The only solution of the initial-value problem y" + x^2y = 0, y(0) = 0, y'(0) = 0 is y(x) = 0, where y(x) represents the unknown function and x represents the independent variable.
To determine the solution of the initial-value problem, we consider the given second-order linear homogeneous differential equation y" + x^2y = 0 along with the initial conditions y(0) = 0 and y'(0) = 0.
First, we solve the differential equation by assuming a solution of the form y(x) = Ax^n, where A is a constant and n is an exponent to be determined. Substituting this into the differential equation, we obtain the characteristic equation n(n-1) + x^2 = 0.
Solving the characteristic equation, we find that the roots are n = 0, which corresponds to the solution y(x) = A, and n = 1, which corresponds to the solution y(x) = Bx. However, when we apply the initial conditions y(0) = 0 and y'(0) = 0, we find that both solutions are equal to zero.
Therefore, the only solution that satisfies both the differential equation and the initial conditions is y(x) = 0, indicating that the function y(x) is identically zero for all values of x.
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Finding a Function to Match a Current Grade: 0.0/1.0 Remaining Time: Unlimited Shape For this week's discussion, you are asked to generate a continuous and differentiable function f(x) with the following properties: - f(x) is decreasing at x=−6 - f(x) has a local minimum at x=−3 - f(x) has a local maximum at x=3 Your classmates may have different criteria for their functions, so in your initial post in Brightspace be sure to list the criteria for your function. Hints: - Use calculus! - Before specifying a function f(x), first determine requirements for its derivative f ′
(x). For example, one of the requirements is that f ′
(−3)=0. - If you want to find a function g(x) such that g(−9)=0 and g(8)=0, then you could try g(x)=(x+9)(x−8). - If you have a possible function for f ′
(x), then use the techniques in Indefinite Integrals this Module to try a possible f(x). You can generate a plot of your function by clicking the plotting option (the page option with a "P" next to your function input). You may want to do this before clicking "How Did I Do?". Notice that the label " f(x)= " is already provided for you. Once you are ready to check your function, click "How Did I Do?" below (unlimited attempts). Please note that the bounds on the x-axis go from -6 to 6 .
To find a function that satisfies the given criteria, we can start by determining the requirements for its derivative, f'(x).
Let's break down the given properties and find the corresponding requirements for f'(x): f(x) is decreasing at x = -6: This means that the slope of the function should be negative at x = -6. Therefore, f'(-6) < 0. f(x) has a local minimum at x = -3: At a local minimum, the slope changes from negative to positive. Thus, f'(-3) = 0. f(x) has a local maximum at x = 3: At a local maximum, the slope changes from positive to negative. Hence, f'(3) = 0.
Now, let's integrate f'(x) to obtain f(x): Integrating f'(x) = -6 < x < -3 will give us a decreasing function on that interval. Integrating f'(x) = -3 < x < 3 will give us an increasing function on that interval. Integrating f'(x) = 3 < x < 6 will give us a decreasing function on that interval. To simplify the process, let's assume that f'(x) is a quadratic function with roots at -6, -3, and 3. We can represent it as: f'(x) = k(x + 6)(x + 3)(x - 3), where k is a constant that affects the steepness of the curve. By setting f'(-3) = 0, we find that k = -1/18.
Therefore, f'(x) = -1/18(x + 6)(x + 3)(x - 3). Integrating f'(x) will give us f(x): f(x) = ∫[-6,x] -1/18(t + 6)(t + 3)(t - 3) dt. Evaluating this integral is a bit complicated. Let's denote F(x) as the antiderivative of f(x): F(x) = ∫[-6,x] -1/18(t + 6)(t + 3)(t - 3) dt. Now, we can find f(x) by differentiating F(x): f(x) = d/dx[F(x)]. To get an explicit equation for f(x), we need to calculate the integral and differentiate the resulting antiderivative. Once you have the equation for f(x), you can plot it on the provided graphing option to verify that it matches the criteria mentioned in the question.
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help now please!!!!!! ^click picture
Answer:
A- lll
B- l
C- lV
D- V
E- ll
Step-by-step explanation:
I think that is the answer.
