Find the Taylor series for F(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x)→0. f(x)=x^4 - 4x^2 + 3, a = 2
[infinity]
a. Σ f^n(2)/n! (x-2)^n = -3 -16(x-2) + 20(x-2)^2 + n=0 8(x-2)^3 + (x-2)^4
[infinity]
b. Σ f^n(2)/n! (x-2)^n = -3 +16(x-2) + 20(x-2)^2 + n=0 8(x-2)^3 + (x-2)^4
[infinity]
c. Σ f^n(2)/n! (x-2)^n = 3 +16(x-2) + 8(x-2)^2 + n=0 20(x-2)^3 + (x-2)^4
[infinity]
d. Σ f^n(2)/n! (x-2)^n = 3 +16(x-2) - 8(x-2)^2 + n=0 20(x-2)^3 - (x-2)^4
[infinity]
e. Σ f^n(2)/n! (x-2)^n = 3 +16(x-2) + 20(x-2)^2 + n=0 8(x-2)^3 + (x-2)^4
Find the associated radius of convergence R.
R = ____
Use a Maclaurin series in this table to obtain the Maclaurin series for the given function. f(x) = 2x cos(1/3x^2) Evaluate the indefinite integral as an infinite series.
∫ cos x -1/x . dx
[infinity]
Σ + C
n=1
Answers
The correct answer is:
\(b. \sum f^n(2)/n! (x-2)^n = -3 +16(x-2) + 20(x-2)^2 + \sum 8(x-2)^3^/^n! + \sum (x-2)^4^/^n!\)
The Maclaurin series is:
\(f(x) = \sum (-1)^n (2x)^(^4^n^+^1^)/(3^(^2^n^)^(^2^n^)!) = \sum (-1)^n (2^(^4^n^+^1^)/3^(^2^n^)^(^2^n^)!) x^(^4^n^+^1^)\)
The indefinite integral of (cos(x) - 1/x) can be expressed as an infinite series.
Which statement about Taylor series for F(x) is true?The correct answer for the Taylor series of f(x) = x^4 - 4x^2 + 3 centered at a = 2 is:
\(b. \sum f^n(2)/n! (x-2)^n = -3 +16(x-2) + 20(x-2)^2 + \sum 8(x-2)^3^/^n! + \sum (x-2)^4^/^n!\)
The associated radius of convergence for this series is infinity, since the power series expansion of f(x) exists for all real numbers x.
How to find Maclaurin series for f(x)?To obtain the Maclaurin series for \(f(x) = 2x cos(1/3x^2)\), we can first find the Maclaurin series for \(cos(x^2^/^3^)\) and then multiply by 2x:
\(cos(x^2/3) = \sum (-1)^n (x^2/3)^(2n)/(2n)! = \sum (-1)^n x^(^4^n^)/(3^(^2^n^)^(^2^n^)!)\)
Therefore, the Maclaurin series for \(f(x) = 2x cos(1/3x^2)\) is:
\(f(x) = \sum (-1)^n (2x)^(^4^n^+^1^)/(3^(^2^n^)^(^2^n^)!) = \sum (-1)^n (2^(^4^n^+^1^)/3^(^2^n^)^(^2^n^)!) x^(^4^n^+^1^)\)
How to evaluate the indefinite integral?To evaluate the indefinite integral ∫(cos(x) - 1/x) dx as an infinite series, we can use the Maclaurin series for cos(x) and the power series expansion of \(ln(x) = \sum(-1)^(^n^+^1^) (x-1)^n/n\), which converges for 0 < x ≤ 2. Thus, we have:
\(\int(cos(x) - 1/x) dx = \intcos(x) dx - \int(1/x) dx = \sum(-1)^n x^(^2^n^+^1^)/(2n+1)! - ln|x| + C\)
where C is the constant of integration. We can simplify this expression as follows:
\(\int (cos(x) - 1/x) dx = \sum (-1)^n x^(^2^n^+^1^)/(2n+1)! - ln|x| + C\)
\(= -ln|x| + \sum(-1)^n x^(^2^n^+^1^)/(2n+1)!\)
\(= -ln|x| + \sum(-1)^n (x-1)^(^2^n^+^1^)/(2n+1)x^(^2^n^+^1^)\)
Thus, the indefinite integral of (cos(x) - 1/x) can be expressed as an infinite series.
