2. Assume that each of the following functions has a power series expansion. Find the Maclaurin series for each. Be sure to provide the domain on which the expansion is valid. (a) f(x) = ln(1 + 2) (b) f(x) = xe^{2x}
Answers
The following can be answered by the concept of Maclaurin series
a. Maclaurin series: f(x) = ln(3)
The expansion is valid for all x in the domain of real numbers (-∞, ∞).
b. The Maclaurin series for f(x) = xe²ˣ is: f(x) = x + 2x² + …
The expansion is valid for all x in the domain of real numbers (-∞, ∞).
(a) f(x) = ln(1 + 2)
Since the function is a constant, its Maclaurin series is just the constant value itself.
Maclaurin series: f(x) = ln(3)
The expansion is valid for all x in the domain of real numbers (-∞, ∞).
(b) f(x) = xe²ˣ
To find the Maclaurin series for this function, we need to find its derivatives and evaluate them at x = 0.
f(x) = xe²ˣ
f'(x) = e²ˣ + 2xe²ˣ
f''(x) = 4xe²ˣ + 4e²ˣ
Now, evaluate the derivatives at x = 0:
f(0) = 0
f'(0) = 1
f''(0) = 4
Using the Maclaurin series formula, we get:
f(x) = f(0) + f'(0)x + (f''(0)x²)/2! + …
f(x) = 0 + 1x + (4x²)/2 + …
So, the Maclaurin series for f(x) = xe²ˣ is:
f(x) = x + 2x² + …
The expansion is valid for all x in the domain of real numbers (-∞, ∞).
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Related Questions
a sink drips 2 fifths $\frac{2}{5}$ gallon of water in 4 hours. which rate is the unit rate of water dropped per day?
Answers
The unit rate of water dropped per day is \($\frac{12}{5}$\) gallons per day.
We can start by finding the unit rate of water dropped per hour.
If the sink drips \($\frac{2}{5}$\) gallon of water in 4 hours, then it drips \(\frac{1}{5}$\) gallon of water in 2 hours (since \((\frac{2}{4} = \frac{1}{2}$).\)
The unit rate of water dropped per hour is:
\($\frac{1/5}{2} = \frac{1}{10}$\) gallon per hour.
To find the unit rate of water dropped per day, we can multiply the unit rate per hour by the number of hours in a day:
\($\frac{1}{10} \text{ gallon per hour} \cdot 24 \text{ hours per day} = \frac{12}{5} \text{ gallons per day}$\)
Therefore, the unit rate of water dropped per day is \($\frac{12}{5}$\) gallons per day.
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The unit rate of water dropped per day is 12/5 gallons.
To find the unit rate of water dropped per day for a sink that drips 2/5 gallons of water in 4 hours, we'll follow these
steps:
Determine how many gallons of water are dripped in one hour.
Calculate the gallons of water dripped in 24 hours (one day).
Calculate gallons per hour.
Since the sink drips 2/5 gallons in 4 hours, we'll divide the gallons by the number of hours:
(2/5) / 4 = (2/5) × (1/4) = 2/20 = 1/10
So, the sink drips 1/10 gallons per hour.
Calculate gallons per day.
Now, we'll multiply the gallons per hour by 24 hours to find the gallons per day:
(1/10) × 24 = 24/10 = 12/5
The unit rate of water dropped per day is 12/5 gallons.
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Claire wants to build a patio outside her café. The rectangular space outside of Claire's café is three times as long as it is wide. The area of the space is 48 m2. Claire
would like to build a patio with dimensions 3.5 m by 12.5 m in this space. Will it fit? Explain.
pls help
Answers
Answer:
Step-by-step explanation:
It will fit in the space but it won't cover the entire space. the area is 48m2, but the patio in question will cover 43.75 of the 48m2. Also, it stated that the rectangular space is 3 times as long, and the patio is 3.57x as long as the width. So yes, it will fit, but it will not be a picture perfect fit, there will be some obscenities.
Step-by-step explanation:
Let length of space be lm and width be wm.
l = 3w --(1)
lw = 48 --(2)
sub (1) into (2):
3w(w) = 48
3w^2 = 48
w^2 = 16
w = ± 4 (rej -4 since w > 0)
when w = 4, l = 3(4) = 12
Since dimensions of the patio is 3.5m by 12.5m, and the dimensions of the space is 4m by 12m it does not fit
This is a topic known as simultaneous equations. If you would like to find out more / know when and how to use it, you can check out my Instagram page (learntionary)
Which Graph Matches The Table?
Answers
Answer:
graph W
Step-by-step explanation:
what is the term, coefficient and constants of 6+n2+12d
Answers
Answer: Coefficient is 2 and 12. Constant is 6.
