Prove the following limit:
f(x) = ax^2
lim f(x) =a
x ->1
Answers
The limit of \(f(x) = ax^2\)as x approaches 1 is a.
How to prove that the limit of f(x)?To prove that the limit of \(f(x) = ax^2\) as x approaches 1 is a, we need to show that for any positive number ε, there exists a corresponding positive number δ such that if 0 < |x - 1| < δ, then |f(x) - a| < ε.
Let ε be any positive number. We need to find a corresponding δ such that |f(x) - a| < ε whenever 0 < |x - 1| < δ.
Notice that:
\(|f(x) - a| = |ax^2 - a| = |a(x^2 - 1)| = |a||x + 1||x - 1|\)
Since we are interested in values of x that are close to 1, let us assume that 0 < |x - 1| < 1. Then we have:
|x + 1| < 3
Now let δ be any positive number such that δ < min{1, ε/3|a|}. Then, if 0 < |x - 1| < δ, we have:
|x + 1||x - 1| < 3|δ|
Since δ < ε/3|a|, we also have:
3|δ||a| < ε
Therefore, we have:
|f(x) - a| = |a||x + 1||x - 1| < 3|aδ| < 3|a|ε/3|a| = ε
This shows that for any positive number ε, there exists a corresponding positive number δ such that if 0 < |x - 1| < δ, then |f(x) - a| < ε. Therefore, the limit of \(f(x) = ax^2\)as x approaches 1 is a.
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Problem 1: Bose Einstein Condensation with Rb 87 Consider a collection of 104 atoms of Rb 87, confined inside a box of volume 10-15m3. a) Calculate Eo, the energy of the ground state. b) Calculate the Einstein temperature and compare it with £i). c) Suppose that T = 0.9TE. How many atoms are in the ground state? How close is the chemical potential to the ground state energy? How many atoms are in each of the (threefold-degenerate) first excited states? d) Repeat parts (b) and (c) for the cases of 106 atoms, confined to the same volume. Discuss the conditions under which the number of atoms in the ground state will be much greater than the number in the first excited states.
Answers
Bose-Einstein condensation occurs and the ground state is significantly populated compared to the excited states.
a) To calculate the energy of the ground state, we need to use the formula E = (3/2)NkBT, where N is the number of particles, kB is Boltzmann's constant, and T is the temperature. Since we are dealing with Rb 87 atoms, which are bosons, we also need to consider the Bose-Einstein statistics. In this case, the energy of the ground state is given by Eo = (3/2)NkBTE, where TE is the Einstein temperature. Given that the number of atoms is N = 104, we can calculate Eo using the given values.
b) The Einstein temperature (TE) can be calculated using the formula TE = (2πℏ^2 / (mkB))^(2/3), where ℏ is the reduced Planck constant and m is the mass of the particle. We can calculate TE using the known values for Rb 87.
c) For T = 0.9TE, we can determine the number of atoms in the ground state by calculating the probability of occupation for that state using the Bose-Einstein distribution. The chemical potential (μ) represents the energy required to add an extra particle to the system. By comparing it to the ground state energy, we can determine how close the chemical potential is to the ground state energy. The number of atoms in the first excited states can also be calculated using the Bose-Einstein distribution.
d) By repeating parts (b) and (c) for a larger number of atoms (N = 106) but confined to the same volume, we can analyze the conditions under which the number of atoms in the ground state is much greater than the number in the first excited states. This comparison depends on the values of TE, T, and the number of atoms N.
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consider an interval from 0 to 1 with ten uniformly distributed points in it. what is the distribution of the difference between the 6th and the 5th smallest point?
Answers
Answer: It has to be between 0.1 to 0.3
Step-by-step explanation:
6) Which statement best describes the function notation f(2) = 10? Circle your answer.
a. The output is f(x) when the input is 2.
c. The output is 2 when the input is 10.
b. The output is 10 when the input is 2.
d. The output is f(10) when the input is 2.
Answers
Answer:
b. The output is 10 when the input is 2.
Step-by-step explanation:
2 is the input value (x), and 10 is the output value (y)
The stream of water from a fountain follows a parabolic path. The stream reaches a maximum height of 5 feet, represented by a vertex of (3,5), and lands 6 feet from the water jet, represented by (6,0). Write a function in vertex form that models the path of the stream.