PLEASE HELP I DON'T UNDERSTAND!!!!!! I WILL MARK!!!!!!!!!!!!!!
Two functions are defined by the equations f(x)=5−0.2x and g(x)=0.2(x+5). all statements that are true about the functions.
Answer:
The true statements are -: A f(3)>0 , C g(-1)=0.8 and D g(-1)< f(-1)
Step-by-step explanation:
Given f(x)= 5- 0.2x and g(x)= 0.2(x+5)
(A) f(3) > 0
Putting this value in the given equation -
f(3) =5 -0.2(3)
f(3)=5 - 0.6
f(3) =4.4
Therefore , f(3) > 0 . This is a true statement .
(B) f(3) > 5
We got the value of f(3) from the above equation,
Therefore , f(3) = 4.4 , hence f(3) > 5 option is False.
(C) g(-1) = 0.8
Taking the second given equation ,
g(-1) = 0.2 ( -1+ 5)
g(-1) = 0.2 (-4)
Therefore, g(-1) = -0.8 . Hence , this statement is true.
(D) g(-1) < f(-1)
g(-1) = -0.8
f(-1) =5 - 0.2(-1)
f(-1) = 5 +0.2
f(-1) = 5.2
Therefore , g(-1) < f(-1) , Hence this option is true.
(E) f(0) =g(0)
f(0) = 5 + 0.2(0)
f(0) = 5
g(0) = 0.2(0+5)
g(0) = 0.2 +5
g(0) = 1
Therefore , f(0)[tex]\neq[/tex] g(0) , Hence , the option is false .
Hence , true statements about the functions are - A , C , D .
I've calculated the area inside the circle r=3acos(θ)
and outside the cardioid r=a(1+cos(θ))
The integral for the area becomes:
A = ∫₀ᵃʳᶜᶜᵒˢ(ᵃ/₂) ∫ₐ(₁+ᶜᵒˢ(θ))³ᵃᶜᵒˢ(θ) r dr dθ
To find the area inside the circle r = 3acos(θ) and outside the cardioid r = a(1 + cos(θ)), we can set up a double integral in polar coordinates.
First, let's find the points of intersection between the two curves. The circle r = 3acos(θ) and the cardioid r = a(1 + cos(θ)) intersect when:
3acos(θ) = a(1 + cos(θ))
Simplifying, we get:
3acos(θ) - a(1 + cos(θ)) = 0
2acos(θ) - a = 0
acos(θ) = a/2
θ = arccos(a/2)
Now, let's set up the integral. We want to find the area inside the circle and outside the cardioid, so the region of integration is defined by:
0 ≤ θ ≤ arccos(a/2)
a(1 + cos(θ)) ≤ r ≤ 3acos(θ)
Evaluating this double integral will give us the desired area inside the circle and outside the cardioid.
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f(x) = x2. What is g(x)?
A. g(x) = -2-3
B. (X) = -x2
C. g(x) = x2-3
D. g(x) = -3/2
Answer: B
Step-by-step explanation:
The value g(x) if f(x) = x2 is g(x) = x2-3, the correct option is C.
What is a function?Function is a type of relation, or rule, that maps one input to specific single output.
In mathematics, a function is an expression, rule, or law that describes the relationship between one variable (the independent variable) and another variable (the dependent variable) (the dependent variable). In mathematics and the physical sciences, functions are indispensable for formulating physical relationships.
Linear function is a function whose graph is a straight line
We are given that;
The function f(x)=x2
Now,
In the graph of g(x)
Coordinate x=0
Coordinate y=-3
Plugging in the values
g(x) = x2-y
y=3
g(x)=x2-3
Therefore, by the given function the answer will be g(x)=x2-3.
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For her birthday, Gale received $200. She would like to spend all or some of the money to take
guitar lessons. After an initial $50 fee to cover rental equipment, each guitar lesson will cost $25.
Write an inequality to determine how many lessons Gale could take using her birthday money.
Highlight your inequality in green. Then, solve the inequality.