Step-by-step explanation: Coefficient is the numbers that multiply the variables or letters so 2 and 12 is coefficient. Constants are terms without variables so 6 is a constant.
At the West Street Bakery, 1/6 of the dessert case holds pies. There are 4 equal size shelves.
Which fraction of the West Street Bakery’s dessert case is taken up by 1 pie shelf?
1/4
2/3
1/24
5/6
Answers
Answer:
1/4 or 1/24
Step-by-step explanation: I am not sure but try and if you get it rong I am sorry
Answer:
Im thinking the answer is 1/24.
Step-by-step explanation:
1/6 divided by 4
6 x 4 = 24
4 x 6 = 24
if you put a fraction first, the answer will be a fraction.
I REALLY NEED HELP WITH THIS QUESTION!!!
Answers
it's the fourth option
what you need to do is convert them all to a decimal and compare them to see which one applies :)
decimal form:
3.14, 2.70, 2.28
which shapes could but do not always have perpendicular lines.
Answers
Answer: Right triangles
Step-by-step explanation:
Right triangles have perpendicular sides, rectangles have both perpendicular and parallel sides, but other quadrilaterals might not. A regular pentagon has no parallel or perpendicular sides, but a non-regular pentagon might have parallel and perpendicular sides. It all depends on the polygon.
50 points brainliest fore most straight forward answer
Which answer is the best estimate of the correlation coefficient for the variables in the scatter plot?
−0.8
−0.25
0.25
0.8
Answers
Answer:
The answer is -8 :)
Step-by-step explanation:
Hannah swims in a swimming pool of 25 metres long.
How many laps does she need to complete for a total distance of 0.5 kilometre?
Answers
She needs to complete 20 laps for a total distance of 0.5 kilometer.
1 kilometers = 1000 meters
0.5 kilometers = 500 meters
Length of swimming pool = 25 meters
That means, she covers 25 meters in 1 lap.
Laps to cover 500 meters = 500/25
= 20
Hence, she needs to do 20 laps.
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Describe the long run behavior of f(x)=5(2)x+1:
As x→−[infinity], f(x) =
As x→[infinity], f(x) =
Answers
The long run behavior of the function f(x)=5(2)x+1 is that it approaches 1 as x approaches negative infinity and it approaches infinity as x approaches positive infinity.
The long-term behavior of the function f(x)=5(2)x+1 can be discovered by examining how the function behaves as x gets closer to negative and positive infinity.
As x→−[infinity], f(x) = 5(2)^ -∞+1 = 5(0)+1 = 1
As x approaches negative infinity, the value of the function approaches 1.
As x→[infinity], f(x) = 5(2)^ ∞+1 = 5(∞)+1 = ∞
As x approaches positive infinity, the value of the function approaches infinity.
As a result, the function f(x)=5(2)x+1 behaves in the long run in such a way that it approaches 1 as x approaches negative infinity and infinity as x approaches positive infinity.
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BRAINLEST AND 75 POINTS!!!! PLEASE HELP ASAP!!!
You are now a teacher, and you notice that many of your students are consistently making the dividing-out mistake that appears below. Some of the students even admit to knowing the method was wrong as soon as you point it out.
Create a visual (a short video or a series of images) to help your students stop making this common mistake: fraction numerator up diagonal strike x squared plus 3 x − 4 over denominator up diagonal strike x squared − 2 x − 8 end fraction .
Your lesson should do the following:
Explain why the dividing-out method is incorrect. You may want to start with a simpler expression and work your way up to polynomials. (For example, compare fraction numerator 3 left parenthesis 5 right parenthesis over denominator 3 end fraction and fraction numerator 3 plus 5 over denominator 3 end fraction.)
Explain when you can cancel a number that is in both the numerator and denominator and when you cannot cancel out numbers that appear in both the numerator and the denominator.
Share tricks, reminders, memory devices, or other methods to help students catch themselves before making this common mistake.
Post your video or series of images. Post answers to the following questions:
A. Why do you think the mistake shown here is such a common one?
B. Have you ever made this mistake before? What helped you stop making this mistake? What will help you stop making this mistake in the future?
Answers
Answer:
A: Division can be very confusing sometimes. Lots of people get confused by this. This is a problem because It will give you the wrong answer.
B: Yes I have made this mistake. I wrote down the steps and kept that piece of paper.
