a) Does the following improper integral converge or diverge? Show your reasoning -20 6 re-21 dt (b) Apply an appropriate trigonometric substitution to confirm that L'4V1 –c?dx == 47 7T (c) Find the general solution to the following differential equation. dy (+ - 2) = 3, 1-2, 1 da
Answers
(a) The improper integral ∫[0,∞] \((xe^(-2x)dx)\) converges.
(b) To evaluate the integral ∫[0,1] \((4\sqrt{1-x^2}dx)\), we can use the trigonometric substitution x = sin(θ).
(c) The general solution to the given differential equation is y = ln|x + 2| - ln|x - 1| + C.
(a) To determine if the improper integral ∫[0,∞] \((xe^{-2x}dx)\) converges or diverges, we can use the limit comparison test.
Let's consider the function f(x) = x and the function g(x) = \(e^{-2x}\).
Since both f(x) and g(x) are positive and continuous on the interval [0,∞], we can compare the integrals of f(x) and g(x) to determine the convergence or divergence of the given integral.
We have ∫[0,∞] (x dx) and ∫[0,∞] \((e^(-2x) dx)\).
The integral of f(x) is ∫[0,∞] (x dx) = [\(x^2/2\)] evaluated from 0 to ∞, which gives us [∞\(^2/2\)] - [\(0^2/2\)] = ∞.
The integral of g(x) is ∫[0,∞] \((e^{-2x} dx)\) = \([-e^{-2x}/2]\) evaluated from 0 to ∞, which gives us [\(-e^{-2\infty}/2\)] - [\(-e^0/2\)] = [0/2] - [-1/2] = 1/2.
Since the integral of g(x) is finite and positive, while the integral of f(x) is infinite, we can conclude that the given integral ∫[0,∞] (\(xe^{-2x}dx\)) converges.
(b) To evaluate the integral ∫[0,1] (4√(\(1-x^2\))dx), we can make the trigonometric substitution x = sin(θ).
When x = 0, we have sin(θ) = 0, so θ = 0.
When x = 1, we have sin(θ) = 1, so θ = π/2.
Differentiating x = sin(θ) with respect to θ, we get dx = cos(θ) dθ.
Now, substituting x = sin(θ) and dx = cos(θ) dθ in the integral, we have:
∫[0,1] (4√(\(1-x^2\))dx) = ∫[0,π/2] (4√(1-\(sin^2\)(θ)))cos(θ) dθ.
Simplifying the integrand, we have √(1-\(sin^2\)(θ)) = cos(θ).
Therefore, the integral becomes:
∫[0,π/2] (4\(cos^2\)(θ)cos(θ)) dθ = ∫[0,π/2] (4\(cos^3\)(θ)) dθ.
Now, we can integrate the function 4\(cos^3\)(θ) using standard integration techniques:
∫[0,π/2] (4\(cos^3\)(θ)) dθ = [sin(θ) + (3/4)sin(3θ)] evaluated from 0 to π/2.
Plugging in the values, we get:
[sin(π/2) + (3/4)sin(3(π/2))] - [sin(0) + (3/4)sin(3(0))] = [1 + (3/4)(-1)] - [0 + 0] = 1 - 3/4 = 1/4.
Therefore, the value of the integral ∫[0,1] (4√(\(1-x^2\))dx) is 1/4.
(c) To find the general solution to the differential equation (\(x^2 + x - 2\))(dy/dx) = 3, for x ≠ -2, 1, we need to separate the variables and integrate both sides.
(dy/dx) = 3 / (\(x^2 + x - 2\)).
∫(dy/dx) dx = ∫(3 / (\(x^2 + x - 2\))) dx.
Integrating the left side gives us \(y + C_1\), where \(C_1\) is the constant of integration.
To evaluate the integral on the right side, we can factor the denominator:
∫(3 / (\(x^2 + x - 2\))) dx = ∫(3 / ((x + 2)(x - 1))) dx.
Using partial fractions, we can express the integrand as:
3 / ((x + 2)(x - 1)) = A / (x + 2) + B / (x - 1).
Multiplying both sides by (x + 2)(x - 1), we have:
3 = A(x - 1) + B(x + 2).
Expanding and equating coefficients, we get:
0x + 3 = (A + B)x + (-A + 2B).