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Related Questions
SOMEONE PLEASE HELP MEEE !!!
Answers
Answer:
see explanation for answers
Step-by-step explanation:
prism = 12*4 + 5*4 + 12*4 + 5*4 + 12*5 + 12*5 = 256 ft²
lateral surface area of pyramid = 4*1/2*5*5 = 50 ft²
area of square base of the pyramid = 5*5 = 25 ft²
total exposed surface area of the figure = 256 + 50 - 25 = 281 ft²
Rewrite the following equation in slope-intercept form. Y + 5 = 1 7 ( x + 7 )
Answers
Answer: y = 17x + 114
Step-by-step explanation:
The equation for the slope-intercept form is y = mx + b.
Arrange the equation so that it resembles y = mx + b.
You will do this by multiplying and subtracting so y is on the left side of the equation and mx + b is on the right side of the equation.
y + 5 = 17(x + 7)
y + 5 = 17x + 119
y + 5 - 5 = 17x + 119 - 5
y = 17x + 114
Answer:
Y = 17x + 114
Step-by-step explanation:
1. Y + 5 = 17 (x+7)
2. Y + 5 = 17x + 119 [Multiply the numbers in parenthesis by 17.]
3. Y = 17x + 114. [To keep the balance and move the 5 over, subtract it from 119.]
Can I get help with this?
Answers
Answer:
AC = 120
Step-by-step explanation:
<D = 360 - 95 - 145
D =120
Central Angle = Intercepted Arc
AC = <D = 120
Find the area!!!!!!!!!!
Answers
Answer:
87.5
Step-by-step explanation:
Area = 11+7
--- x 7
2
OR
Area = a+b/2 x h
A point is reflected across the x-axis. The new point is located at (4.75, -2.25) Where was the original point located.
Answers
Answer:
(4.75, 2.25)
Step-by-step explanation:
Given the coordinate (x,y). If this coordinate is reflected over the x axis, the resulting coordinate will be (x, -y)
Note that the y coordinate was negated.
Let the original point needed be (x, y)
If the new point is located at (4.75, -2.25)
Since the y coordinate was negated, then;
-y = -2.25
y = 2.25
x = 4,75 (x coordinate remains the same)
Hence the original point is (4.75, 2.25)
if there were 1580 people who participated in the survey then how many people said they had 4 members in their family?
Answers
Answer:
632
Step-by-step explanation:
1580*0.4=632
Tyrone is an investment broker who tells his clients that some investments are a moderate risk and others are an aggressive risk. He wants to know the difference between the annual return on the investments in each category. He sampled 17 investments from each category. There was a mean annual return of 6.2% on the investments in the moderate sample, with a standard deviation of 9.3%. The aggressive investments had a sample mean of 7.3% and a standard deviation of 12.4%. Assume that the conditions for inference have been met, and that Tyrone will use the conservative degrees of freedom corresponding to a sample size of 17. Leta - Am be the difference in mean annual return (in percents) of the groups of investments. Which of the following is a 90% confidence interval for p - um? Choose 1 answer: a. (-5.49, 7.69) b. (-5.464, 7.664)c. (-5.084, 7.284) d. (-3.926, 6.126) e. (-2.659,4.859)
Answers
the 90% confidence interval for p - um is (-0.816, 2.916), which can be rounded to (-0.82, 2.92). The closest option to this interval is (c) (-5.084, 7.284).
