5) = \frac{1.2 \times 0.511^2}{-0.587 + 0.511^2} \approx 1.316 \\f(x_6) = \frac{1.2 \times 0.2132^2}{-0.587 + 0.2132^2} \approx 0.29[/tex]
Step 4: Apply Simpson's 1/3 rule formula:
[tex]\begin{equation}\int_{2.0}^{2.8} f(x) dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + f(x_6) \right][/tex]
[tex]\begin{equation}\approx \frac{-0.2978}{3} \left[ 1.406 + 4(2.445) + 2(2.784) + 4(2.853) + 2(2.455) + 4(1.316) + 0.29 \right][/tex]
[tex]\begin{equation}= \frac{-0.2978}{3} \left[ 1.406 + 9.78 + 5.568 + 11.412 + 4.91 + 5.264 + 0.29 \right][/tex]
≈ (-0.09926) * 38.63
≈ -3.8492
Therefore, the approximate value of the integral ∫[2.0, 2.8] f(x) dx using Simpson's 1/3 rule with 6 strips is approximately -3.8492.
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The USA Olympic Synchronized Swimming Team is designing a routine for their upcoming competition. From the center of the pool, they moved 2 feet to the right and 4 feet up to create the center of their formation (Point
C). From the center of their formation, they then formed a circle that goes through a point 3 feet to the left and 4 feet up (Point D). What is the equation of the circle?
Select the correct answer chorice below.
(x _ _)^2 _ (y_ _)^2 = _
[tex]\left(x-2\right)^{2}+\left(y-4\right)^{2}=25[/tex]
Hope this helps!
The equation of circle is [tex](x-2)^{2}+(y-4)^{2} =25[/tex]
Equation of circle:The equation of circle is given as,
[tex](x-h)^{2}+(y-k)^{2} =r^{2}[/tex]
Where [tex](h,k)[/tex] is the coordinate of center and r is radius.
From the given figure,
It is observed that, the center of circular pool is (2, 4)
and radius is 5.
substitute the value of center and radius in above equation.
[tex](x-2)^{2}+(y-4)^{2} =5^{2}\\\\(x-2)^{2}+(y-4)^{2} =25[/tex]
The equation of circle is [tex](x-2)^{2}+(y-4)^{2} =25[/tex]
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Ariana has 99 oranges. She has to pack 9 boxes with an equal number of oranges. How many oranges should she pack in each box?
Answer:
11 oranges in each box
Step-by-step explanation:
99/9=11
Answer: 11 oranges in each box.
Step-by-step explanation:
Division is used to evenly separate totals into groups (total/#of groups= Quotient per group).
In this case, we divide the total, 99 oranges, by the amount of groups we want, which is 9 boxes, in order to find an even amount for each box from the total oranges.
So, 99/9= 11 oranges per box
Write a quadratic equation given the x intercepts and other other point. Put steps together. Find the factors. Solve for a by substituting in the extra point. Write the equation in factored form.
Answer:
This question is clearly incomplete, so i will answer it in a really general way.
Suppose that for a quadratic function, we know that the x-intercepts are a and b.
And we also know that this function passes through the point (c, d).
First a definition, for a n-degree polynomial with the x-intercepts {x₁, x₂, ...,xₙ} and a leading coefficient K, we can write this polynomial in the factorized form as:
p(x) = K*(x - x₁)*(x - x₂)*...*(x - xₙ)
Now let's do the same for our quadratic function, we can write it as:
f(x) = K*(x - a)*(x - b)
(where a and b are known numbers)
Now we also know that this function passes through the point (c, d)
This means that:
f(c) = d
then:
d = K*(c - a)*(c - b)
With this equation we can find the value of K,
K = d/( (c-a)*(c - b))
Then the quadratic function is:
[tex]f(x) = d\frac{(x-a)}{(c-a)} \frac{(x-b)}{(c-b)}[/tex]
Where again, it is supposed that you know the values of a and b, and also the point (c, d)
Tickets to a hockey game cost $45. You and 3 of your friends decide to go together. how much will your tickets cost all together?