Equating the coefficients of like terms, we have:
A + B = 0,
- A + 2B = 3.
Solving this system of equations, we find A = -3 and B = 3.
3 / ((x + 2)(x - 1)) = (-3 / (x + 2)) + (3 / (x - 1)).
∫(3 / (\(x^2 + x - 2\))) dx = -3∫(1 / (x + 2)) dx + 3∫(1 / (x - 1)) dx.
-3ln|x + 2| + 3ln|x - 1| + C2,
where C2 is another constant of integration.
Therefore, the general solution to the differential equation is:
y = -3ln|x + 2| + 3ln|x - 1| + C,
where C = C1 + C2 is the combined constant of integration.
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Related Questions
Solve the equation: 4 - 2y + 3 = - 9. What is y equal to?
Answers
Answer:
b
Step-by-step explanation:
In the given equation 4 - 2y + 3 = - 9, y is equal to 8.
The equation is a statement in which the values of two mathematical expressions are equal (indicated by the sign =).
To find the value of y from the given equation
∴ 4 - 2y + 3 = - 9
=> 4+3 -2y = -9
=> 7 - 2y = -9
=> 7 + 9 = 2y
=> 16 = 2y
=> 16/2 = y
=> 8 = y
=> y = 8
So, the value of y in the above equation 4 - 2y + 3 = -9 is 8.
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how would you solve this equation?
Answers
Answer:
19,188
Step-by-step explanation:
hope this helps ;)
A ferris wheel is 45 meters in diameter and boarded from a platform that is 4 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. What is the Amplitude? meters What is the Midline? y = meters What is the Period? minutes How High are you off of the ground after 4 minutes? meters
Answers
If the wheel rotates clockwise, the height function is h(t) = 28.5 + 22.5sin(270 - 36t).
The function's central axis for the height of the Ferris wheel's center is given by h = f(t).
= 6 + 45 ÷ 2 = 6 + 22.5 = 28.5 meters.
The wheel rotates at a speed of 36 degrees per minute since the period of revolution is 10 minutes.
The wheel's direction of rotation—clockwise or counterclockwise—is not specified.
Let's look at both situations:
The present angle, assuming it spins clockwise, is equal to 270 – 36t degrees, where t is the duration in minutes.
Then the height function h = f(t)
= 28.5 + 22.5sin(a)
= 28.5 + 22.5sin(270 - 36t)
If the wheel rotates anti-clockwise, the current angle is
b = 270 + 36t degrees.
Then the height function
h = r(t)
= 28.5 + 22.5sin(b)
= 28.5 + 22.5sin(270 + 36t)
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I wil Mark BRANLIEST!! HELP!!!! Consider the graphs of linear function f(x) = 3x, quadratic function g(x) = 3x2, and exponential function h(x) = 2x. Which statement about the functions f, g, and h is correct?
A. As x increases, the value of f(x) will eventually exceed the values of g(x) and h(x).
B. As x increases, the value of g(x) will eventually exceed the values of f(x) and h(x).
C. As x increases, the value of h(x) will eventually exceed the values of f(x) and g(x).
D. As x increases, the values of both f(x) and g(x) will eventually exceed the value of h(x).
Here the Graph: https://www.savvasrealize.com/community/proxy/assessment/68bd51a0599744be92849bd2e08be180/images/bdbcc6b2-8670-4ddc-95a2-a48b6d6da348
Answers
Answer:
option C
Step-by-step explanation:
Given :
\(f(x) = 3x \\\\g(x) = 3x^2\\\\h(x) = 2^x\)
We will just put values for x and check the function :
Let x = 1
f(x) = 3
g(x) = 3
h(x) = 2
Let x = 10,
f(x) = 30
g(x) = 300
h(x) = 1024
Let x = 100
f(x) = 300
g(x) = 30000
h(x) = \(1.27 \times 10^{30}\)
Clearly , as x increases value of h(x) exceeds the value of f(x) and g(x).
how do you solve -4 + y = 6 and y= 7x - -6 by substitution?
