Answer:
[tex]f(2\pi) = 32\pi^3 - 2[/tex]
Step-by-step explanation:
Main steps:
Step 1: Use integration to find a general equation for f
Step 2: Find the value of the constant of integration
Step 3: Find the value of f for the given input
Step 1: Use integration to find a general equation for f
If [tex]f'(x) = g(x)[/tex], then [tex]f(x) = \int g(x) ~dx[/tex]
[tex]f(x) = \int [12x^2 - sin(x)] ~dx[/tex]
Integration of a difference is the difference of the integrals
[tex]f(x) = \int 12x^2 ~dx - \int sin(x) ~dx[/tex]
Scalar rule
[tex]f(x) = 12\int x^2 ~dx - \int sin(x) ~dx[/tex]
Apply the Power rule & integral relationship between sine and cosine:
Power Rule: [tex]\int x^n ~dx=\frac{1}{n+1}x^{n+1} +C[/tex]sine-cosine integral relationship: [tex]\int sin(x) ~dx=-cos(x)+C[/tex][tex]f(x) = 12*(\frac{1}{3}x^3+C_1) - (-cos(x) + C_2)[/tex]
Simplifying
[tex]f(x) = 12*\frac{1}{3}x^3+12*C_1 +cos(x) + -C_2[/tex]
[tex]f(x) = 4x^3+cos(x) +(12C_1 -C_2)[/tex]
Ultimately, all of the constant of integration terms at the end can combine into one single unknown constant of integration:
[tex]f(x) = 4x^3 + cos(x) + C[/tex]
Step 2: Find the value of the constant of integration
Now, according to the problem, [tex]f(0) = -2[/tex], so we can substitute those x,y values into the equation and solve for the value of the constant of integration:
[tex]-2 = 4(0)^3 + cos(0) + C[/tex]
[tex]-2 = 0 + 1 + C[/tex]
[tex]-2 = 1 + C[/tex]
[tex]-3 = C[/tex]
Knowing the constant of integration, we now know the full equation for the function f:
[tex]f(x) = 4x^3 + cos(x) -3[/tex]
Step 3: Find the value of f for the given input
So, to find [tex]f(2\pi)[/tex], use 2 pi as the input, and simplify:
[tex]f(2\pi) = 4(2\pi)^3 + cos(2\pi) -3[/tex]
[tex]f(2\pi) = 4*8\pi^3 + 1 -3[/tex]
[tex]f(2\pi) = 32\pi^3 - 2[/tex]
Answer:
[tex]f(2 \pi)=32\pi^3-2[/tex]
Step-by-step explanation:
Fundamental Theorem of Calculus[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]
If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.
Given:
[tex]g(x)=12x^2-\sin x[/tex][tex]f(0)=-2[/tex]If g(x) is the first derivative of f(x), we can find f(x) by integrating g(x) and using f(0) = -2 to find the constant of integration.
[tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex] [tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $\sin x$}\\\\$\displaystyle \int \sin x\:\text{d}x=-\cos x+\text{C}$\end{minipage}}[/tex]
[tex]\begin{aligned} \displaystyle f(x)&=\int f'(x)\; \text{d}x\\\\&=\int g(x)\;\text{d}x\\\\&=\int (12x^2-\sin x)\;\text{d}x\\\\&=\int 12x^2\; \text{d}x-\int \sin x \; \text{d}x\\\\&=12\int x^2\; \text{d}x-\int \sin x \; \text{d}x\\\\&=12\cdot \dfrac{x^{(2+1)}}{2+1}-(-\cos x)+\text{C}\\\\&=4x^{3}+\cos x+\text{C}\end{aligned}[/tex]
To find the constant of integration, substitute f(0) = -2 and solve for C:
[tex]\begin{aligned}f(0)=4(0)^3+\cos (0) + \text{C}&=-2\\0+1+\text{C}&=-2\\\text{C}&=-3\end{aligned}[/tex]
Therefore, the equation of function f(x) is:
[tex]\boxed{f(x)=4x^3+ \cos x - 3}[/tex]
To find the value of f(2π), substitute x = 2π into function f(x):
[tex]\begin{aligned}f(2 \pi)&=4(2 \pi)^3+ \cos (2 \pi) - 3\\&=4\cdot 2^3 \cdot \pi^3+1 - 3\\&=32\pi^3-2\\\end{aligned}[/tex]
Therefore, the value of f(2π) is 32π³ - 2.
