Answer:
[tex]\displaystyle \int {\frac{sin(x)}{2x}} \, dx = \frac{1}{2}\sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n + 1}}{(2n + 1)^2(2n)!}} + C[/tex]
General Formulas and Concepts:
Calculus
Integration
Integrals[Indefinite Integrals] Integration Constant CSequences
Series
Taylor Polynomials
MacLaurin PolynomialsPower Series
Power Series of Elementary FunctionsTaylor Series: [tex]\displaystyle P(x) = \sum^{\infty}_{n = 0} \frac{f^n(c)}{n!}(x - c)^n[/tex]Integration of Power Series:
[tex]\displaystyle f(x) = \sum^{\infty}_{n = 0} a_n(x - c)^n[/tex] [tex]\displaystyle \int {f(x)} \, dx = \sum^{\infty}_{n = 0} \frac{a_n(x - c)^{n + 1}}{n + 1} + C_1[/tex]Step-by-step explanation:
*Note:
You could derive the Taylor Series for sin(x) using Taylor polynomials differentiation but usually you have to memorize it.
We are given the integral and are trying to find the infinite series of it:
[tex]\displaystyle \int {\frac{sin(x)}{2x}} \, dx[/tex]
We know that the power series for sin(x) is:
[tex]\displaystyle sin(x) = \sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n + 1}}{(2n + 1)!}[/tex]
To find the power series for [tex]\displaystyle \frac{sin(x)}{2x}[/tex], divide the power series by 2x:
[tex]\displaystyle \frac{sin(x)}{2x} = \sum^{\infty}_{n = 0} \bigg[ \frac{(-1)^nx^{2n + 1}}{(2n + 1)!} \cdot \frac{1}{2x} \bigg][/tex]
Simplifying it, we have:
[tex]\displaystyle \frac{sin(x)}{2x} = \sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n}}{2(2n + 1)!}[/tex]
Rewrite the original integral:
[tex]\displaystyle \int {\frac{sin(x)}{2x}} \, dx = \int {\sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n}}{2(2n + 1)!}} \, dx[/tex]
Integrate the power series:
[tex]\displaystyle \int {\frac{sin(x)}{2x}} \, dx = \sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n + 1}}{2(2n + 1)(2n + 1)!}} + C[/tex]
Simplify the result:
[tex]\displaystyle \int {\frac{sin(x)}{2x}} \, dx = \frac{1}{2}\sum^{\infty}_{n = 0} \frac{(-1)^nx^{2n + 1}}{(2n + 1)^2(2n)!}} + C[/tex]
And we have our final answer.
Topic: AP Calculus BC (Calculus I + II)
Unit: Power Series
a construction company is planning to bid on a building contract. the bid costs the company $1900. the probability that the bid is accepted is 1/10. if the bid is accepted, the company will make $94000 minus the cost of the bid.
a. what is the expected value in this situation?
The expected value for the construction company if they bid on the contract is $7,500, calculated by weighing the potential profit and loss with their respective probabilities.
Explanation:The question is asking us to calculate the expected value of the construction company's bid on a building contract. To find the expected value, we need to multiply each outcome by its probability and sum these amounts.
There are two possible outcomes:
The expected value (EV) is calculated as follows:
EV = (Probability of acceptance × Net profit when accepted) + (Probability of not being accepted × Loss when not accepted)
EV = (1/10 × $92,100) + (9/10 × -$1,900)
EV = $9,210 - $1,710
EV = $7,500
The expected value for the construction company if they bid on the contract is $7,500.
If the diameter of the duct is 5.2 inches what's the area of the circular hole
solution for 5y+2y = -24
what percent of 67.5 is 7.02
7.02 is approximately 10.4 percent of 67.5.
To find what percent 7.02 is of 67.5,
Since we know that,
A figure or ratio that may be stated as a fraction of 100 is a percentage. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the word percent means. The letter "%" stands for it.
We can use a proportion.
Let's call the unknown percent "x."
The proportion would be:
7.02/67.5 = x/100
We can cross-multiply to solve for x:
(7.02 * 100) / 67.5 = x
Simplifying this expression, we get:
x = 10.4
Therefore, 7.02 is approximately 10.4% of 67.5.