Answer:
The inequality that can be used to determine the number of guitar lessons Gale could take using her birthday money is:
[tex]25x + 50 \leqslant 200[/tex]
The solution to the inequality is:
[tex]x \leqslant 6[/tex]
She can take at most, 6 guitar lessons
Step-by-step explanation:
Total amount Gale received is $200
For rental equipment, she paid $50
Each guitar lesson will cost $25
Let x represent the number of lessons she could take, then the total amount she will spend must bot exceed $200
The guitar lesson will cost a total of 25x
So, her total spending will be:
[tex]25x + 50[/tex]
This must be at most $200
[tex]25x + 50 \leqslant 20[/tex]
Solving the above inequality:
[tex]25x \leqslant 200 - 50 = 150[/tex]
[tex]x \leqslant \frac{150}{25} = 6[/tex]
Glen is driving to the mega
mall that is 30 miles away from his
mom's house. On Glen's map app, his mom's house and the mega
mall are 5 inches apart. What is the scale on Glen's map?
A. 1 inch = 150 miles
B. - 1 inch = 35 miles
C. 1 inch = 25 miles
D. 1 inch = 6 miles
Answer:
D
Step-by-step explanation:
•°|NO LINKS PLEASE, LOOKING FOR REAL ANSWERS|°•
What is the formula for finding the surface area of a square pyramid?
Answer:
a2 + 2al (or) a2 + √a24+h2 a 2 4 + h 2
Step-by-step explanation:
Answer:
SA=L²+2L√(L²/4+H²)
Step-by-step explanation:
Where H is the height and L is the length
Help please I’ll give brainlest
Answer:
1m²
Step-by-step explanation:
Answer:
A = 157.3 units²
Step-by-step explanation:
A = 1/2(6.9)(8 x 5.7) = 157.3 units²
Find the 82nd term of the following sequence.
-32, -27, -22, ....
Why is it important to know the background of a poet.
Answer:
To understand their poems better
Step-by-step explanation:
First you have a general knowledge on why the author made the poem
Second you now have more info to use in order to understand the poems
Third is the same reason why there are people in history books.
Which of the following shows the polynomial below written in descending
order?
Answer:
A. 4x¹² + 9x⁷ + 3x³ -x
Step-by-step explanation:
Hi!
==================================================================
To write a polynomial in descending order, we write the terms with higher degrees, or exponents, first.
3x³ + 9x⁷ -x + 4x¹²
4x¹² has the highest degree, so it is written as the first term.
⇒4x¹²
9x⁷ has the next highest degree, so it is written next.
⇒4x¹² + 9x⁷
3x³ has the next highest degree, so it is written next.
⇒4x¹² + 9x⁷ + 3x³
-x has the lowest degree, so it is written last.
⇒4x¹² + 9x⁷ + 3x³ -x
4x¹² + 9x⁷ + 3x³ -x
==================================================================
Hope I Helped, Feel free to ask any questions to clarify :)
Have a great day!
-Aadi x
the arc measure of a sector in a given circle is doubled. will the area of the sector also be doubled? explain your reason
Answer:
Yes. See explanation below.
Step-by-step explanation:
The central angle and the degree arc measure of a sector of a circle are equal. Doubling the arc measure, doubles the central angle measure and vice versa.
Area of the the original sector:
[tex] A_{sector} = \dfrac{n}{360^\circ}\pi r^2 [/tex]
where n = measure of the central angle of the sector
Since the central angle and the arc measure of the sector are equal, changing the arc measure has the same effect as changing the central angle measure.
Let's double the central angle to 2n which is the same as doubling the arc measure.
Area of the sector with a doubled central angle or a doubled arc measure:
[tex] A_{sector} = \dfrac{2n}{360^\circ}\pi r^2 [/tex]
Now we divide the area of the doubled sector by the area of the original sector.
[tex]\dfrac{\frac{2n}{360^\circ}\pi r^2}{\frac{n}{360^\circ}\pi r^2} =[/tex]
Simplify:
[tex]= \dfrac{2n}{n} \times \dfrac{360^\circ \pi r^2}{360^\circ \pi r^2}[/tex]
[tex] = \dfrac{2n}{n} [/tex]
[tex] = 2 [/tex]
The ratio of the areas is 2, so the area of the sector is indeed doubled.
Answer: Yes.
please help.I don’t understand
Answer:
y=13 degrees
Step-by-step explanation:
This is an isosceles triangle, we know this because NO and NM are equal.