Answers
Answer:
by doing the math
Step-by-step explanation:
do (7x)-(-6) then the other problem and find out witch one is more
Answer:
x=4/7 and y=10
Step-by-step explanation:
To solve for x, we must isolate y in one of the equations to then insert it into the other equation. We can take the first equation, -4 + y = 6, and add 4 to both sides. This gives us y = 4+6, or y = 10. Since y is a variable, we can subsititute 10 into the second equation, y = 7x - (-6), giving us 10 = 7x - (-6). The minus negative six is equal to plus six. This gives us 10 = 7x +6. Now we can solve for x. Subtract 6 from both sides; 4 = 7x. Now to find x, we have to divide both sides by 7. This gives us x = 4/7.
use lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter p is a square. let the sides of the rectangle be x and y and let f and g represent the area (a) and perimeter (p), respectively. find the following.
Answers
we get x=y =0 which gives 0 perimeter or x=y this implies rectangle must be a square
1. Method of Langrage Multipliers:
To find the extremum values of f(x,y) subject to constraint g(x,y) = k
find all values of x,y and λ, such that :
Δλ(f,x) = λΔg(x,y)
And g(x,y) = k
2. let the two side of the rectangle be x and y
therefore
f(x,y) = xy And g(x,y)= 2(x+y)=p
fₓ=λgₓ => y=2λ ----------------- 1
fy = λgy => x = 2λ----------------2
using equation and 1 and 2
λ=0, but this is not possible because tis implies x = y = 0, which gives 0 perimeter
or
x=y
Hence the rectangle must be a square
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Write the equation of the line that passes through the points (2,-6)and (8,−2). Put your answer in fully reduced point-slope form, unless it is a vertical or horizontal line.
please help meeee
Answers
Answer:
Step-by-step explanation:
45 % of z is 72. Find the value of z
Answers
9/20z=72
z=72/ 9/20
z=160
please answer this question
Answers
The value of m<2 = 107
Given:
m<SOX = 160
m<1 = x+14
m<2 = 3x - 10
x + 14 + 3x - 10 = 160
4x + 14 - 10 = 160
4x + 4 = 160
4x = 160 - 4
4x = 156
divide by 4 on both sides
4x/4 = 156/4
x = 39
m<2 = 3*39 - 10
= 117 - 10
= 107
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What conjecture can you make about the sum of the first 19 odd numbers?
Answers
Answer: 1, 3, 5, 7, 9, . . . . , 37. Therefore, 361 is the sum of first 19 odd numbers.
Step-by-step explanation:
Find f
f ''(t) = 2et + 7 sin t, f(0) = 0, f(π) = 0
Answers
After solving the differential equation f ''(t) = 2et + 7 sin t to find function t we get f(t) = (1/3)et³ - 7 sin t + (1/3)e(π)³t.
We need to integrate the given second derivative twice with respect to t.
First, we integrate f''(t) with respect to t to obtain f'(t):
f'(t) = ∫(2et + 7 sin t) dt
= et^2 - 7 cos t + C1
where C1 is the constant of integration.
Next, we integrate f'(t) with respect to t to obtain f(t):
f(t) = ∫(et^2 - 7 cos t + C1) dt
= (1/3)et^3 - 7 sin t + C1t + C2
where C2 is the constant of integration.
Using the initial conditions f(0) = 0 and f(π) = 0,
we can solve for C1 and C2:
f(0) = (1/3)e(0)^3 - 7 sin 0 + C1(0) + C2 = 0
Therefore, C2 = 0
f(π) = (1/3)e(π)^3 - 7 sin π + C1(π) + C2 = 0
(1/3)e(π)^3 - C1 = 0
C1 = (1/3)e(π)^3
Therefore, the solution to the differential equation is:
f(t) = (1/3)et³ - 7 sin t + (1/3)e(π)³t.
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1. Deborah finds that the theoretical probability of flipping "heads" on a fair coin was
50%. After she flipped the fair coin 100 times, she calculated that she Flipped "heads"
45 times. What is the percent difference in theoretical and experimental probability?
Answers
Answer:
29
Step-by-step explanation:
If you calculate the difference with the 45 and percentage you will get 29 including with the X and - should be east
Junior has one gallon of red paint and one gallon of blue paint. He takes one cup of red paint and adds it to the gallon of blue paint. After mixing it thoroughly, he takes out one cup of the mixed paint. Inside this cup the ratio of paint that is blue is a/b as a reduced common fraction. What is the value of A+B
Answers
One possible way to approach the problem is to use ratios and proportions. Let's start by finding the ratio of red paint to blue paint in the final mixed paint cup.