Learn more about the percent visit:
https://brainly.com/question/24877689
#SPJ6
the function f(x) varies inversely with x and f(x)=8 when x=3
what is f(x) when x=2
12
14
22
48
Answer:
[tex]12[/tex]
Step-by-step explanation:
we know that
A relationship between two variables, x, and y, represent an inverse variation if it can be expressed in the form [tex]y*x=k[/tex] or [tex]y=k/x[/tex]
In this problem we have
For [tex]x=3[/tex] -----> [tex]f(x)=8[/tex]
so
Find the value of k
[tex]f(x)*x=k[/tex]
substitute the value of x and f(x)
[tex]8*3=k[/tex]
[tex]k=24[/tex]
therefore
the equation is equal to
[tex]f(x)=24/x[/tex]
For [tex]x=2[/tex] -------> substitute in the equation and find the value of f(x)
[tex]f(x)=24/2=12[/tex]
What value of b will cause the system to have an infinite number of solutions?
Answer:
Step-by-step explanation:
b=6
resultado de x/12 = 9/x+2
what is 1.375 as a fraction
Final answer:
To convert 1.375 to a fraction, express it as 1,375/1,000 and simplify by dividing by the greatest common divisor, which is 125, to get 11/8.
Explanation:
To convert the decimal 1.375 to a fraction, we will follow a few steps. First, count the number of decimal places, which in this case is three. This tells us that the denominator of the fraction should be a power of 10; specifically, 1,000. So we write 1.375 as 1,375/1,000. Next, we simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. In this case, the GCD of 1,375 and 1,000 is 125. When we divide both the top and the bottom by 125, we get 11/8, which is the simplest form of the fraction.
Converting decimals to fractions involves aligning the decimal with the appropriate power of 10 in the denominator and simplifying by finding the greatest common divisor. This systematic approach ensures the representation of the decimal as a simplified fraction, such as 11/8 in the case of 1.375.
A store has a contest to guess the mass of 10,000 peanuts if a peanut has a mass of about 0.45 g what would be a reasonable guess with a mass of 10,000 peanuts
The mass of 10,000 peanuts is 4,500 g or 4.5 kg.
The question asks for a reasonable guess for the mass of 10,000 peanuts, given that each peanut has a mass of about 0.45 g. To calculate this, we simply multiply the number of peanuts by the mass of each peanut:
Mass of one peanut = 0.45 g
Number of peanuts = 10,000
Total mass = 0.45 g/peanut x 10,000 peanuts
Now we will carry out the multiplication:
Total mass = 4,500 g
Since we are dealing with a large quantity of peanuts, it is more convenient to express this mass in kilograms:
4,500 g is equivalent to 4.5 kg (since 1 kg = 1,000 g).
what is the ratio in lowest terms of 14 miles to 84 miles
-n^2+16n-63/n^2-2n-35
Find a whole number solution that makes the following true
a.) Divisible by 2 and 3 but not 6
b.) Divisible by 5 and 10 but not 2
1. In the on-line convex-hull problem, we are given the set Q of n points one point at a time. After receiving each point, we compute the convex hull of the points seen so far. Obviously, we could run Graham’s scan once for each point, with a total running time of O(n2lg n). Show how to solve the on-line convex-hull problem in a total of O(n2) time.
2. Show how to implement the incremental method for computing the convex hull of n points so that it runs in O(n lg n) time.
3 chairs and a table cost 149 4 chairs and a table cost 177 how much does each item cost?
Suppose that sin(θ)=6/10 and θ is in the second quadrant. Use trigonometric identities to find the following quantities exactly.
You weigh a gallon of 2% milk in science class and learn that it is approximately 8.4 pounds. You pass the milk to the next group, and then realize that you teacher wants it in kilograms, not pound. Explain how you can adjust your answer without weighing the milk again. Then give you the weight in kilograms.
Find The last two digit of 36^2015
Answer:
76
Step-by-step explanation:
The last two digits of powers of 36 repeat in a pattern of length 5:
36^1 mod 100 = 36
36^2 mod 100 = 96
36^3 mod 100 = 56
36^4 mod 100 = 16
36^5 mod 100 = 76
36^6 mod 100 = 36
The last two digits of the power 2015 are the same as those of the 5th power: 76.
If linear correlation between two variable is negative, what can be said about the slope of the regression line
raise g
to the 7th power,
then subtract h
from the result
To carry out the student's request, you would first raise 'g' to the 7th power to obtain g^7, then subtract 'h' from this result to get the final expression 'g^7 - h'.
Explanation:To solve the expression mentioned by the student, you need to carry out two mathematical operations in sequence. The first operation is raising 'g' to the 7th power, which can be expressed as g^7. The second operation is subtracting 'h' from the result of the first operation. This can be represented mathematically as g^7 - h.
For example, if g equals 2 and h equals 3, then you would first calculate 2^7 which equals 128. Then you would subtract 3 from 128, yielding an answer of 125.