In an isosceles triangle, the base angles are congruent. In this case, they are angle NOM and angle NMO.
We also know that the sum of the interior angles of a triangle are equal to 180.
With this information, we can make an equation by gathering all the interior angles:
8y+2(3y-1)=180
Solve for y.
8y+6y-2=180
14y-2=180
14y=182
y=13
3.
Convert the recurring decimal to fraction
A) 1.45°
Answer:
A
Step-by-step explanation:
A flywheel (I = 185.0 kg m2) rotating counterclockwise at 350.0 rev/min is brought to rest by friction in 5.0 min. What is the frictional torque on the flywheel (in N m)? (Indicate the direction with the sign of your answer
The frictional torque on the flywheel is `22.58 N.m` in the clockwise direction.
The formula for angular velocity is given by;`ω = (2π / T)`.Where;ω = angular velocity of the object. T = time period (in seconds).`I = 185.0 kg m2` represents the moment of inertia of the flywheel.`ω = 350.0 rev/min = (350.0 * 2π) / 60 = 36.61 rad/s` represents the initial angular velocity of the flywheel.
The flywheel is brought to rest by friction in `5.0 min = 5.0 * 60 = 300 seconds`.
The formula for the angular acceleration is given by;`α = (ωf - ωi) / t`. Where;`α` = angular acceleration of the object.`ωi` = initial angular velocity.`ωf` = final angular velocity of the object.`t` = time taken (in seconds).
At rest, the final angular velocity of the flywheel is zero.
Therefore;`α = (- ωi) / t`.The formula for torque is given by;`τ = I * α`.Where;τ = torque exerted on the object.I = moment of inertia of the object.α = angular acceleration of the object.
Substituting the values;`τ = I * α = 185.0 * (-36.61) / 300 = -22.58 N.m`.
Therefore, the frictional torque on the flywheel is `22.58 N.m` in the clockwise direction.
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A shelf is designed so it will fit in a 90º corner between two walls. The shelf has dimensions, rounded to the nearest tenth, as shown. A triangle has side lengths 11 centimeters, 11 centimeters, and hypotenuse of StartRoot 242 EndRoot centimeters. [Not drawn to scale] Will the shelf fit snugly in a 90º corner?
Answer:
Yes, it will fit snugly in a 90º corner
Step-by-step explanation:
To do this, we simply need to check if the given sides of the shelf is right-angled.
So, we have:
[tex]Hypotenuse = \sqrt{242[/tex]
[tex]Side\ 1 = Side\ 2 = 11[/tex]
To check for right-angle triangle, we make use of:
[tex]Side\ 1^2 + Side\ 2^2 = Hypotenuse^2[/tex]
This gives:
[tex]11^2 + 11^2 = \sqrt{242}^2[/tex]
[tex]121 + 121 = \sqrt{242}^2[/tex]
[tex]242 = \sqrt{242}^2[/tex]
[tex]242 = 242[/tex]
This shows that the given sides of the shelf is a right-angled triangle.
Hence, it will fit the wall
Answer:
A
Step-by-step explanation:
find the slope of the line through each pair of points
a. (8,-7) and (5,-3)
b. (-5,9) and (5,11)
c. (-8,-4) and (-4,-9)
Answer:
I think the answers would be :
a . 4/3
b. 1/5
c . - 5/4
hope it helps u ^^
!!!!NEED HELP ASAP DUE SOON!!!!!This table of values represents a linear function. Enter an equation that represents the function defined by this table of values.
Answer:
y=3x+6
Step-by-step explanation:
Just test this on like a graph or something and you will see it works. Let me know if there is a fault in my answer. Thanks! Have a good day.
THE ANSWER IS NOT LETTER b. 2
Independent Practice
Now practice solving some problems.
Which number is a solution of the inequality ?
4−m/m ≥ 5
A.
0.5
B.
2
C.
–4
D.
0.75
Answer:
A.