Will mark brainliest
Answers
\( \frac{19}{6} \pi \times \frac{360}{75} = \frac{19}{6} \pi \times \frac{24}{5} = \frac{76}{5} \pi\)
\( {x}^{2} \times \pi = \frac{76}{5} \pi\)
\(x = \frac{2 \sqrt{95} }{5} = 3.9\)
Jayden has two jobs. one week his paycheck was the same for both jobs. as a server he eared $9. 64 per hour plus $82. 35 in tips working at a jewelry store he earned $14. 86 per hour plus $17. 10 in sales
bonuses.
part a
select the equation that can be used to determine the number of hours, h, that jayden worked the week his paycheck was the same
for both jobs.
o 24. 5h
= 99.45
0 91 99
= 31. 96h
0 82. 35h + 9.64
0 9. 64h + 82. 35
17.10h + 14.86
14.86h + 17.10
part b
be
the total amount of money that jayden made when both paychecks were the same is $
Answers
the total amount of money that jayden made when both paychecks were the same is $99.45
To determine the number of hours Jayden worked the week his paycheck was the same for both jobs, we can solve an equation. The equation we'll use is 14.86h + 17.10 = 99.45. We can subtract 17.10 from both sides, leaving 14.86h = 82.35. Then we cane b divide both sides of the equation by 14.86, which will give us the result of h = 5.55 hours. The total amount of money that Jayden made when both paychecks were the same is 99.45. This can be confirmed by combining his earnings for both jobs, which would be 9.64h + 82.35 + 14.86h + 17.10 = 99.45.
Jayden worked 5.55 hours in the week his paycheck was the same for both jobs, and he earned a total of $99.45.
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8 + (7 + 1) 2 ÷ 4 ⋅ 1 over 2 whole to the power of 4
Answers
Answer:24divided by 16/64
Step-by-step explanation:
8x[7+1]2/4x4/16
8x8/2/4x4/16
48/2/4x4/16
48/2/16/64
24/16/64
Evaluate the following when a = 4 and b = 7 (a+b)^2 - (a^2 + b^2)
Answers
Answer:
56
Step-by-step explanation:
(4+7)²-(4²+7²)=121-65
=56
Answer:
56
Step-by-step explanation:
(4 + 7)² - (4² + 7²) = 11² - (16 + 49) = 121 - 65 = 56
can someone help me solving this equation for x please
Answers
Answer:
±2/11
Step-by-step explanation:
Take the square root of both sides of the equation:
\(\sqrt{x^2}=\sqrt{\dfrac{4}{121}}\\\\\boxed{x=\pm\dfrac{2}{11}}\)
_____
Here, we need to recognize that (-2/11)^2 = (2/11)^2 = 4/121, so both the positive and negative roots of 4/121 are solutions to the equation.
Anyone out there can solve this I will give 17 points!!!
Lauren is twice Kristy’s age. The sum of Lauren and Kristy’s ages is 51. Which equation represents the sum of their ages?
A K + K + 2 = 51 B K + 2K + 2 = 51 C 2K = 51 D K + 2K = 51
Answers
Answer:
D. K+2K=51
Step-by-step explanation:
Since Lauren is 2 times Kristy's age, Lauren's age=2K
so, if the sum of the two is 51, then 2K+K=51
If you want to solve it, 2K+K=3K=51
51/3=K
17=K
34=L
Answer:
D K + 2K = 51
Step-by-step explanation:
Lauren is twice Kristy's age so it would be 2K cause you want to multi[ly, and then you have to add Kristy's age in the equation because their ages added together is 51
if 5 oz of a solution contains 4 1/2 oz of water, how many ounces of water are in 20 oz of solution?
Answers
5 oz of a solution contains 4 1/2 oz of water. 20 oz of solution contains 18 oz of water.
We can set up a proportion to solve this problem:
5 oz of solution contains 4 1/2 oz of water, which means that 5 - 4 1/2 = 1/2 oz of the solution is not water.
So the ratio of water to solution is:
4 1/2 oz / 5 oz = x oz / 20 oz
where x is the number of ounces of water in 20 oz of solution.
Cross-multiplying, we get:
5 * x = 4 1/2 * 20
x = (4 1/2 * 20) / 5
x = 18 oz
Therefore, 20 oz of solution contains 18 oz of water.
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Martin charges $10 for every 5 bags of leaves he rakes
last weekend he raked 22 bags of leaves how much money did he earn?
.
Answers
hi i have an exam coming up
Answers
The equation that we want is:
80 = g*5
How to write the equation?Here we have the sentence:
"80 is the product of Greg's age and 5"
And we want to write that as an equation.
If we define g as Greg's age, then we can write that the product between g and 5 must be equal to 80, then the equation will be just:
80 = g*5
We can even solve that to get:
80/5 = g
16 = g
Gregg is 16 years old.