Answers
The function in vertex form that models the path of the stream is:
\(y = (-5/9)(x - 3)^2 + 5\)
What are vertex?A location where two or more straight lines or curves intersect is referred to as a vertex . A vertex in geometry is sometimes referred to as a corner or an intersection point. The phrase is frequently used to refer to the highest or lowest point on a curve or surface, the place where two lines intersect to make an angle, and the point where two sides of a polygon meet.
The vertex form of the equation of a parabola is given by:
\(y = a(x - h)^2 + k\)
where (h, k) represents the vertex of the parabola.
Using the given vertex and the point (6,0), we can find the value of "a" and plug it into the vertex form to get the equation of the parabolic path of the water stream.
Step 1: Find the value of "a"
Since the vertex is (3,5), we know that:
h = 3
k = 5
Using the point (6,0), we can find the value of "a" as follows:
\(0 = a(6 - 3)^2 + 5\)
\(0 = 9a + 5\)
\(9a = -5\)
\(a = -5/9\)
Step 2: Plug in the values of h, k, and a into the vertex form:
\(y = (-5/9)(x - 3)^2 + 5\)
Therefore, the function in vertex form that models the path of the stream is:
\(y = (-5/9)(x - 3)^2 + 5\)
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Which function displays this end behavior p
Answers
Answer:
Option D.y=-2x³-1 will be your answer
Step-by-step explanation:
hope it helps...
have a great day!!
Find the angle between vector bold lower u equals 3 bold lower I plus start root 3 end root bold lower j and vector bold lower v equals negative 2 bold lower I minus 5 bold lower j to the nearest degree. A. 82° B. 38° C. 142° D. 98°
Answers
Answer:
C. 142°
Step-by-step explanation:
You want the angle between vectors u=3i+√3j and v=-2i-5j.
AngleThere are a number of ways the angle between the vectors can be found. For example, the dot-product relation can give you the cosine of the angle:
u•v = |u|·|v|·cos(θ) . . . . . . where θ is the angle of interest
You can find the angles of the vectors individually, and subtract those:
u = |u|∠α
v = |v|∠β
θ = α - β
When the vectors are expressed as complex numbers, the angle between them is the angle of their quotient:
\(\dfrac{\vec{u}}{\vec{v}}=\dfrac{|\vec{u}|\angle\alpha}{|\vec{v}|\angle\beta}=\dfrac{|\vec{u}|}{|\vec{v}|}\angle(\alpha-\beta)=\dfrac{|\vec{u}|}{|\vec{v}|}\angle\theta\)
This method is used in the calculation shown in the first attachment. The angle between u and v is about 142°.
A graphing program can draw the vectors and measure the angle between them. This is shown in the second attachment.
__
Additional comment
The approach using the quotient of the vectors written as complex numbers is simply computed using a calculator with appropriate complex number functions. There doesn't seem to be any 3D equivalent.
The dot-product relation will work with 3D vectors as well as 2D vectors.
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when a coin is tossed three times, what is the probability that all three tosses are heads? the possible outcomes for the coin tosses are{hhh,ttt,htt,hht,thh,tth,hth,tht}
Answers
The probability of getting all three tosses to head is 1/8.
We have,
When a coin is tossed, there are two possible outcomes:
heads (H) or tails (T).
Since there are three tosses, the total number of possible outcomes.
2³ = 8.
The probability of getting heads on one toss is 1/2.
The probability of getting heads on all three tosses is the product of the probabilities of getting heads on each individual toss:
P(HHH) = P(H) x P(H) x P(H) = (1/2) x (1/2) x (1/2) = 1/8.
Therefore,
The probability of getting all three tosses to head is 1/8.
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Simplify (-3/5) divided by (7/6) HELP ASAP
Answers
Answer:
-18/35
Step-by-step explanation:
(-3/5)/(7/6)=
(-3/5)x(6/7)=
(-3x6)/(5x7)=
-18/35
Which of the following functions exhibit the end behavior f(x) —>infinite as x—> -infinite and f(x) —> -infinite as x—>infinite? Select all that apply. Please help me out! Nobody wants to help :(
Answers
The end behavior f(x) → ∞ as x → -∞ means that x-value is negative and y-value is positive. If this is the case, the other end part of the function is located in the 2nd quadrant.