0.5
Step-by-step explanation:
Take a factor out of the square root:
a) √6x^2, where x≥0
b)√9a^3
d)√50b^4
plz help 30 points will give brainliest
Answer:
Question A)
[tex]=\sqrt{6}x[/tex]
Question B)
[tex]=3a\sqrt{a}[/tex]
Question C)
[tex]=5\sqrt{2}b^2[/tex]
Step-by-step explanation:
A)
We are given:
[tex]\sqrt{6x^2}\, \text{ where } x\geq 0[/tex]
We can rewrite the expression:
[tex]=\sqrt{6}\cdot \sqrt{x^2}[/tex]
The square root and square will cancel each other out. Thus:
[tex]=\sqrt{6}x[/tex]
B)
We are given:
[tex]\sqrt{9a^3}[/tex]
Rewrite:
[tex]=\sqrt{9}\cdot \sqrt{a^3}[/tex]
Note that the square root of 9 is simply 3. We can also factor the second part:
[tex]=3\cdot \sqrt{a^2\cdot a}[/tex]
Rewriting:
[tex]=3\cdot\sqrt{a^2}\cdot\sqrt{a}[/tex]
Simplify:
[tex]=3a\sqrt{a}[/tex]
C)
We are given:
[tex]\sqrt{50b^4}[/tex]
Rewrite. Note that 50 = 25(2):
[tex]=\sqrt{25}\cdot \sqrt{2}\cdot \sqrt{b^4}[/tex]
Simplify. We can rewrite the factor as:
[tex]=5\cdot \sqrt{2}\cdot \sqrt{(b^2)^2}[/tex]
The square and square root will cancel out. Thus:
[tex]=5\sqrt{2}b^2[/tex]
The probability of flu symptoms for a person not receiving any treatment is 0.021. In a clinical trial of a common drug used to lower cholesterol. 26 of 1133 people treated experienced flu symptoms. Assuming the drug has no effect on the likelihood of flu symptoms, estimate the probability that at least 26 people experience fiu symptoms. What do these results suggest about flu symptoms as an adverse reaction to the drug? (a) P(X226)
Using binomial probability, the probability that at least 26 people experience flu symptoms is 0.8788 and It indicates a potential adverse reaction to the drug, and further investigation or monitoring may be required to assess the relationship between the drug and flu symptoms.
What is the probability that at least 26 people experience flu symptoms?To estimate the probability that at least 26 people experience flu symptoms, we can use the binomial distribution. The probability of flu symptoms for a person not receiving any treatment is given as 0.021.
Let's denote the number of people treated who experience flu symptoms as X, which follows a binomial distribution with parameters n (number of trials) and p (probability of success).
In this case, n = 1133 (number of people treated) and p = 0.021 (probability of flu symptoms for a person not receiving treatment).
We want to find P(X ≥ 26), which is the probability that at least 26 people experience flu symptoms. To calculate this probability, we need to sum the probabilities of the complementary event (X < 26), which is the probability that fewer than 26 people experience flu symptoms.
Using statistical software or a binomial distribution table, we can calculate the probability as follows:
P(X ≥ 26) = 1 - P(X < 26)
Using the binomial distribution, we can calculate the probability of X < 26 as:
P(X < 26) = Σ[P(X = k)] for k = 0 to 25
Now, let's calculate the probability:
P(X < 26) = Σ[C(n, k) * p^k * (1 - p)^(n - k)] for k = 0 to 25
However, calculating this probability manually for each value of k can be cumbersome. Therefore, I'll use statistical software to estimate the probability.
Using the binomial distribution with parameters n = 1133 and p = 0.021, we find:
P(X < 26) ≈ 0.1212
Therefore, the estimated probability that at least 26 people experience flu symptoms is:
P(X ≥ 26) ≈ 1 - 0.1212 ≈ 0.8788
This suggests that the observed number of people experiencing flu symptoms (26) is higher than what would be expected if the drug had no effect on the likelihood of flu symptoms.
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identify the pattern then write the next three terms in each sequence 2,8,32,128
Step-by-step explanation:
the difference is the previous term multiplied by 4 to get the next term
2,8,32,128,512,2048,8192
Hope that helps :)
Solve 7 sin(2x) = 6 for the two smallest positive solutions A and B, with A
To solve the equation 7 sin(2x) = 6 for the two smallest positive solutions A and B, we can use algebraic techniques and trigonometric properties.