The point estimate for the difference in mean annual return is:
a - m = 7.3 - 6.2 = 1.1
To find the margin of error for a 90% confidence interval, we use the t-distribution with 16 degrees of freedom (conservative df) and a 5% significance level (90% confidence level):
t* = 1.745
The margin of error is then:
ME = t* * sqrt[(s1^2/n1) + (s2^2/n2)]
= 1.745 * sqrt[(0.093^2/17) + (0.124^2/17)]
= 0.916
The 90% confidence interval is:
(a - m) ± ME
1.1 ± 0.916
Thus, the 90% confidence interval for p - um is (-0.816, 2.916), which can be rounded to (-0.82, 2.92). The closest option to this interval is (c) (-5.084, 7.284).
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ances
A hole in the ground is 3 feet deep. A pile of dirt next to the hole is 3 feet high. Which list
best uses the values -3.0, and 3 to describe these elevations?
ground level:
bottom of the hole:
top of the dirt pile:
3 feet
3 feet
O feet
ground level:
bottom of the hole:
top of the dirt pile:
3 feet
0 feet
3 feet
ground level:
bottom of the hole:
top of the dirt pile:
0 feet
3 feet
-3 feet
ground level:
bottom of the hole:
top of the dirt pile:
O feet
-3 feet
3 feet
Answers
simplify
(9q) (-4q^3)
Answers
Statistics from the Port Authority of New York and New Jersey show that 83% of the vehicles using the Lincoln Tunnel that connects New York City and New Jersey use E-ZPass to pay the toll rather than stopping at a toll booth. Twelve cars are randomly selected.
Click here for the Excel Data File
a. How many of the 12 vehicles would you expect to use E-ZPass? (Round your answer to 4 decimal places.)
b. What is the mode of the distribution (the mode is the value with the highest probability)? What is the probability associated with the mode? (Round your answer to 4 decimal places.)
c. What is the probability eight or more of the sampled vehicles use E-ZPass? (Round your answer to 4 decimal places.)
Answers
Rounding to 4 decimal places, we expect 9.9600 of the 12 vehicles to use E-ZPass.
Rounding to 4 decimal places, the mode is 10 and the probability associated with the mode is 0.2822.
Rounding to 4 decimal places, the probability of eight or more of the sampled vehicles using E-ZPass is 0.9975.
a. We can use the binomial distribution to model the number of vehicles out of 12 that use E-ZPass. The probability of a vehicle using E-ZPass is 0.83, so the expected number of vehicles out of 12 that use E-ZPass is:
E(X) = np = 12 x 0.83 = 9.96
Rounding to 4 decimal places, we expect 9.9600 of the 12 vehicles to use E-ZPass.
b. The mode of the binomial distribution is the value of X that maximizes the probability mass function (PMF). In this case, the PMF is given by:
P(X = x) = (12 choose x) * 0.83^x * 0.17^(12-x)
We can find the mode by calculating the PMF for each possible value of X and identifying the value(s) with the highest probability. Alternatively, we can use the formula for the mode of the binomial distribution, which is:
mode = floor((n+1)p)
where n is the number of trials and p is the probability of success.
In this case, the mode is:
mode = floor((12+1) x 0.83) = floor(10.08) = 10
The probability associated with the mode is:
P(X = 10) = (12 choose 10) * 0.83^10 * 0.17^2 = 0.2822
Rounding to 4 decimal places, the mode is 10 and the probability associated with the mode is 0.2822.
c. The probability of eight or more of the sampled vehicles using E-ZPass can be found using the binomial distribution:
P(X >= 8) = 1 - P(X < 8) = 1 - P(X <= 7)
Using a binomial calculator or a cumulative binomial table, we can find:
P(X <= 7) = 0.0025
Therefore:
P(X >= 8) = 1 - 0.0025 = 0.9975
Rounding to 4 decimal places, the probability of eight or more of the sampled vehicles using E-ZPass is 0.9975.
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find the equation of the parabola below:
Answers
The equation of the parabola is y = x² + 6x + 8.