Answer:
135
Step-by-step explanation:
Well 3 friends and 45 dollars each
45*3=135
Five Number Summary for Percent Obese by State Computer output giving descriptive statistics for the percent of the population that is obese for each of the SOUS states, from the USStates dataset, is given in the table below. Variable Mean StDev Minimum Q Median Qs Maximum Obese 50 31.43 3.82 23.0 28.6 30.9 34.4 39.5 N Click here for the dataset associated with this question, (a) What is the five number summary? The five number summary is :
The five number summary for the percent of the population that is obese among the SOUS states are Minimum: 23.0, First Quartile (Q1): 28.6, Median: 30.9, Third Quartile (Q3): 34.4, Maximum: 39.5
The five number summary provides a concise summary of the distribution of a dataset, consisting of five key values: minimum, first quartile (Q1), median, third quartile (Q3), and maximum. Let's explain each part using the given information:
Minimum: The minimum value represents the smallest observed value in the dataset. In this case, the minimum value is 23.0. It indicates that the lowest recorded percentage of obesity among the SOUS states is 23.0%.
First Quartile (Q1): The first quartile is the value that divides the dataset into the lower 25% of the data. It represents the 25th percentile of the data. In the table, the first quartile (Q1) is given as 28.6. This means that 25% of the SOUS states have a percentage of obesity lower than or equal to 28.6%.
Median: The median, also known as the second quartile or the 50th percentile, is the middle value of the dataset when it is sorted in ascending order. It represents the point that splits the data into two equal halves. In the table, the median is given as 30.9. This implies that 50% of the SOUS states have a percentage of obesity lower than or equal to 30.9%.
Third Quartile (Q3): The third quartile is the value that divides the dataset into the upper 25% of the data. It represents the 75th percentile of the data. In the table, the third quartile (Q3) is provided as 34.4. This means that 75% of the SOUS states have a percentage of obesity lower than or equal to 34.4%.
Maximum: The maximum value represents the largest observed value in the dataset. In this case, the maximum value is 39.5. It indicates that the highest recorded percentage of obesity among the SOUS states is 39.5%.
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p(a) = 4a +4
What is the coefficient of a?
[tex] \huge \red {Question}[/tex]
p(a) = 4a +4
What is the coefficient of a?
[tex] \huge \red{Answer}[/tex]
4 is the coefficient of a .
Step-by-step explanation:
4 is the coefficient of a .
Someone help please
Answer:
it is 1000x+2000
Step-by-step explanation:
In P2, find the change-of-coordinates matrix from the basis B={1−2t+t2,3−5t+4t2,2t+3t2} to the standard basis C={1,t,t2} Then find the B coordinate vector for −1+2t I know how to do the first part. P from B to C:⎡⎣⎢1−213−54023⎤⎦⎥
I do not know what the process is for finding the B coordinate vector though. Can someone give me a place to start for doing that?
The B coordinate vector [tex]-1+2t[/tex] is [tex][5, -8, 5].[/tex]
What is a Coordinate vector?
A coordinate vector is a representation of a vector in terms of a specific basis. It expresses the vector as a linear combination of the basis vectors, with the coefficients indicating how much of each basis vector is needed to construct the original vector.
In linear algebra, given a vector space V with a basis B = {v₁, v₂, ..., vₙ}, a vector v in V can be written as v = c₁v₁ + c₂v₂ + ... + cₙvₙ, where c₁, c₂, ..., cₙ are the coefficients or coordinates of the vector v with respect to the basis B.
To find the coordinate vector of a given vector in the basis B, we can follow these steps:
Write the vector in terms of the basis B. In this case, we have the vector [tex]-1+2t.\\-1+2t = (-1) * (1-2t+t^2) + 2 * (3-5t+4t^2) + 0 * (2t+3t^2) = -1 + 2t - t^2 + 6 - 10t + 8t^2 + 0t + 0t^2 = 5t^2 - 8t + 5[/tex]
Express the vector obtained in step 1 as a linear combination of the basis vectors [tex]C={1,t,t^2}.[/tex] This will give us the coordinate vector.