The end behavior f(x) → -∞ as x → ∞ means that the x-value is positive and the y-value is negative. The other end part of the function is located in the 4th quadrant.
Looking at the choices, Option 1 and 4 exhibits these characteristics.
(5+root8)^2
give your answer in the form b+c root 2
Answers
The solution is: (5+√8)² = 89 + 10√8, in the form b+c root 2.
Here, we have,
given that,
the expression is:
(5+√8)²
we know that,
the algebraic formula is:
( a + b)² = a² + 2ab + b²
so, here, we get,
(5+√8)²
=5² + 2*5*√8 + √8²
=25 + 10√8 + 64
=89 + 10√8
Hence, The solution is: (5+√8)² = 89 + 10√8, in the form b+c root 2.
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The graph of f(x) and g(x) are shown below. How many solutions does the system of equations have?
Click pic to see whole problem
Answers
Answer:
Step-by-step explanation:
Solving systems of equations gives the points of intersection when the equations are graphed.
The answer is 3.
Isabel’s competition,who sells books and no other products, charges a shipping fee of $1.99 per book plus a fixed fee of $3 for each order. Let x be the number of books purchased and f(x) be the price of shipping one order of books. Write a function that expresses f(x) and explain the value of f(x) for x = 0. (5 points)
Answers
Answer:
Step-by-step explanation:
for the function i get f(x)=1.99x+3
f(x) = 1.99x + 3 for x>0
x≥1
X cannot get to zero it is no allowed by the function I defined
i tried my best
Using an integrating factor, solve y-y-5 CD- in the method for solving a first-order linear differential equation, the first step is to put the equation in the standard form y alty bit). is the given equation in the standard form? No Yes Identify a(t) and bit)
Answers
The value of a(t) is -1 and b(t) is 55 + \(e^t\)
No, the given equation y' - y = 55 + \(e^t\) is not in the standard form of a first-order linear differential equation.
In the method for solving a first-order linear differential equation, an integrating factor is a function used to transform the equation into a form that can be easily solved.
For an equation in the standard form y' + a(t)y = b(t), the integrating factor is defined as:
μ(t) = e^∫a(t)dt
To solve the equation, you multiply both sides of the equation by the integrating factor μ(t) and then simplify. This multiplication helps to make the left side of the equation integrable and simplifies the process of finding the solution.
To put it in standard form, we need to rewrite it as y' + a(t)y = b(t).
Comparing the given equation with the standard form, we can identify:
a(t) = -1
b(t) = 55 + \(e^t\)
Therefore, The value of a(t) is -1 and b(t) is 55 + \(e^t\)
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which equation results from applying the secant and tangent segment theorem to the figure?12(a 12)
Answers
This simplifies to:
a² = 12x + 144
This is the equation that results from applying the Secant-Tangent Segment Theorem to the given figure.
Let's first define the terms and state the Secant-Tangent Segment Theorem.
1. Equation: A mathematical statement that shows the equality of two expressions.
2. Secant: A line that intersects a circle at two points.
3. Tangent: A line that touches a circle at only one point, without crossing it.
Secant-Tangent Segment Theorem: If a secant and a tangent are drawn from a point outside a circle, the product of the length of the secant segment and its external part is equal to the square of the length of the tangent segment.
Let's represent the given lengths as follows:
- Length of tangent segment = a
- Length of secant segment (inside the circle) = 12
- Length of the external part of the secant segment (outside the circle) = x
According to the Secant-Tangent Segment Theorem, the equation is:
(a)(a) = (12)(x + 12)
This simplifies to:
a² = 12x + 144
This is the equation that results from applying the Secant-Tangent Segment Theorem to the given figure.
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Kelsey uses the similarity statement ARST - ARUV to describe the triangles below.
S
3.0 cm
5.4 cm
4.0 cm
R
V
2.7 cm
T
8.0 cm
1.5 cm
U
Which best describes the accuracy of her statement?
Answers
Answer:
The answer is A: Accurate. Each pair of sides corresponds with a common ratio of 2.
Step-by-step explanation:
Got it on the quiz on edg 2021 :D
Q.4 What is the difference between price floors and price ceiling? Give example and illustrate graphically in support of your answer.