What is vertex form and quadratic form?The primary distinction between the two forms is that the vertex form informs us of the parabola's vertex while the standard form informs us of the quadratic equation's coefficients.
We can finish the square to change a parabola from standard form to vertex form. If the equation is in vertex form, it is simple to determine what the parabola's vertex is (h, k).
The standard form equation of a parabola:
y = a(x - h)² + k
where (h, k) is the vertex of the parabola.
Now, the x-intercepts at -2 and -4 thus:
y = a(x + 2)(x + 4)
Now, y-intercept at (0, 8) thus:
8 = a(0 + 2)(0 + 4)
8 = 8a
a = 1
Substitute the value of a we have:
y = (x + 2)(x + 4)
Hence, the equation of the parabola is y = x² + 6x + 8.
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if am making half of a recipe then I should divide it by two true or false
Answers
True
To obtain half of something (1/2) you have to divide by 2
Suppose the probability of having your name drawn for a door prize is 0.01. What are the odds of having your name drawn? Express your answer in the form a:b.
Answers
For the given event and probability:
\(\text{Odds}=\frac{0.01}{1-0.01}=\frac{0.01}{0.99}=1\colon99\)Stan owns 45 of an acre of land. Of that, 23 of the land is fenced. How many acres of land is fenced?.
Answers
Stan only owns 4/5 of an acre and 2/3 of that acre of a piece of property. He thus owns 8/15 acres of enclosed property.
What is an expression?A grouping of numeric variables or arithmetic operations using symbols such as addition, subtraction, multiplication, and division is called an expression.
Stan only owns a piece of property that is 4/5 acres in size, and 2/3 of those acres are fenced, therefore we must take that into account when calculating the overall amount of land that is fenced.
We would multiply 4/5, the amount Stan possesses, by 2/3, the amount fenced, to get the amount of his property that is fenced.
=> 4/5 X 2/3
=> 8/15
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Madelyn buys 3 bottles of orange juice at the corner store for a total cost of $2.31.
If each bottle costs the same amount, how much is each bottle of juice?
Answers
Answer:
$0.77
Step-by-step explanation:
To find how much each bottle of juice costs, divide 2.31 by 3:
2.31/3
= 0.77
So, each bottle of juice is $0.77
Answer: $0.77
Step-by-step explanation:
the total cost of three bottles is $2.31 because each bottle cost the same amount we can divide by 3 to get our answer.
Which best represents the solution to this system of equations?
Answers
Answer:
a. (3,-2) that is the answer
, I already study this lesson
The sum of two numbers is 91. The greater number is 7 more than the smaller number. Which equation can be used to solve for the smaller number?
x(x + 7) = 91
x(x − 7) = 91
x + (x + 7) = 91
x − (x + 7) = 91
Answers
Answer:
x+(x+7)=91
Step-by-step explanation:
Hope this helps :)
Answer: your answer for the smaller number is 38.5 and the larger number is 52.5.
Step-by-step explanation:
To make sure it's right, I just add the numbers together.
Your Problem:
91 ÷ 2 = 45.5
45.5 + 7 = 52.5
45.5 - 7 = 38.5
52.5 + 38.5 = 91
A bank promises to pay 5.5% pa compound interest on
deposits of £2337 if saved for 3 years. How much will the
savings be worth at the end of the period mentioned? Write
your answer correct to 2 decimal places.
Answer:
Answers
Answer:
£407.20Step-by-step explanation:
Deposit = £2337Interest = 5.5% pa compound Time = 3 yearsSavings:
2337*((1 + 5.5/100)^3 - 1) = 2337*((1.055)^3 - 1) =407.20Allison purchased 2.12 kilograms of food pellets for the animals in the petting zoo. Allison gave 177 grams of food pellets to the rabbits and 276 grams to the chickens. She gave 0.1 kilogram of pellets to the ducks and 619 grams to the goats. Allison gave all of the remaining food to the calves. How many grams of food pellets did Allison give to the calves?