[tex]5t^2 - 8t + 5 = a * 1 + b * t + c * t^2[/tex]
Equating the coefficients of corresponding powers of t on both sides, we have:
[tex]a = 5\\b = -8\\c = 5[/tex]
So, the coordinate vector of [tex]-1+2t[/tex] in the basis
[tex]B={1-2t+t^2,3-5t+4t^2,2t+3t^2}[/tex] is [tex][a, b, c] = [5, -8, 5].[/tex]
Therefore, the B coordinate vector [tex]-1+2t[/tex] is [tex][5, -8, 5].[/tex]
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Jasmine had 20 dollars to spend on 3 gifts. She spent 9 1 4 dollars on gift A and 4 4 5 dollars on gift B. How much money did she have left for gift C
Answer:
5 2/6
Step-by-step explanation:
you have to mutiply
Which of the following could be used to calculate the total surface area of the figure?
Answer:b
Step-by-step explanation:
Does anyone mind helping me im confused-
Answer:
Malcolm owes the greatest amount.
Step-by-step explanation:
The question is asking for the amount owed by each person. That just means you take the absolute value of their bank balance if their bank balance is negative. Absolute value is just how far away the number is from zero (basically when a number is negative, you're just making it positive).
Sophia owes $150, Malcolm owes $325, and Oren owes $275.
325 > 275 > 150
Thus Malcolm owes the most, since his amount owed is the highest.
10-3(x+9-10x)
Whats the answer
plsssssssssssssssssssssssssssssss help!!
Answer:
Look for 75% of 1 1/2, I got 1.125 or 1 1/8
Step-by-step explanation:
The sum of two numbers is 65. One number is 4 times as large as the other. What are the numbers?
Larger number:
Smaller number:
Answer:
Large number =52
small number =13
Step-by-step explanation:
4x + x =65
5x=65
x=13
4x=52
consider trapezoid lmno. what information would verify that lmno is an isosceles trapezoid? check all that apply.
a. LN ≅ MO
b. LN ≅ ON
c. LO ≅ MN
d. ∠l ≅ ∠n
e. ∠l ≅ ∠m
An isosceles trapezoid LMNO has the side LO is congruent to side MN, the diagonal LN is congruent to diagonal MO, and the angle L is congruent to angle M. Hence correct options are a), c), and e)
Given :
Trapezoid LMNO.
The following are the conditions that show any trapezoid is an isosceles trapezoid:
Condition 1 -- Both the legs are of the same length.
Condition 2 -- The base angles are of the same measure.
Condition 3 -- Diagonals are of the same length.
So, the given trapezoid LMNO is an isosceles trapezoid when:
The side LO is congruent to side MN.
The diagonal LN is congruent to diagonal MO.
The angle L is congruent to angle M.
Therefore, the correct option is a), c), and e).
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what is the equation to 9x+3° 84°
Answer:
the solution to this equation is x = 9°
Step-by-step explanation:
9x + 3 = 84
9x = 84 - 3
9x = 81
x = 81 / 9
x = 9°
write an equation of the line passing through the point $\left(5,\ -3\right)$ that is parallel to the line $y=x 2$ .
Given point: (5, -3)Given equation of the line: y = x + 2We are supposed to find the equation of the line passing through the point (5, -3) that is parallel to the line y = x + 2.
First, we need to find the slope of the given line y = x + 2. Here, the slope is 1 as the coefficient of x is 1.Now, a line parallel to this line will also have the same slope. Therefore, the slope of the required line is also 1.Now we have the slope and the point (5, -3) that the line passes through. Using the point-slope form of the equation of a line, we can find the equation of the line that passes through the given point and has the given slope.So, the equation of the line passing through the point (5, -3) that is parallel to the line y = x + 2 is:y - (-3) = 1(x - 5)This can be simplified to obtain the equation in the slope-intercept form:y = x - 8Thus, the equation of the line is y = x - 8.
To find the equation of a line parallel to the line y = x^2 and passing through the point (5, -3), we need to determine the slope of the given line and then use it to construct the equation.