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True or False. If it is true, briefly explain. Otherwise, give a counterex- ample. [4 marks] (a) Any spanning set of a subspace S of R" is a basis for S. (b) If a matrix A can be reduced to a reduced row echelon form R, then col(A) = col(R). (c) If a matrix A can be reduced to a reduced row echelon form R, then row(A) = row(R). (d) The dimnesion of NulA is the number of variables in the equation AX = 0.
Answers
A spanning set of a subspace S of R^n is not always a basis for a) False. A spanning set may not be linearly independent, which means it may not form a basis for the subspace. For example, in R², {(1,0), (0,1), (1,1)} is a spanning set for the subspace S={(x,y)∈R² : x=y}, but it is not linearly independent, so it is not a basis for S.
b) True. Row operations do not change the column space of a matrix, so if A can be reduced to R by row operations, then the columns of A and R span the same space. Moreover, R is in reduced row echelon form, which means that the columns of R form a basis for col(A).
c) True. Row operations do not change the row space of a matrix, so if A can be reduced to R by row operations, then the rows of A and R span the same space. Moreover, R is in reduced row echelon form, which means that the rows of R form a basis for row(A).
d) True. The null space of A is the set of all solutions to the homogeneous equation AX=0. By the rank-nullity theorem, dim(NulA)=n-r, where n is the number of variables and r is the rank of A. Since A is in reduced row echelon form, the number of nonzero rows is equal to the rank of A, which means that r is the number of pivot variables, which is the same as n-d, where d is the number of free variables. Therefore, dim(NulA)=d=n-r.
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A family pays $19 each month for Internet access. Next month, the cost will increase by 5% because of an equipment fee. After this increase, what will be the cost for the Internet access?
Answers
Step-by-step explanation:
5% of $19 = 19*5/100 = 19/20 = 0.95
5% increase in cost will make the total cost as,
$19 + $0.95 = $19.95
Mai says that equations A and B have the same solution.
- Equation A:-3(x + 7) = 24
- Equation B:2 + 7 =-8
Which statement explains why this is true?
A: Adding 3 to both sides of Equation A gives x + 7 = -8.
B: Applying the distributive property to Equation A gives x + 7 = -8.
C: Subtracting 3 from both sides of Equation A gives X + 7 = -8.
D: Dividing both sides of Equation A by-3 gives x + 7 = -8.
Answers
Answer: D
Step-by-step explanation: Took It
The correct option will be "option D".
It is given that both the equations A and B have the same solution.
We have to find out that which of the given options are correct.
What will be the value if 3 is added to the both sides of the equation ' 3x - 6 = 4x - 8.
The equation will become , 3x - 3 = 4x -5
As given in the question ;
equation 1 and equation 2 have same solution.
Let's check which option is correct.
A) Adding 3 to both sides of Equation A i.e., -3 (x + 7) = 24
=3 x - 21 + 3 = 24 + 3
-3x - 18 = 27
x + 6 = -9
So option A is not correct
B) Applying the distributive property to Equation A
a × ( b + c ) = ( a + b ) × c
-3 ( x + 7 ) = ( 1 + 7 ) × -3
x + 7 = 8
So , option B is not correct.
C) Subtracting 3 from both sides of Equation A
- 3x - 21 -3 = 24 - 3
-3x - 24 = 21
x + 8 = 7
So, option C is not correct.
D) Dividing both sides of Equation A by -3
[ -3 (x + 7) ] / 3= 24 / 3
-x - 7 = 8
x + 7 = -8
So , option D is correct.
Thus , correct option will be "option D".
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HELP!! PLEASE!!!!!
Solve the equation by taking square roots
2-8n^2=-198
The best answer gets Brainiest!!
Answers
Answer:
Isolate the variable by dividing each side by factors that don't contain the variable.
n=5, -5
Answer:
\(n=\frac{3i\sqrt{22}-2}{-8}\)
Step-by-step explanation:
\((2-8n)^2=-198\)
\(\sqrt{(2-8n)^2} =\sqrt{-198}\)
\(2-8n=i\sqrt{198}\)
\(2-8n=3i\sqrt{22}\)
\(-8n=3i\sqrt{22} -2\)
\(n=\frac{3i\sqrt{22}-2}{-8}\)
I hope this helps!
determine the probability of event if the odds for (i.e., in favor of) are 21 9.
Answers
The probability of the event, given the odds in favor of 21 to 9, is 0.7
How can we calculate the probability of the event based on the given odds?To calculate the probability of an event given the odds in favor, we can use the formula:
Probability = Odds in Favor / (Odds in Favor + Odds against)
In this case, the odds in favor are given as 21 to 9. This means that for every 21 favorable outcomes, there are 9 unfavorable outcomes.
