a) log n = O(n) is false because in logarithmic functions, the growth rate is much slower than any polynomial function like n, n², n³, etc. Hence, it is not true that logarithmic functions grow at the same rate as polynomial functions.
b) n² + 3 = O(n³) is true. The big O notation tells us that n² + 3 grows at most as fast as n³ for large values of n. Thus, it is true that n² + 3 = O(n³).c) n³ + 2 = O(n) is false. The big O notation tells us that n³ + 2 grows at most as fast as n for large values of n. This is not true, as n grows much faster than n³ + 2 for large values of n.
Hence, it is not true that n³ + 2 = O(n).d) nº = O(nb) is true because any constant function grows at most as fast as any power function. Since nº is a constant function, it grows at most as fast as any power function nb. Hence, it is true that nº = O(nb).
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The following differential equation describes the movement of a body with a mass of 1 kg in a mass-spring system, where y(t) is the vertical position of the body in meters) at time t. y" + 4y + 5y = -21 To determine the position of the body at time t complete the following steps. (a) Write down and solve the characteristic (auxiliary) equation. (b) Determine the complementary solution, yc, to the corresponding homogeneous equation, y" + 4y' + 5y = 0. (c) Find a particular solution, Yp, to the nonhomogeneous differential equation, Y" + 4y' + 5y = e-21. Hence state the general solution to the nonhomogeneous equation as y = Ye + yp. (d) Solve the initial value problem if the initial position of the body is 1 m and its initial velocity is zero.
After considering the given data we conclude that
a) r = -2 + i and r = -2 - i value generated after solving the auxiliary equation
b)[tex]y_c = c_1e^{(-2t)} cos(t) + c_2e^{(-2t)} sin(t)[/tex] complementary solution
c) general solution to the nonhomogeneous equation is [tex]y = y_c + Y_p = c_1e^{(-2t)} cos(t) + c_2e^{(-2t)} sin(t) + (1/26)e^{(-21)} .[/tex]
d) initial value problem is [tex]y(t) = e^{(-2t)} (cos(t) + 2sin(t))/2 + (1/26)e^{(-21)} .[/tex]
To determine the position of the body at time t, we can apply the given differential equation to solve for y(t). To do this, we can follow the steps below:
(a) the characteristic (auxiliary) equation is
The characteristic equation is obtained by setting the coefficients of the differential equation to zero, which gives us the equation [tex]r^2 + 4r + 5 = 0.[/tex]Solving this quadratic equation, we get r = -2 + i and r = -2 - i.
(b) evaluate the complementary solution, yc, to the corresponding homogeneous equation, [tex]y" + 4y' + 5y = 0.[/tex]
The complementary solution is given by[tex]y_c = c_1e^{(-2t)} cos(t) + c_2e^{(-2t)} sin(t),[/tex]where [tex]c_1[/tex] and [tex]c_2[/tex] are constants determined by the initial conditions.
(c) Calculate a particular solution, Yp, to the nonhomogeneous differential equation, [tex]Y" + 4y' + 5y = e^{(-21)}[/tex]. Hence state the general solution to the nonhomogeneous equation as [tex]y = y_c + y_p.[/tex]
To describe a particular solution, we can use the method of undetermined coefficients and guess a solution of the form [tex]Yp = Ae^{(-21)}[/tex], where A is a constant to be determined.
Staging this into the differential equation, we get A = 1/26. Therefore, the particular solution is [tex]Y_p = (1/26)e^{(-21)}[/tex]. The general solution to the nonhomogeneous equation is [tex]y = y_c + Y_p = c_1e^{(-2t)} cos(t) + c_2e^{(-2t)} sin(t) + (1/26)e^{(-21)}[/tex]
(d) Solve the initial value problem if the initial position of the body is 1 m and its initial velocity is zero.
Applying the initial conditions, we can solve for the constants [tex]c_1[/tex] and [tex]c_2[/tex] in the complementary solution. Since the initial velocity is zero, we have [tex]y'(0) = -2c_1 + c_2 = 0[/tex], which gives us [tex]c_2 = 2c_1.[/tex]
Applying the initial position, we have [tex]y(0) = c_1 = 1[/tex], which gives us [tex]c_2 = 2.[/tex]Therefore, the solution to the initial value problem is [tex]y(t) = e^{(-2t)} (cos(t) + 2sin(t))/2 + (1/26)e^{(-21)}[/tex]
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As an estimation we are told £3 is €4.
Convert £40.50 to euros.
Answer:
As per the conversion in the question, £40.50 = €54
Step-by-step explanation:
Here, we want to make a conversion;
from the question;
£3 = €4
£40.50 = €x
Thus;
3 * x = 4 * 40.5
x = (4 * 40.5)/3
x = €54
What is the volume of a box with 385 cubes and 11/3 x 5/3 x 7/3
Answer:
y
=
x
2
−
2
x
,
y
=
x
Step-by-step explanation:
-23 = x - 23
A. Infinite number of solutions
B. No solution
C. 0
D. 5
Answer:
x=0
Step-by-step explanation:
have a nice day or evening
The fixed costs for a company are $1,596.00 per month, and their variable cost per unit is $3.20. Suppose the company insists on producing 140 units, the selling price per unit required to break even is $
When the company insists on producing 140 units then the selling price per unit required to break even is approximately $14.60.