Step 1: Find the ratio of red to blue paint in each of the original paints.Junior has one gallon of red paint and one gallon of blue paint. Since one gallon is equal to 16 cups, we have:Red paint: 16 cups red paintBlue paint: 16 cups blue paint
Step 2: Mix one cup of red paint with one gallon of blue paint.In the second step, Junior takes one cup of red paint and adds it to the gallon of blue paint. Now we have:Red paint: 15 cups red paintBlue paint: 17 cups blue paint
Step 3: Take out one cup of the mixed paint.In the third step, Junior takes out one cup of the mixed paint. Now we have: Red paint: 15 cups red paint Blue paint
16 cups blue paint
The ratio of blue paint to total paint (blue paint plus red paint) in the final mixed paint cup is'
16/31, or 16/31 = a/b, '
where a and b are integers that have no common factors other than 1.To find the value of A+B, we need to determine the values of a and b.
We can do this by setting up a proportion:
16/31 = a/b
Multiplying both sides by the product of the denominators:
16b = 31a
Solving for a: a = (16b)/31 Since a is an integer, b must be divisible by 31.
The smallest such value of b is 31, which gives a = 16.A + B = 16 + 31 = 47Therefore, the value of A+B is 47.
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pls help me with this if you know it also to explain it thanks Solve the equation -21 + 70n = 77 for n.
Answers
Answer:
n = 1 2/5
Step-by-step explanation:
-21 + 70n = 77
Add 21 to each side
-21+21 + 70n = 77 +21
70n = 98
Divide each side by 70
70n/70 = 98/70
n =98/70
n = 70/70 +28/70
n = 1 + 2/5
n = 1 2/5
Answer:
This would also be known as 7/5
Step-by-step explanation:
The person above included the mixed number but I just thought it would be nice to save some of the math for some people. Have a good day !!
Evaluate: (4/25)^1/2
Answers
Find the original price of a salt lamp that is $60 after a 25% discount.
Answers
The original price of the salt lapm os $80.
How to find the original price of the salt lamp?Let's assume that the original price of the salt lamp is P, if we apply a discount of x (a percentage in decimal form) then the new price will be:
P' = P*(1 - x)
Here we know that the discount is of 25%, then x = 0.25
And the final price is $60, then we can write:
$60 = P*(1 - 0.25)
Now we can solve this for P.
$60 = P*0.75
$60/0.75 = P
$80 = P
That is the original price.
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At what point will the graphs intersect? y=3x/2-5 and 3x-4y=8
Answers
Answer:
(4,1)
Step-by-step explanation:
Hope this helps
Suppose you deposit $2,000 in a savings account that pays interest at an annual rate of 5% if no money is added or withdrawn from the account, how much will be in the account after 4 years?
Answers
Answer:
$2431.01
Step-by-step explanation:
2000×(1 + 0.05)^4 = 2431.0125
5. In one year, there were 125 major new automobile types available. If you look at the miles per gallon (MPG) for these vehicles, the distributions are roughly normal. The means and standard deviations are given by: Mean Standard Deviation City MPG 22.37 4.77 Highway MPG 29.09 5.46 a. The Honda Civic had a rating of 45 highway MPG. What is the probability that your car has a worse highway gas mileage than the Civic? b. The Lincoln Continental had a rating of 17 city MPG. What percent of cars are likely to have a better city gas mileage than the Continental?
Answers
Answer:
The answer is "99.82% and 86.99%".
Step-by-step explanation:
In point a:
\(=P(X < 45)\\\\= P(Z < \frac{45-29.09}{5.46}) \\\\=P(Z < 2.9139)\\\\=0.9982\\\\=99.82\%\)
In point b:
\(=P(X >17)\\\\= P(Z > \frac{17-22.37}{4.77}) \\\\=P(Z > -1.1258)\\\\=0.8699\\\\=86.99\% \\\\\)
Somebody help me please..