To calculate the probability, we divide the number of favorable outcomes (21) by the total number of outcomes (21 + 9 = 30):
Probability = 21 / 30 = 7/10
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Use the point and the slope to graph each line. Write the equation of the line. The line contains point (2,-2) and is perpendicular to a line with slope 1
Answers
Answer:
x + y = 0
Step-by-step explanation:
Slope 1
b = -2 - (-1)(2)
b = -2 +2
b = 0
y = -x
The equation of the line will be y = -x. The graph of the line is given below.
What is the equation of a perpendicular line?If the slope of the line is m, then the slope of the perpendicular line will be negative 1/m.
The slope of the line is 1. Then the slope of the perpendicular line will be - 1.
y = - x + c
Then the line is passing through (2, -2), then the value of the c will be
- 2 = - 2 + c
c = - 2 + 2
c = 0
The equation of the line will be y = -x.
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Need Help ASAP
On Monday, the high temperature in Frostbite, Alaska was -5 degrees. On Tuesday it rose 14 degrees. On Wednesday it dropped 20 degrees. What was the temperature on Wednesday?
Answers
Answer:
-11°
Step-by-step explanation:
-5+14=9-20=-11
not quite sure if it's the answer since it's quite simple, but I take my chances
An open-top container is to be made from a 13-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side. What size square should be cut out of each corner to get a container with the maximum volume?
Answers
To maximize the volume of an open-top container made from a 13-inch by 48-inch piece of plastic, you need to determine the optimal size of the squares to be cut out from each corner. Let 'x' be the side length of the square removed from each corner. After cutting, the dimensions of the container will be:
- Length: 48 - 2x
- Width: 13 - 2x
- Height: x
The volume of the container can be calculated using the formula: V = L * W * H. the dimensions, we get:
V(x) = (48 - 2x)(13 - 2x)(x)
To find the maximum volume, we need to identify the value of 'x' that maximizes V(x). This can be achieved using calculus, by finding the critical points where the derivative of the function V(x) is zero or undefined.
Differentiating V(x) with respect to x and setting the derivative equal to zero, we can solve for the optimal value of 'x'. After performing these calculations, we find that the optimal size of the square to be cut out from each corner is approximately 1.52 inches. By removing 1.52-inch squares from each corner and folding up the flaps, the open-top container will have the maximum volume.
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if f(x) = exg(x), where g(0) = 2 and g'(0) = 4, find f '(0).
Answers
The derivative of f(x) = eˣ * g(x) at x = 0 is equal to 6.
The derivative of a function f(x) at a point x = a can be interpreted as the instantaneous rate of change of the function at that point. To find the derivative of f(x) = eˣ * g(x), we can use the product rule for derivatives, which states that if f(x) = u(x) * v(x), then f'(x) = u'(x) * v(x) + u(x) * v'(x).
In this case, u(x) = eˣ and v(x) = g(x), so u'(x) = eˣ and v'(x) = g'(x). Therefore, f'(x) = eˣ * g(x) + eˣ * g'(x) = eˣ * (g(x) + g'(x)).
We are asked to find f'(0), so we can substitute x = 0 into the expression for f'(x) to get f'(0) = e⁰ * (g(0) + g'(0)) = e⁰ * (2 + 4) = 6. This means that the rate of change of f(x) at x = 0 is equal to 6.
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Complete question:
if f(x) = eˣg(x), where g(0) = 2 and g'(0) = 4, find f '(0).
What critical value of " should be used for a 95% confidence interval for the population mean based on a random sample of 30 observations? Find the t-table here. 1* = 2.042 * = 2.045 " = 2.147 * = 2.150
Answers
The critical value (t₍₃₀,₀.₀₅₎) for a 95% confidence interval, based on a random sample of 30 observations, is t₍₃₀,₀.₀₅₎ = 2.042.
Find the critical value?To determine the critical value, we refer to the t-table with degrees of freedom (df) equal to n - 1, where n represents the sample size. In this case, the sample size is 30, so the degrees of freedom is 30 - 1 = 29.
For a 95% confidence interval, we need to consider the two-tailed critical region. Since the area in each tail is 0.025 (0.05/2), we look for the corresponding value in the t-table at a significance level of α/2 = 0.025 for df = 29.
The closest value to 0.025 in the table is 2.042.
Therefore, the critical value (t₍₃₀,₀.₀₅₎) for a 95% confidence interval based on a random sample of 30 observations is t₍₃₀,₀.₀₅₎ = 2.042.
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