Area model division 246÷3
Answer:
it is right
Step-by-step explanation:
Where's the vertical asymptote?
why is 50 + 7 x 5 = 85
"The answer is: 50 + 7 x 5 = 85 is correct due to the order of operations.
To understand why 50 + 7 x 5 equals 85, we must follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). According to this rule, multiplication and division are performed before addition and subtraction.
Let's break down the expression step by step:
1. Identify the multiplication part of the expression: 7 x 5.
2. Perform the multiplication first: 7 x 5 = 35.
3. Now, we have 50 + 35, since we have already computed 7 x 5.
4. Finally, perform the addition: 50 + 35 = 85.
So, following the order of operations, we correctly get that 50 + 7 x 5 equals 85."
explain how you can determine if four points lie on a single parabola
On Monday David travels m miles. On Tuesday he travels 35 more than he did on Monday. On Wednesday he travels a total of twice the distance he travelled Monday. Work out and expression of the total distance he travels during the three days
To find the total distance David travels during the three days, we add up the distances he travels on each day: m + (m + 35) + 2m = 4m + 35 miles.
Explanation:To find the total distance David travels during the three days, we need to add up the distances he travels on each day.
On Monday, David travels m miles.
On Tuesday, he travels 35 more than he did on Monday, so he travels m + 35 miles.
On Wednesday, he travels twice the distance he traveled on Monday, so he travels 2m miles.
To find the total distance, we add up the distances on all three days:
Total distance = m + (m + 35) + 2m = 4m + 35 miles.
For a particular event, 845 tickets were sold for a total of $4162. If students paid $4 per ticket and non students paid $6 per ticket, how many student tickets were sold?
Final answer:
Using a system of equations, we found that 454 student tickets were sold at the event.
Explanation:
To determine how many student tickets were sold for the event, we need to set up a system of equations based on the information given. Let's define x as the number of student tickets sold at $4 each, and y as the number of non-student tickets sold at $6 each.
The total number of tickets sold (student plus non-student) is given as 845 tickets:
x + y = 845
The total amount of money made from selling these tickets is $4162:
4x + 6y = 4162
Now we have a system of two equations:
x + y = 845
4x + 6y = 4162
We can solve this system by multiplying the first equation by 4:
(4)(x + y) = (4)(845)
4x + 4y = 3380
Now, we subtract this new equation from the second equation given by the total amount made to eliminate y:
4x + 6y - (4x + 4y) = 4162 - 3380
2y = 782
Dividing both sides of this equation by 2 gives us:
y = 391
Now, substitute y back into the first equation to find x:
x + 391 = 845
x = 845 - 391
x = 454
So, 454 student tickets were sold.
Mountain Ski Corp. was set up to take large risks and is willing to take the greatest risk possible. Lakeway Train Co. is more typical of the average corporation and is risk-averse.
Projects Returns:
Expected Value Standard
Deviation
A $ 527,000 $ 834,000
B 682,000 306,000
C 74,000 135,000
D 140,000 89,000
a-1. Compute the coefficients of variation. (Round your answers to 3 decimal places.)
Coefficient of
Variation
Project A
Project B
Project C
Project D
a-2. Which of the following four projects should Mountain Ski Corp. choose?
Project A
Project B
Project C
Project D
b.
Which one of the four projects should Lakeway Train Co. choose based on the same criteria of using the coefficient of variation?
Project A
Project B
Project C
Project D
Please help me with substitution! I'm terrible at it. :(
-3x+5y=-11
y=2x-5
Danielle and David each have a job. Danielle earns $15 an hour. She also receives a $50 weekly bonus. Danielle wrote an equation to determine how much she earns in a given week. y=15x+50y=15x+50 David earns $12 an hour and receives an $80 weekly bonus. David made a graph to show how much he earns in a given week. How many hours will it take Danielle and David to earn the same amount of money? Enter your answer in the box below.
Answer:10
Step-by-step explanation:
the sales tax rate for the state of washington was 5.1% . What is the sales tax on a $5900 car in Washington?
identify the decimal labeled with the letter A on the scale below
The decimal labeled with the letter A on the scale is 0.75.
What is number line?A number line can be defined as a straight line with numbers placed at equal intervals or segments along its length.
Here, the number starts with zero.
There are 8 division between 0 and 1.
So, first one represent 1/8= 0.125
second one =0.25
third one= 0.375
fourth one= 0.50
fifth one= 0.625
sixth one= 0.75
Hence, the decimal labeled with the letter A on the scale is 0.75.
learn more about number line here:
https://brainly.com/question/17172919
#SPJ2