Answers
Explanation:
(1 + b/2) - 3a + 3
(1 + b + 6/2) - 3a
= 2b - 3a + 7/2
(Pls do solution or take picture)In the figure, ACEF is a rectangle and BC = CD DE The area of rectangle ACEF is 80 cm. Find the area of the triangle BDF.
Answers
Answer:
28 cm²
Step-by-step explanation:
The area of Δ BDF is the area of the rectangle subtract the area of the 3 white triangles.
Since the area of the rectangle = 80 cm² and
area = length × breadth, then
breadth = 80 ÷ 10 = 8 cm
Thus CE = EF = 8 cm and CD = DE = BC = 4 cm
Area of Δ DEF = 0.5 × 10 × 4 = 20 cm²
Area of Δ ABF = 0.5 × 8 × (10 - 4) = 0.5 × 8 × 6 = 24 cm²
Area of Δ BCD = 0.5 × 4 × 4 = 8 cm²
Thus
area of Δ BDF = 80 - (20 + 24 + 8) = 80 - 52 = 28 cm²
the table shows ordered pairs of the function y=16+0.5x. Which ordered pair could be the missing values represented by (x,y)?
Answers
Answer:
(2,17)
Step-by-step explanation:
This is because x is the input and y is the output. All you have to do is create a value for x the substitute it in the equation. Next solve for y 0.5 times 2 is 1, plus 16 is 17. Then just put your values in an order pair.
Answer:
8,20
Step-by-step explanation:
41 in the ratio 1:6?
Answers
Answer:
94884959969449i9959595959595
Step-by-step explanation:
the difference between two numbers is 9. three times the smaller is equal to twice the larger. what are the numbers?
Answers
Answer: 27 and 18
Step-by-step explanation:
x - y = 9 → x = 9 + y
3y = 2x
3y = 2(9 + y)
3y = 18 + 2y
3y - 2y = 18
y = 18
2x = 3(18)
2x = 54
x = 27
x - y = 9
27 - 18 = 9
9 = 9
What is the range of this table?
Answers
Answer: B
Step-by-step explanation:
Range will always be Y
Domain will always be X
Determine the constant rate of change for each table.
1. Age (yr) Height (in.)
Answers
9514 1404 393
Answer:
2 3/4Step-by-step explanation:
The rate of change in each case can be found by dividing right-column differences by left-column differences.
Table 1
(56 -54)/(10 -9) = 2/1 = 2 . . . the constant rate of change
__
Table 2
(3 -0)/(4 -0) = 3/4 . . . the constant rate of change
_____
Additional comment
Often, in math problems of this sort, you fill in a blank with a number. Above, we have computed those numbers. In real life, it is useful to consider the units that go with that number. The units are divided in the same way the numerical values are.
Table 1: 2 in/yr
Table 2: 3/4 °C/h
the question is on the image
Answers
Answer:
(i) - rectangular prism
(Ii) - triangular prism
(iii) - square pyramid
Step-by-step explanation:
can someone give me the step by step for 10
Answers
x^2 = (16)^2 + (16)^2
x^2 = 256 + 256
x^2 = 512
x = 16√2
Which equation is the Pythagorean theorem
Answers
Answer:
B
Step-by-step explanation:
a cook uses 1 3/4 teaspoons if salt to make 3 1/2 pounds of pasta. what is the unit rate in teaspoons per pound , at which the cook uses salt to make pasta?
Answers
The unit rate in teaspoons per pound would be 1/2 teaspoons per pound.
What is the unit rate?
A unit rate means a rate for one or something. We write this as a ratio with a denominator of one. For example, if you ran 70 yards in 10 seconds, you ran on average 7 yards in 1 second. Both of the ratios, 70 yards in 10 seconds and 7 yards in 1 second, are rates, but the 7 yards in 1 second is a unit rate.
The number of teaspoons:
\(=1\frac{3}{4}\\\\=\frac{4+3}{4}\\\\=\frac{7}{4}\)
Amount of pasta:
\(=3\frac{1}{2}\\\\=\frac{6+1}{2}\\\\=\frac{7}{2}\)
Now the unit rate = number of teaspoons/ amount of pasta
\(=\frac{7/4}{7/2}\\\\=\frac{1}{2}\\\)
Hence, the unit rate in teaspoons per pound would be 1/2 teaspoons per pound.
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Answer:
Step-by-step explanation:
1/2