Your friend has a bag of yellow and purple candy. He wants to use them to play a game with you.
Purple is worth 2 points and yellow is worth three points. Pick 9 candies from the bag
Create a system of equations where the combination of the candies gives you a total of 22 points. Make sure you label and define your variables.
The system of equations where the combination of the candies gives you a total of 22 points are:
x + y = 9
2x + 3y = 22
How can we create system of equations?To create the equations, we shall first define the variables:
x = the number of purple candies
y = the number of yellow candies
Next, we shall use the given information that purple candies are worth 2 points and yellow candies are worth 3 points.
Then, we would make sure that the total number of candies selected is 9.
The first equation represents the total number of candies selected:
x + y = 9
The second equation represents the total number of points obtained:
2x + 3y = 22
Therefore, the system of equations is:
x + y = 9
2x + 3y = 22
This is a system of linear equations.
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In how many distinct ways can the digits in the number 2,563,183,083 be arranged?
(If there are any zero digits then assume they can be placed in any position)
How do I help my student understand this?
Cell 1 is in the "7th grade" row and "Pets" column. Have your student circle these labels or highlight them. Then it should be clear why choice E is the final answer.
Another way to rephrase it: the 35 people in cell 1 represent the number of 7th graders with pets.
Please if you know the answer put the steps on thank you.
Answer:
1. # of people who predicted they would pass = 30
2. # of people who predicted they would fail = 20
3. # of people who predicted they would pass and actually passed = 27
4. # of people who predicted they would pass and actually failed = 3
5. # of people who predicted they would fail and actually passed = 11
6. # of people who predicted they would fail and actually failed = 9
Step-by-step explanation:
1. # of people who predicted they would
The total number of people who took the test is 50.The number of people who predicted they would pass is 30.The number of people who predicted they would fail is 20 (since 50 - 30 = 20)Let x be the number of people who predicted fail and actually passed the test. Since three times as many people who passed predicted pass than predicted fail, we know that 3x is the number of people who predicted pass and actually passed the test. Therefore, the total number of people who passed the test is x + 3x = 4x, and we know that 36 people passed the test, so 4x = 36, and x = 9.Since x is the number of people who predicted fail and actually passed the test, then the number of people who predicted fail and actually failed the test is 20 - x = 20 - 9 = 11.The number of people who predicted pass and actually passed the test is 3x = 3(9) = 27.The number of people who predicted pass and actually failed the test is 30 - 27 = 3.I also filled in the frequency table by extracting it from Brainly and drawing on it to show how the math works and fits in the table.
Adam works as an account manager for a local insurance company. He makes $3500 per month, after taxes. He sticks to a strict monthly budget to make sure that he is able to pay for all of his expenses each month. Look at the list below of the items Adam must budget for each month. He gives a specific percentage of his income to each item. Use his monthly income and the percentage of each item to determine how much Adam will need to budget for each expense. 30% for rent/mortgage $ 15% for Insurances $ 12% for food $ 8% for utilities $ 10% for savings $ 5% for fun $ 7% for clothing $ 3% for personal items $ 10% charitable giving $ How much money will Adam have left over after budgeting for each item on the list? $
Answer:
see below
Step-by-step explanation:
Given a $3500 net income per month and a list of budget percentages, you want to know the budget amounts for each item, and the amount left over.
AmountsEach budget amount is the product of the net income and the associated budget percentage. Those products are ...
rent: $1050insurance: $525food: $420utilities: $280savings: $350fun: $175clothing: $245personal: $105charity: $350Left overThe sum of these amounts is $3500, so there will be $0 left over.