Answers
A price floor is a law that limits the minimum price at which a good, service, or factor of production can be sold while a price ceiling is a regulation that limits the maximum price at which a good, service, or factor of production can be sold
Price floors are commonly implemented to support producers, while price ceilings are typically put in place to protect consumers from higher prices that might result from shortages or monopolies.
Example of Price Floor:Agricultural subsidies are a common example of price floors. Government price floors ensure that farmers receive a minimum price for their crops.
If the market price of wheat falls below the government-established price floor, the government may buy the excess supply at the guaranteed price, ensuring that farmers are able to make a profit. If there is a price floor, the minimum price is set above the equilibrium price.
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Can I get some help please??
Answers
Answer:
\( - 108\)
Step-by-step explanation:
\( - 6 \times 18 = b \\ - 108 = b\)
4(8x−1) = 19+32x
I need help with a question I am kinda stuck on please someone help and please give me some steps for it at least some.
Answers
Answer: 0 = 23
Almost positiveee :))
Step-by-step explanation:
EASY Math Homework EASY
Answers
1a) 2x + 3y = 24
To solve this first equation, plug in provided values until you get a true statement. In this case, option 2 is correct.
2(3) + 3(6) = 24
6 + 18 = 24
24 = 24
1b) y > x + 2
To solve this first equation, plug in provided values until you get a true statement. In this case, option 1 is correct.
7 > 4 + 2
7 > 6
1c) x - 3y ≤ 5
To solve this first equation, plug in provided values until you get a true statement. In this case, option 3 is correct.
0 - 3(7/2) ≤ -2
0 - 10.5 ≤ -2
True
1d) Needs options
Answer:1a) 2x + 3y = 24 is (3,6)
1b) y > x + 2 is (7,4)
1c) x - 3y ≤ 5 is (0,-2)
1d) what are the options?
Step-by-step explanation:
If 5x+2=52, then what does x equal?
Answers
Answer:
x=10
Step-by-step explanation:
If we subtract 2 from 52 we get 50 and ten mutiplys into 50.
Need help with math problem please!!
Answers
Answer:
m + 2
2 dollars + one dollar per minute
find the surface area of each solid to the nearest tenth, if necessary. these bothered
Answers
Answer:
266is the answer too it u times etc
Select all the correct answers.
The distance from Earth to the moon is 384,400 kilometers. What is this distance expressed in scientific notation?
3.844E5 kilometers
3.844 × 105 kilometers
3.844E-5 kilometers
3.844E-6 kilometers
3.844 × 106 kilometers
3.844E6 kilometers
3.844 × 10-6 kilometers
3.844 × 10-5 kilometers
Answers
Answer:
3.844 x 10^5
3.844E-5
3.844 × 10-5 kilometers
Step-by-step explanation:
A reporter surveyed 300 randomly selected people of all ages about their opinion of a new song. The results are shown in the table. Which of the following is a valid conclusion about data? select all that apply.
Answers
Answer:
the answers is A and E <3
Step-by-step explanation:
i took this test
Rounded to the nearest tenth, what is the perimeter of rectangle ABCD? rectangle ABCD with diagonal AC, labeled as measuring 20 centimeters, angle BAC measuring 30 degrees, angle BCA measuring 60 degrees, angle DCA measuring 30 degrees, and angle DAC measuring 60 degrees A. 48.3 cm B. 54.6 cm C. 60.0 cm D. 173.2 cm
Answers
Answer:
54.6 cm : B
Step-by-step explanation:
Mathematically the area of a rectangle is 2( l + b)
So in practical terms we only need the measure of two sides as we have 2 equal sides for a rectangle.
We can see that we have two right angles on each sides
Now to find b, we can use trigonometric ratios
For each of the rectangles, the diagonal represents the hypotenuse
Mathematically;
sin 30 = b/20
From triangle ABC
b = 20 sin 30 = 20 * 0.5 = 10 cm
Now, to get l
Mathematically;
Sin 60 = l/20
l = 20 sin 60 = 17.32 cm
So the perimeter is;
2(10 + 17.32)
2(27.32) = 54.64 cm which is 54.6 to the nearest tenth
what is (fxg)(x)
f(x)=x^3-4x+2
g(x)=x^2+2
Answers
Answer: x^6+6x^4+8x^2+2
Step-by-step explanation:
Since g comes after f then you will take g's equation and plug it into the f equation so it would turn out to be if you plugged it in
(x^2+2)^3-4(x^2+2)+2 which would equal x^6+6x^4+8x^2+2
how to get rid of a fraction with a variable in the denominator
Answers
To get rid of a fraction with a variable in the denominator, multiply both the numerator and denominator by that variable. This technique is very useful in simplifying complex fractions and solving equations involving fractions. To get rid of a fraction with a variable in the denominator
To get rid of a fraction with a variable in the denominator, you can use the technique of multiplying both the numerator and the denominator by the variable that is in the denominator. This will result in the variable canceling out from the denominator, leaving only the numerator.