Answers
Answer: 948 grams
Step-by-step explanation:
Kilograms of food purchased = 2.12 kilograms = 2120 grams
Number of grams of food pellets to the rabbits = 177
Number of grams given to the chickens = 276
Number of grams of pellets given to the ducks = 0.1kg = 100 grams
Number of grams given to the goats = 619
Then, grams of food pellets that Allison give to the calves will be:
= 2120 - (177 + 276 + 100 + 619)
= 2120 - 1172
= 948 grams
Therefore, the grams of food pellets that Allison give to the calves is 948 grams.
Note that 1000 grams = 1 kilogram
x + 4y = 13
x - y = 3
systems of equations I need help on a couple of these like ASAP
Answers
Slope:-1/4
Y intercept: (0,13/4)
X intercepts:(13,0)
B.)
X intercepts:(3,0)
Y intercepts:(0,-3)
Solve :-7x-6y=4
When x=-3y+8
Answers
And solve for y:
-7(-3y+8)-6y=4
21y- 56-6y =4
15y =60
Y = 60/15 = 4
And then sub y=4 in to our eqn for x to get:
x = -3(4) +8
x = -4
So we have solutions to our equations, x =-4 and y = 4 or in Cartesian form (-4,4)
replace x with x,
-7(-3y+8)-6y=4
21y-56-6y=4
now collect like terms,
21y-6y=4+56
15y=60
divide both sides by 15,
y=4
now replace y with 4 to find x,
x=-3(4)+8
x=-4
Le cratère d’un volcan a la forme d’un cône ; le fond du cratère est comblé de sédiments, un lac constitue la partie supérieure.
1) Le cône rempli de sédiments est une
réduction du grand cône. Calculer le
coefficient de réduction. DEJA REPONDU
2) Calculer le volume du cratère. DEJA REPONDU; le volume fait 2661845 m
3) En déduire le volume des sédiments.
4) Quel est le volume d’eau dans ce lac ?
Answers
Jennifer has already driven 160 miles on her trip to her grandparents house for every hour she travels she drives an additional 55 miles which equation represents the relationship between the Total miles driven y,and the number of hours ,x.
Answers
Answer:
y = 55x + 160
Step-by-step explanation:
The computation of the equation that represents the relationship between the total miles driven y and the number of hours x is shown below:
she starts at 160 miles so it would be y-intercept
So the equation form would be
"b" in y = mx + b
where,
m = 55
and b = 160
Therefore the equation would be
y = 55x + 160
The same is to be considered
Express 13 + 5-14 as a complex number.
Answers
Answer:
the answer is 4
Step-by-step explanation:
Answer:
13+5-14 is equaled to 4 so if i do 18-14 is will still give me 4 because 13+5 is 18 so 18- 14 is 4
Please help?!?!
Thanjss
Answers
Answer:
Step-by-step explanation:
y= mx+ b helps you find the equations of the lines
y= x+2 is the line with slope m=1, and y-intercept b=2
y= -4x-3 is the line with slope m=-4, and y-intercept b=-3
y > x+2 because is a dotted line
y ≤ -4x-3 because is a solid line
Answer:
aha i see yo bbg
Step-by-step explanation:
(lipbite)
A pencil holder shaped like a triangular prism is shown in the picture. The height of the pencil holder is 12 cm, and the volume of the pencil holder is 216 ㎤.
What is the area of the base
of the pencil holder in
square centimeters?
Answers
Answer:
18cm
Step-by-step explanation:
Answer:
18 cm
Step-by-step explanation:
SCM 300–Exam 1Total Annual Inventory Cost Equation-TC =DC+(Q/2) H+(D/Q) S-D = Annual Demand-C = Cost per unit-H = Annual holding cost per unit-S =
Answers
As per the data Economic Order Quantity is the given equation TC = DC + (Q/2) H + (D/Q) S is 1118 units.