__
Additional comment
The list is stored in the variable 'b' by the first line in the calculator. Then these values are multiplied by the net income to get the budget amounts.
A scale drawing of a square has a sidle length of 9 inches. The drawing has a scale of 1 in. : 7 mi. Find the actual permitted and area of the square
Answer: To find the actual perimeter and area of the square, we need to use the scale given in the problem, which is 1 inch on the drawing represents 7 miles in real life.
The side length of the square on the drawing is 9 inches, so in real life, the side length would be:
9 inches x 7 miles/inch = 63 miles
Therefore, the actual perimeter of the square would be:
4 x 63 miles = 252 miles
To find the actual area of the square, we can use the formula:
area = side length x side length
In this case, the side length is 63 miles, so:
area = 63 miles x 63 miles = 3,969 square miles
So the actual perimeter of the square is 252 miles, and the actual area of the square is 3,969 square miles.
Step-by-step explanation: can i get brainliest :D
Find the value of x.
The value of x in the chord is 3 units.
How to find line segment when chord intersect?The chord intersection theorem states that the products of the lengths of the line segments on each chord are equal.
Therefore,
6(x + 5) = 4(2x + 6)
Open the brackets
6x + 30 = 8x + 24
subtract 8x from both sides of the equation
6x - 8x + 30 = 24
-2x + 30 = 24
subtract 30 from both sides of the equation'
-2x + 30 - 30 = 24 - 30
-2x = -6
divide both sides by -2
x = -6 / 2
x = 3
Therefore,
x = 3
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PLEASE HELP I have 30 minutes! What is the end behavior of this equation and how did you get it?
The limit of the function will be:
lim f(x) = ∞ as x → ∞
lim f(x) = 0 as x → -∞
How to explain the functionThe limit of f(x) as x goes towards infinity is infinite while the limit of f(x) approaches 0 when x tends to a negative value. Mathematically speaking,
lim f(x) = ∞ as x → ∞
lim f(x) = 0 as x → -∞
We can comprehend such behavior because the function's base is greater than 1 which explains how its growth accelerates as x increases and diminishes quickly as x declines towards zero.
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What is 15% of 100
Please help
Answer:
Step-by-step explanation:15
Does the infinite geometric series diverge or converge?
1/5 + 1/10 + 1/20 +1/40 + …
Answer:
the answer is C
Step-by-step explanation:
the answer is C it converges
Prove the following this : Let H be a subgroup of a group G . The left cosets of H constitute a partition of G ; the right cosets of H constitute a partition of G ?
To prove that the left cosets of H constitute a partition of G, we need to show that they satisfy two conditions:
1) Each element of G belongs to some left coset of H.
2) Any two distinct left cosets of H are disjoint.
Let g be any element of G. Then, by definition of a left coset, there exists an element h of H such that g = h·x for some x in G. Therefore, g belongs to the left coset H·x. Since this holds for any element g of G, we have shown that the left cosets of H cover all of G, satisfying the first condition.
Now, let H·x and H·y be two distinct left cosets of H. Suppose there exists an element z in both H·x and H·y.
Then, z = [tex]h_{1}[/tex]·x = [tex]h_{2}[/tex]·y for some [tex]h_{1}[/tex], [tex]h_{2}[/tex] in H.
Solving for x,
x = [tex]h_{1}^{-1}[/tex]· [tex]h_{2}[/tex] · y.
Since H is a subgroup, [tex]h_{1}^{-1}[/tex]· [tex]h_{2}[/tex] is also in H, and therefore x belongs to H·y.
Thus, H·x is contained entirely in H·y, and the two cosets cannot intersect.
This satisfies the second condition, and hence the left cosets of H constitute a partition of G.
A similar argument can be used to show that the right cosets of H also constitute a partition of G. This completes the proof.
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I will give brainlyest
Determine if the relationship between x and y is linear or not linear. Explain.
Answer: To determine if the relationship between x and y is linear, we need to graph the data and look for a straight-line pattern.