Identify the variable in the denominator and the value of its exponent. For example, in the fraction 1/(x^2), the variable is x and the exponent is 2. Multiply both the numerator and denominator by the same power of the variable that is present in the denominator. In our example, multiply the numerator and denominator by x^2: (1 * x^2)/(x^2 * x^2). simplify the resulting expression by canceling out common terms between the numerator and denominator. In this case, x^2 in the numerator and denominator cancel out, leaving 1/(x^2) as the simplified answer without a variable in the denominator.
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Each server can deliver nine waffles at a time. It takes each server 10 minutes per trip. How many servers will you need to deliver 300 waffles in one hour? How many trips will the servers have to make all together?
Answers
Based on the number of waffles that each server can deliver at a time, the number of servers needed for 300 waffles in an hour is 6 servers
The number of trips the servers will have to make is 33 trips
How to find the number of servers?If each server can deliver nine waffles at a time and they take 10 minutes per trip, the number of waffles they can deliver in an hour is:
= Number of waffles per trip x Number of minutes in an hour / Number of minutes per trip
= 9 x 60 / 10
= 54 waffles
If there are 300 waffles to be delivered, the number of servers needed is:
= 300 / 54
= 5.56
= 6 servers
The number of trips that the servers will make altogether is:
= Number of servers x trips per hour for each server
= 5.5555 x 60 / 10
= 33 trips
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Write an equivalent double integral with the order of integration reversed.
¹∫₀ ᵗᵃⁿ⁻¹ˣ∫₀ dy dx
A) π/⁴∫₀ ¹∫ₜₐₙ ᵧ dy dx
B) π/⁴∫₀ π/²∫ₜₐₙ₋₁ᵧ dy dx
C) π/⁴∫₀ ¹∫ₜₐₙ₋₁ᵧ dy dx
D) π/⁴∫₀ π/²∫ₜₐₙ ᵧ dy dx
Answers
An equivalent double integral with the order of integration reversed can be written as π/⁴∫₀ ¹∫ₜₐₙ ᵧ dy dx.
To understand why this is the correct answer, let's analyze the given integral ¹∫₀ ᵗᵃⁿ⁻¹ˣ∫₀ dy dx and how the order of integration can be reversed.
In the original integral, we have the limits of integration as ᵗₐₙ⁻¹ to ₀ for x and ₀ to ᵧ for y. To reverse the order of integration, we need to switch the order of the integrals and reverse the limits accordingly. Therefore, the equivalent double integral with the order of integration reversed becomes ∫₀ ᵧ ¹∫ₜₐₙ ᵡ dx dy.
Now, let's analyze the options given. Option A has the limits of integration for the inner integral as ₀ to ᵧ and the outer integral as π/⁴ to ₀, which matches our reversed order of integration. Therefore, the correct answer is A) π/⁴∫₀ ¹∫ₜₐₙ ᵧ dy dx.
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Can someone help me with this
Answers
Answer:
a = 17
c = 9
Step-by-step explanation:
2a + 3c = 61
3a + 5c = 96
3a = 96 - 5c (solve for 'a'. Then you can substitute)
a = 32 - \(\frac{5}{3}\) c
2(32 - \(\frac{5}{3}\)c) + 3c = 61 (Substitute)
64 - \(\frac{10}{3}\)c + 3c = 61 (\(\frac{10}{3}\) = 3 \(\frac{1}{3}\). So subtracting, adding a negative, leaves \(\frac{-1}{3}\))
-\(\frac{1}{3}\) c= -3
c = 9
2a + 3(9) = 61
2a + 27 = 61
2a = 34
a = 17