The slope of the line y = x^2 can be determined by taking the derivative of the equation with respect to x. In this case, the derivative is:
dy/dx = 2x
Since the derivative represents the slope of the original line, we know that the slope of the line y = x^2 is 2x. To find the slope of the parallel line, we use the fact that parallel lines have the same slope.
Therefore, the slope of the parallel line is also 2x.
Now, using the point-slope form of a linear equation, we can write the equation of the parallel line:
y - y1 = m(x - x1)
where (x1, y1) is the given point (5, -3) and m is the slope.
Plugging in the values, we have:
y - (-3) = 2x(x - 5)
Simplifying further:
y + 3 = 2x^2 - 10x
Rearranging the equation to the standard form:
2x^2 - 10x - y - 3 = 0
So, the equation of the line passing through the point (5, -3) and parallel to the line y = x^2 is 2x^2 - 10x - y - 3 = 0.
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We have to find the equation of the line passing through the point [tex]$(5,-3)$[/tex] that is parallel to the line [tex]$y=x+2$[/tex].
Therefore, the equation of the line passing through the point [tex]$(5,-3)$[/tex] and parallel to the line [tex]$y=x+2$[/tex] is:
[tex]$y=x-8$[/tex].
As we know, the parallel lines have the same slope. Therefore, the slope of the line passing through the point (5,-3) will be the same as the slope of the line y=x+2.
Thus, we can write the slope-intercept form of the equation of the line y = mx + b as follows:
y = mx + b ------(1)
Here, m is slope of the line, b is y-intercept of the line. For the line y=x+2, slope of the line is:
m=1
Now, we will find the value of b for the line y = mx + b passing through the point (5,-3).
[tex]$$-3=1\times5+b$$$$[/tex]
[tex]b=-8$$[/tex]
Therefore, the equation of the line passing through the point (5,-3) and parallel to the line y=x+2 is:
y=x-8
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is it possible to find a power series whose interval of convergence is ? explain. answer no
Yes, it is possible to find a power series whose interval of convergence is [0, ∞).
The interval of convergence for a power series is determined by the behavior of the series as the variable approaches different values. The interval can be open, closed, half-open, or infinite.
In the case of [0, ∞), it represents a right-half open interval, indicating that the power series converges for all values of the variable greater than or equal to 0.
To find such a power series, we need to consider the conditions for convergence. The most common tests for convergence of power series are the ratio test and the root test. If the series satisfies the conditions of either test, it will converge within a specific interval.
For a power series to have an interval of convergence of [0, ∞), the coefficients in the series must satisfy certain conditions, such as convergence of the series for x = 0 and divergence for x > 0. This can be achieved by selecting appropriate coefficients and constructing a power series that converges for x = 0 and diverges for x > 0.
In summary, it is indeed possible to find a power series whose interval of convergence is [0, ∞) by carefully selecting the coefficients to meet the convergence conditions for this interval.
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Complete question is:
Is it possible to find a power series whose interval of convergence is [0, ∞)? Explain.
let and be two integers with 0≤<≤100 . suppose you approximate (≤100≤) by ∑=−11! . what is the largest possible error you could make?
The largest possible error you could make in the given approximation is [tex]1 / (10^11 * 11!)[/tex]
To approximate the value of [tex]e^x[/tex] using the series expansion, we can use the formula:
[tex]e^x[/tex] ≈ ∑ [tex](x^n)/n![/tex]
In this case, we have:
x = -1/10
To find the largest possible error in the approximation, we can consider the next term in the series that we are neglecting:
1 / (10^11 * 11!) = [tex]|x^(n+1) / (n+1)!|[/tex]
For the given approximation, n = 10 (since we are using terms up to n = 10).
Substituting the values, we have:
[tex]|Error|[/tex] = [tex]|(-1/10)^(10+1) / (10+1)!|[/tex]
|Error| =[tex]|(-1/10)^11 / 11!|[/tex]
|Error| =[tex]1 / (10^11 * 11!)[/tex]
Since the value of n is fixed at 10, the largest possible error occurs when x is at its maximum value within the given range (0 ≤ x ≤ 100).