If the graph shows a straight-line pattern, then the relationship is linear. If the graph shows a curve or a non-linear pattern, then the relationship is not linear. So it is linear
Step-by-step explanation: can i get brainliest :D
Which equation can be used to solve for x
Answer:
(a) 7(x +3.5x) = 56
Step-by-step explanation:
You want an equation to solve for x, the length of a morning run, if the evening run is 3.5 times as long, and these runs total 56 miles when done every day of the week.
Daily runThe morning run is given as x.
The evening run is 3.5 times as long, so is 3.5x
The total mileage each day is (x +3.5x).
Weekly totalIn 7 days, the mileage will be 7 times the daily mileage. That total is given as 56 miles:
7(x +3.5x) = 56
Out of 600 people sampled, 42 had kids. Based on this, construct a 95% confidence interval for the true population proportion of people with kids.
Use GeoGebra to calculate! Give your answers as decimals, to three places
Answer:
So we can be 95% confident that the true proportion of people with kids in the population is between 0.044 and 0.096.
Step-by-step explanation:
To construct a confidence interval, we need to use the formula:
CI = p ± zsqrt((p(1-p))/n)
Where:
CI = confidence interval
p = sample proportion (in this case, 42/600 = 0.07)
z = the z-score corresponding to the desired level of confidence (in this case, 1.96 for a 95% confidence interval)
n = sample size (in this case, 600)
Plugging in the numbers, we get:
CI = 0.07 ± 1.96sqrt((0.07(1-0.07))/600)
Simplifying this, we get:
CI = 0.07 ± 0.026
Therefore, the 95% confidence interval for the true population proportion of people with kids is:
0.044 ≤ p ≤ 0.096
If x=3t+t^2 and y=7−2t, find an equation y=mx+b of the tangent to the curve at t=2.
m=
b=
The values of m and b after evaluation is -2 and 23 under the condition that If x=3t+t² and y=7−2t, and tangent to the curve at t=2.
In orde to evaluate the equation of the tangent to the curve at t=2, we have to calculate the slope of the curve at t=2.
The slope of the curve at t=2 is presented by the derivative of y with respect to x at t=2.
y = 7 - 2t
x = 3t + t²
dy/dx = -2
Then, the slope of the tangent to the curve at t=2 is -2.
Now we have to evaluate the point on the curve at t=2.
x = 3(2) + (2)² = 10
y = 7 - 2(2) = 3
So, the point on the curve at t=2 is (10,3).
Applying point-slope form of equation of line:
y - y₁ = m(x -x₁)
Here,
m = slope and (x₁,y1) is a point on the line.
Staging m=-2 and (x₁,y₁)=(10,3),
y - 3 = -2(x - 10)
Applying simplification
y = -2x + 23
Then, the equation of the tangent to the curve at t=2 is y=-2x+23.
So, m=-2 and b=23.
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-3k - 7 ≤ 17 help!!!!!!!!!!!!!!!
Factor the expression. x^2-12x-45
Answer:
(x + 3)(x - 15)
Step-by-step explanation:
To factor the expression x^2 - 12x - 45, we need to find two numbers that multiply to -45 and add to -12. Let's use the method of decomposition to find these two numbers:
x^2 - 12x - 45
We need to find two numbers that multiply to -45 and add to -12. Let's try -15 and 3:
x^2 - 12x - 45 = x^2 - 15x + 3x - 45
Now we can group the first two terms and the last two terms:
x^2 - 15x + 3x - 45 = x(x - 15) + 3(x - 15)
We can see that both terms have a common factor of (x - 15), so we can factor this out:
x(x - 15) + 3(x - 15) = (x + 3)(x - 15)
Therefore, the expression x^2 - 12x - 45 can be factored as (x + 3)(x - 15).
Refer to the attachment ^-^
Hope Helpful ~
Can someone help me with this please
The measure of x is 36.
We have,
The angle opposite to equal sides is equal.
And,
The sum of the triangle.
27 + 90 + (x + 27)= 180
27 + 90 + x + 27 = 180
x = 180 - 144
x = 36
Thus,
The measure of x is 36.