The solutions A and B are approximately equal to A ≈ 0.287 and B ≈ 1.569, respectively.
To explain the solution, let's begin by rearranging the equation: sin(2x) = 6/7. Since the range of the sine function is between -1 and 1, the equation has solutions only if 6/7 is within this range. We can find the corresponding angles by taking the inverse sine (arcsin) of 6/7. Using a calculator, we find that the arcsin(6/7) is approximately 0.942.
However, this gives us only one of the solutions. To find the other solution, we can use the periodicity of the sine function. We know that sin(θ) = sin(π - θ), where θ is the angle in radians. Therefore, the second solution is π - 0.942, which is approximately 2.199. However, since we're looking for the smallest positive solutions, we need to consider only the values between 0 and 2π. Thus, the two smallest positive solutions are A ≈ 0.287 and B ≈ 1.569.
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In quadratic functions/equations; roots/solutions/x-intercepts/zeros all represent the same values/points.
True
False
Answer:
True
Step-by-step explanation:
Solutions to quadratic equations are often called
roots / zeros / x- intercepts
Carla wants to save $55.50 to buy a new video game. Carla babysits her niece once a week and earns the same amount of money each week. After every time she babysits she donates $2 from the money she earned to the local food bank. Carla calculates that it will take her 6 weeks to save enough to buy her video game. Write and solve an equation to determine how much money Carla earns per week.
Answer:
$11.25
Step-by-step explanation:
55.50 divided by 6 = 9.25
9.25 + 2 = 11.25
Vince is saving for a new mobile phone. The least expensive model Vince likes costs $225.90. Vince has saved $122.35. He used this solution to determine how much more he needs to save.
225.90 less-than-or-equal-to 122.35 + a. 225.90 minus 122.35 less-than-or-equal-to 122.35 minus 122.35 + a. 103.55 less-than-or-equal-to a.
Vince says that based on the solution, he should save a maximum of $103.55.
Is Vince correct?
Vince is correct because he found the correct solution to the inequality.
Vince is correct because he should save at least $103.55.
Vince is not correct because he wrote the wrong inequality to represent the situation.
Vince is not correct because he should have interpreted the solution as having to save a minimum of $103.55.
Answer:
Vince is not correct because he should have interpreted the solution as having to save a minimum of $103.55.
Step-by-step explanation:my
my sister which is in college helped me with one.
Answer: its D
Step-by-step explanation:
7) Find the perimeter of triangle ABC. Round all answers to the nearest tenth.[6 points)
A
Please help!!! I’ll mark brain list too!
Answer: 14.6
Step-by-step explanation:
I made a square around the triangle which I then counted the squares, found the Pythagorean theorem, and then added the missing sides together
Solve the initial value problem below using the method of Laplace transforms
y" + 5y' + 6y-24 e t, y(0) -5, y'(0)-19 Click here to view the table of Laplace transforms Click here to view the table of properties of Laplace transforms y(t)= __
(Type an exact answer in terms of e.)
To solve the given initial value problem using the method of Laplace transforms, we'll take the Laplace transform of both sides of the differential equation. Let's denote the Laplace transform of the function y(t) as Y(s).
The Laplace transform of the second derivative y" is s²Y(s) - sy(0) - y'(0), where y(0) and y'(0) are the initial conditions given.
The Laplace transform of the first derivative y' is sY(s) - y(0).
The Laplace transform of the term 6y is 6Y(s).
The Laplace transform of the term -24e^t can be found using the table of Laplace transforms.
Applying the Laplace transform to the entire differential equation, we get:
s²Y(s) - sy(0) - y'(0) + 5(sY(s) - y(0)) + 6Y(s) - 24/(s-1) = 0
Substituting the initial conditions y(0) = -5 and y'(0) = -19, we have:
s²Y(s) + 5sY(s) + 6Y(s) - 5s + 19 - 24/(s-1) = 0
Now, we can solve this equation for Y(s). Once we find Y(s), we can take the inverse Laplace transform to obtain y(t), the solution to the initial value problem.
Since the given question doesn't specify a particular form for Y(s), I'm unable to provide the exact solution y(t) in terms of e.
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