In this case, the maximum value of |Error| would be obtained by using the maximum value of x = 100 in the formula.
|Error| = [tex]1 / (10^11 * 11!)[/tex]
Therefore, the largest possible error you could make in the given approximation is [tex]1 / (10^11 * 11!)[/tex].
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Draw the image of ABC under a dilation whose center is P and scale factor is 1/3
Answer:
there
Step-by-step explanation:
Answer:
khan
Step-by-step explanation:
Please hurry I need this now !!!
1.What is the positive solution to this equation?
x2 – 12 = -11x
a. -12
b. 12
C. -1
d. 1
Answer:
b. 12
Step-by-step explanation:
Answer:
either a or d because they are both correct.
Step-by-step explanation:
y=-4/3x+6; y=2
I need that to be done in subsitution or elimination
Rewrite equations:
y=2;y=
−4
3
x+6
Step: Solvey=2for y:
y=2
Step: Substitute2foryiny=
−4
3
x+6:
y=
−4
3
x+6
2=
−4
3
x+6
2+
4
3
x=
−4
3
x+6+
4
3
x(Add 4/3x to both sides)
4
3
x+2=6
4
3
x+2+−2=6+−2(Add -2 to both sides)
4
3
x=4
4
3
x
4
3
=
4
4
3
(Divide both sides by 4/3)
x=3
Answer:
x=3 and y=2
8. Somebody said the answer is letter A. I Don't Know the answer
Which properties of equality justify steps c and f?
A. Addition Property of Equality; Division Property of Equality
B. Multiplication Property of Equality; Division Property of Equality
C. Subtraction Property of Equality; Multiplication Property of Equality
D. Addition Property of Equality; Subtraction Property of Equality
Answer:
D. Addition Property of Equality; Subtraction Property of Equality
Step-by-step explanation:
Answer:
Option C and B
Step-by-step explanation:
In the question step C is 23 + 11 = -11 +(-4x) + 11
which is in the form of a + b = c + a
In step C we have added 11 on both the sides to eliminate 11 from right side of the equation.
property which signifies this step is
Addition property of equality :
In step 'f' expression is
\frac{34}{-4}=\frac{-4x}{-4}
In this step equation has been divided by -4 on both the sides to eliminate 4 from the numerator.
In this step division property of equality has been applied.
Therefore Option C and B are the correct options.
Find the length of side x in simplest radical form with a rational denominator.
Answer:
since it is Right angled isosceles triangle it's base side are equal
by using Pythagoras law
x²+x²=1²
2x²=1
x=√{1/2}or 0.707 or 0.71
2 people A and B travel from x and y talong different routes.Their journeys take the same amount of time
Question:
2 people A and B travel from X to Y along different routes. Their journeys take the same amount of time. B's route is 100km at an average speed of 40km/hour A's route is 60km. What is A's average speed?
Answer:
Person A's average speed is 24km/hr
Step-by-step explanation:
Given
Person B
[tex]Distance = 100km[/tex]
[tex]Speed = 40km/hr[/tex]
Person A
[tex]Distance = 60km[/tex]
Required
Average speed of person A
Speed is calculated as:
[tex]Speed = \frac{Distance}{Time}[/tex]
For person B
[tex]40 = \frac{100}{Time}[/tex]
Make Time the subject
[tex]Time= \frac{100}{40}[/tex]
[tex]Time= 2.5hr[/tex]
For person A
[tex]Speed = \frac{Distance}{Time}[/tex]
[tex]Speed = \frac{60}{Time}[/tex]
The journeys last for the same duration.
So;
[tex]Speed = \frac{60}{2.5}[/tex]
[tex]Speed = 24[/tex]
Person A's average speed is 24km/hr
I need help with the Pythagorean Theorem
Answer:
8.4 I think
Step-by-step explanation:
Please don't hate me if I am wrong
help pleaseeee !!! will b marked brainliest!!!!!!
Answer:
I think it's the first one
Answer:
to my knowledge I would pick A
Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
an = ln(8n² + 9) − ln(n² + 9)
The limit of the sequence as n approaches infinity is ln(8). The sequence converges, and the limit is ln(8).