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Calculate, to the nearest cent, the Present Value of an investment that will be worth $1,000 at the stated interest rate after the stated amount of time.
3 years, at 1.3% per year, compounded weekly (52 times per year).
PV=?
[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 1000\\ P=\textit{original amount deposited}\\ r=rate\to 1.3\%\to \frac{1.3}{100}\dotfill &0.013\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{weekly, thus fifty two} \end{array}\dotfill &52\\ t=years\dotfill &3 \end{cases}[/tex]
[tex]1000 = P\left(1+\frac{0.013}{52}\right)^{52\cdot 3} \implies 1000=P(1.00025)^{156} \\\\\\ \cfrac{1000}{(1.00025)^{156}}=P\implies 961.76\approx P[/tex]
simplify the following:
Answer:
[tex]\dfrac{\left(12x-56\right)x}{\left(x-4\right)\left(-x^2+20x-32\right)}[/tex]
Step-by-step explanation:
Simplify
[tex]\dfrac{\dfrac{16}{x-2}-\dfrac{4}{x-4}}{\dfrac{16}{x}-\dfrac{x-4}{x-2}}[/tex]
The first-level numerator is
[tex]\dfrac{16}{x-4}-\dfrac{4}{x-2}\\\\\\= \dfrac{16(x-4) - 4(x - 2)}{(x-2)(x-4)}\\\\\\= \dfrac{16x -64 - 4x + 8}{(x-2)(x-4)}\\\\\\= \dfrac{12x -56}{(x-2)(x-4)}[/tex]
The first-level denominator is
[tex]\dfrac{16}{x}-\dfrac{x-4}{x-2}\\\\\\= \dfrac{16(x-2) - x(x-4)}{x(x-2)}\\\\\\\\= \dfrac{16x - 32 -x^2 +4x}{x(x-2)}[/tex]
[tex]= \dfrac{-x^2+20x-32}{x\left(x-2\right)}[/tex]
Therefore
[tex]\dfrac{\dfrac{16}{x-2}-\dfrac{4}{x-4}}{\dfrac{16}{x}-\dfrac{x-4}{x-2}}\\\\\\= \dfrac{12x -56}{(x-2)(x-4)} \div\dfrac{-x^2+20x-32}{x\left(x-2\right)} \\\\\\[/tex]
Use the fraction rule: [tex]\dfrac{a}{b} \div \dfrac{d}{c} = \dfrac{a}{b} \cdot \dfrac{d}{c}[/tex]
[tex]= \dfrac{12x -56}{(x-2)(x-4)} \cdot \dfrac{x(x-2)}{-x^2 +20x -32}}[/tex]
The (x-2) term cancels out resulting in:
[tex]\dfrac{\left(12x-56\right)x}{\left(x-4\right)\left(-x^2+20x-32\right)}[/tex]
2. Although Kevin has money, he is not spending it.
Complex or compound -complex
Answer:
complex
Step-by-step explanation:
The given sentence "Although Kevin has money, he is not spending it." is a complex sentence.
This is because it contains one independent clause, "he is not spending it," which can stand alone as a complete sentence. The other part of the sentence, "Although Kevin has money," is a dependent clause because it cannot stand alone as a complete sentence.
The dependent clause "Although Kevin has money" introduces a contrast or concession to the independent clause that follows it. Therefore, this sentence is an example of a complex sentence that uses a dependent clause to add meaning and complexity to the independent clause.
Each container holds 3 L 456 mL of water. How much water is in 206 identical containers?
Answer:
711 L 936 mL
Step-by-step explanation:
3 L 456 ml times 206=711L 936mL
I hope this helps
Please help will give a lot of points
Answer:
how many points will you give?