Annual ordering cost per order
The Total Annual Inventory Cost (TC) equation calculates the total cost of maintaining inventory over a given period, typically a year. The equation considers the cost of ordering inventory, holding inventory, and the cost of lost sales due to stockouts.
The equation is as follows: TC = DC + (Q/2) H + (D/Q) S where:
DC is the Annual Demand, the total quantity of products that must be purchased or manufactured to meet customer demand over a given period.
C is the Cost per unit, the cost to purchase or manufacture a single unit of the product.
H is the Annual holding cost per unit, the cost of storing inventory for a given period. This cost includes storage fees, insurance, and the opportunity cost of using funds that could have been invested elsewhere.
S is the Annual ordering cost per order, the cost of placing an order for inventory, including shipping fees and the cost of preparing the order.
Q is the Economic Order Quantity, the optimal order quantity that minimizes the total cost of ordering and holding inventory.
The first part of the equation, DC, calculates the cost of purchasing the necessary inventory to meet demand. The second part of the equation, (Q/2) H, calculates the cost of holding inventory. The third part of the equation, (D/Q) S, calculates the cost of placing orders for inventory.
If the question asks calculate EOQ, then
Annual Demand = 15,000 units
Unit Cost = $10
Holding Cost = 0.30 or 30% of unit cost per unit per year
Cost to place order = $125D = 15,000H = (0.30) (10) = 3C = 10S = 125
EOQ= √ (2(125) (15,000)/ (3) =1118 UNITS
Thus, Economic Order Quantity is 1118 units.
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sally would like a 90 average on the five math tests this semester. her scores so far are 80, 82, 92, 98. what grade must she earn on her 5th and last test to achieve the 90 average?
Answers
Answer:
Step-by-step explanation: To find out what grade Sally needs to earn on her fifth and last math test to achieve a 90 average, we use the following steps:
Step 1: Add the total points Sally has received: 80 + 82 + 92 + 98 = 352.
Sally has taken 4 tests so far.
Step 2: Find the total marks required for a 90 average on 5 tests: 90 x 5 = 450.
Step 3: Find the score Sally needs to achieve on her fifth test by subtracting the points earned from the total required points: 450 - 352 = 98.
Therefore, Sally needs to earn a grade of 98 on her 5th and last test to achieve a 90 average on all five tests this semester.
Test the series for convergence or divergence. cos(na) 7/8 n = 1 n a. converges b. diverges
Answers
The given series, cos(na) 7/8ⁿ n = 1 n a., diverges.
The given series is Σ(cos(na) × 7/8ⁿ) with n = 1 to infinity. To test for convergence or divergence, we can use the Ratio Test.
Let's examine the limit as n approaches infinity of the absolute value of the ratio between consecutive terms:
L = lim (n→∞) |((cos((n+1)a) × 7/8⁽ⁿ⁺¹⁾) / (cos(na) × 7/8ⁿ))|
This simplifies to:
L = lim (n→∞) (|cos(na) × 7/8ⁿ| / |cos((n+1)a) × 7/8⁽ⁿ⁺¹⁾|)
The 7/8 terms will cancel out:
L = lim (n→∞) (|cos(na)| / |cos((n+1)a)|) × (8/7)
Since the cosine function oscillates between -1 and 1, the limit does not exist, and we can't draw any conclusion about convergence or divergence using the Ratio Test.
However, note that the series is the product of the sequence cos(na) and the geometric sequence 7/8ⁿ. The geometric sequence converges (because its common ratio 7/8 is between -1 and 1), but cos(na) oscillates between -1 and 1. The product of these sequences does not converge to a single value, so the series diverges.
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The price of an item has risen to $301 today. Yesterday it was $140 Find the percentage increase.
Answers
Answer: 115%
Step-by-step explanation: ur welcome
Answer:
53.2%
Step-by-step explanation:
301x53.2%= 160.132 = 160.13
301-160.13=140.87
About 53.2%