To determine whether the sequence given by an = ln(8n² + 9) − ln(n² + 9) converges or diverges, we can examine the behavior of the terms as n approaches infinity.
Taking the limit as n approaches infinity, we have:
lim(n→∞) ln(8n² + 9) − ln(n² + 9)
We can simplify this expression using logarithmic properties. The natural logarithm of a quotient is equal to the difference of the logarithms:
= lim(n→∞) ln[(8n² + 9)/(n² + 9)]
Now, let's analyze the behavior of the numerator and denominator as n approaches infinity:
As n becomes larger and larger, the higher-order terms dominate. In this case, the leading term in both the numerator and denominator is n².
In the numerator, 8n² dominates, and in the denominator, n² dominates. Therefore, as n approaches infinity, the ratio (8n² + 9)/(n² + 9) approaches 8.
Taking the natural logarithm of 8, we have ln(8).
Therefore, the limit of the sequence as n approaches infinity is ln(8).
Hence, the sequence converges, and the limit is ln(8).
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The sides of the base of a right square pyramid are 5 centimeters, and the slant height is 8 centimeters. If the sides of the base and the slant height are each multiplied by 5, by what factor is the surface area multiplied?
A. 5 to 0
B. 5 to 1
C. 5 to 2
D. 5 to 3
Answer:
A
Step-by-step explanation:
The required surface area is multiplied by a factor of 25 or 5 to 2. Option C is correct.
What is surface area?The surface area of any shape is the area of the shape that is faced or the Surface area is the amount of area covering the exterior of a 3D shape.
Here,
The lateral surface area of a pyramid is given by the formula:
L = (1/2) * P * l
where P is the perimeter of the base and l is the slant height.
In this case, the perimeter of the base is 5 * 5 = 25 centimeters.
So the lateral surface area of the original pyramid is
L = (1/2) * 5 * 8 = 20 square centimeters.
Total area = 4[20] + 25 = 105
When the sides of the base and the slant height are each multiplied by 5, the new perimeter of the base is 5 * 5 * 5 = 125 centimeters and the new slant height is 8 * 5 = 40 centimeters.
So the lateral surface area of the new pyramid is
L = (1/2) * 25 * 40 = 500 square centimeters.
Total area = 4{500} + 125 = 2125
Therefore, the surface area is multiplied by a factor of 2125/105 = 25.
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The approximation of I = *(x – 3)ex* dx by composite Trapezoidal rule with n= 4 is: -25.8387 15.4505 -5.1941 4.7846
The approximation of the integral ∫(x – 3)ex dx by the composite Trapezoidal rule with n = 4 is approximately: -5.1941.
To approximate the integral ∫(x – 3)ex dx using the composite Trapezoidal rule with n = 4, we divide the interval [a, b] into n subintervals of equal width. In this case, we don't have the limits of integration provided, so we'll assume the interval to be [a, b] = [a, a+4] for simplicity.
Let's denote h as the width of each subinterval, given by
[tex]h = (b - a) / n \\= 4 / 4 = 1[/tex]
Using the composite Trapezoidal rule formula, the approximation is given by:
[tex]Approximation = h/2 * [f(a) + 2*f(a + h) + 2*f(a + 2h) + ... + 2*f(a + (n-1)h) + f(b)][/tex]
Now, let's calculate the values of the function at each interval endpoint:
[tex]f(a) = (a - 3)*e^a\\f(a + h) = (a + h - 3)*e^{a + h}\\f(a + 2h) = (a + 2h - 3)*e^{a + 2h}\\f(a + 3h) = (a + 3h - 3)*e^{a + 3h}\\f(b) = (b - 3)*e^b[/tex]
[tex]Approximation = (1/2) * [(a - 3)*e^a + 2*(a + h - 3)*e^{a + h} + 2*(a + 2h - 3)*e^{a + 2h} + 2*(a + 3h - 3)*e^{a + 3h} + (b - 3)*e^b][/tex]
[tex]= -5.1941[/tex]
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