Step-by-step explanation:
(1/2 + 1/3) + [1/4 + (2/3)] ÷ 2/5 -4 x 5/6 ÷ 3/7
Answer:
Step-by-step explanation:
first, take 1/2+1/3 + 1/4+2/3
1/2+1/3=5/6 , 1/4+2/3=11/12
5/6+11/12=22/12
2/5-4=-18/5
-18/5*5/6=-3
22/12/3=22*3/12
=11/2
11/2/3/7=11/2*7/3
=77/6=12.8
How to write Two times the sum of y and 5 is 24
Then solve
Answer:
Step-by-step explanation:
2(y+5)=24
Can you help me with this question?
Answer:
C
Step-by-step explanation:
It mentions that they charge 1.20 per kg of lettuce
In tetrahedron $ABCO,$ $\angle AOB = \angle AOC = \angle BOC = 90^\circ.$ A cube is inscribed in the tetrahedron so that one of its vertices is at $O,$ and the opposite vertex lies on face $ABC.$ Let $a = OA,$ $b = OB,$ and $c = OC.$ Show that the side length of the cube is
\[\frac{abc}{ab + ac + bc}.\]
[asy]
import three;
size(180);
currentprojection = orthographic(6,3,2);
real a, b, c, s;
triple A, B, C, O;
a = 6;
b = 3;
c = 2;
s = a*b*c/(a*b + a*c + b*c);
A = (a,0,0);
B = (0,b,0);
C = (0,0,c);
O = (0,0,0);
draw(O--A,dashed);
draw(O--B,dashed);
draw(O--C,dashed);
draw(A--B--C--cycle);
draw((0,0,s)--(s,0,s)--(s,0,0)--(s,s,0)--(0,s,0)--(0,s,s)--cycle,dashed);
draw((s,s,0)--(s,s,s),dashed);
draw((s,0,s)--(s,s,s),dashed);
draw((0,s,s)--(s,s,s),dashed);
label("$A$", A, SW);
label("$B$", B, E);
label("$C$", C, N);
dot("$O$", O, NW);
dot((s,s,s));
[/asy]
pls help me
Answer:
Step-by-step explanation:To solve this problem, we can use similar triangles. Let the side length of the cube be s, and let the midpoints of AB, AC, and BC be P, Q, and R, respectively. Then OP, OQ, and OR are the diagonals of the faces of the cube, and so they are equal to .Consider triangle AOB. By the Pythagorean theorem, we have , so . Since angle AOB is 90 degrees, we can use similarity to see that triangle OAP is similar to triangle OAB, so . Solving, we find that .Using similar triangles, we can find that . Since the sum of the squares of the lengths of the diagonals of a cube is equal to three times the square of the side length, we have
$$(0P2 + 0Q? + OR?) = 35388 Substituting the values we found for $OP$, $OQ$, and $OR$, we haveSimplifying and solving for , we find that . Thus, the side length of the cube inscribed in the tetrahedron is .
Express 1:0.2:0.75 in their simple form
Answer:
20 : 4 : 15
Step-by-step explanation:
1 : 0.2 : 0.75
1 : 2/10 : 75/100
1 : 20/100 : 75/100
(×100)
100 : 20 : 75
(÷5)
20 : 4 : 15
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Javier and Naida are discussing the elevation of Lima, Peru.
200
150 Lima
100
50+
0+Sea level
-50-
-100+
-150
-200
The elevation of Lima, Peru is 154 meters.
Javier: Lima is 154] meters from sea level.
Naida: Lima is 154 meters above sea level.
Whose statement is true?
Choose 1 answer:
Answer:
Naida's statement is true
Step-by-step explanation:
Since the elevation of Lima, Peru is a positive number, you can automatically tell that it is above sea level since sea level is zero
Select all the correct answers.
If the measure of angle is which statements are true?
sin (0) = -
The measure of the reference angle is 30°.
cos (0) = √3
The measure of the reference angle is 45°.
tan (0) = -√3
The measure of the reference angle is 60°.
The measures of the trigonometric relations and reference angles are solved
Given data ,
Let the measure of the angle be θ = 2π/3
Now , from the trigonometric relation ,
The tangent of the function tan ( 2π/3 ) = -√3
And , the reference angle of the θ = 2π/3 is given by A = π/3
So , A = 60°
Hence , the reference angle is 60°
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