To calculate the selling price per unit required to break even, we need to consider the fixed costs, variable cost per unit, and the desired production quantity.
Given:
Fixed costs = $1,596.00 per month
Variable cost per unit = $3.20
Production quantity = 140 units
To break even, the total revenue should cover both the fixed costs and the variable costs.
The total cost can be calculated as follows:
Total cost = Fixed costs + (Variable cost per unit × Production quantity)
Plugging in the given values:
Total cost = $1,596.00 + ($3.20 × 140)
Total cost = $1,596.00 + $448.00
Total cost = $2,044.00
To break even, the total revenue should be equal to the total cost.
The selling price per unit required to break even can be calculated as:
Selling price per unit = Total cost / Production quantity
Plugging in the values:
Selling price per unit = $2,044.00 / 140
Selling price per unit ≈ $14.60
Therefore, the selling price per unit required to break even is approximately $14.60.
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20 POINTS HELPIf these two rectangles are similar, what is the measure of the missing length d ?
Answer:
d (or V in the picture) = 16
Step-by-step explanation:
Because the two triangle are similar, the ratios of each of the side lengths should be equal. Therefore, 8/1 should be equal to d/2.
1.) Set up equality of ratios: 8/1 = d/2
2.) Cross multiply: 8*2 = d*1
3.) 16 = d
what is the ratio of the length of red string to blue string written as a unit rate
Answer:2 to 3
Step-by-step explanation:
Answer:
Step-by-step explanation:
2 qt 1 cup = ___ fl oz
A. 40
B. 72
C. 80
D. 136
Answer:
80 fl oz
2qts = 8 cups
100 POINTS!!!!
An expression is shown below:
f(x) = 2x2 − 5x + 3
Part A: What are the x-intercepts of the graph of f(x)? Show your work. (2 points)
Part B: Is the vertex of the graph of f(x) going to be a maximum or minimum? What are the coordinates of the vertex? Justify your answers and show your work. (3 points)
Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)
Answer:
Part A: In order to find the x-intercepts, you should set f(x)=0. Then, . and .
Part B: From the function, it is visible that it is going to be a minimum, because the function is cup-size. Vertex coordinates can be calculated by the formula . Then and . The vertex coordinates are (-1.25, -0.125)
Part C: By using the vertex coordinates, we can construct the vertex of the function and then using the information about the x-intercept we can complete the function. The function looks like in the picture below.
Step-by-step explanation:
Solve the differential equation = -xy, given that when x=0, y=50. You may assume y>0. (4 marks) dx (b) For what values of x is y decreasing?
The solution to the differential equation is [tex]y = e^{-\frac{x^2}{2} + \ln(50)}[/tex]
The value of y is decreasing for x > 0.
Solving the differential equationFrom the question, we have the following parameters that can be used in our computation:
dy/dx = -xy
Rewrite as
dy/y = -x dx
Differentiate both sides
So, we have
ln|y| = -x²/2 + c
Take the exponent of both sides
[tex]|y| = e^{-\frac{x^2}{2} + c}[/tex]
Assume y > 0
So, we have
[tex]y = e^{-\frac{x^2}{2} + c}[/tex]
When x = 0, y = 50
So, we have
[tex]50 = e^{-\frac{0^2}{2} + c}[/tex]
Evaluate
c = ln(50)
So, we have
[tex]y = e^{-\frac{x^2}{2} + \ln(50)}[/tex]
For what values of x is y decreasing?Recall that dy/dx = -xy.
This means that if dy/dx < 0, then y is decreasing.
This means that, y is decreasing for x > 0.
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Can someone plz help this is my third time posting this question and I’m trying to get an 92
21 is the right answer.
180-138=42°
and since it is given in the question that the triangle is isoceles ,so half of 42 is 21°
Answer:
the last three answers is what i think
Step-by-step explanation:
Which sentence is TRUE!!!!!!
The probability of spinning a 9 on Luke's game spinner is . The probability of rolling a 6 on a fair
die is Which shows the probability of spinning a 9 and then rolling a number that is NOT a 6?
Answer:
First, there are
3
numbers (6, 7, 8) greater than
5
on a spinner numbered
1
-
8
(1, 2, 3, 4, 5, 6, 7, 8). Therefore, there is a:
3
8
probability of spinning a number greater than
5
.
However, there is only a 50-50 or
1
2
chance of tossing a tail on a coin.
Therefore the probability of spinning a number greater than
5
AND tossing a tail is:
3
8
×
1
2
=
3
16
Or
3
in
16
Or
18.75%
Step-by-step explanation:
Help me pls I give points
Answer:
c=14.87
Step-by-step explanation:
use pythagorean theroem
10^2 + 11^2 =c^2
solve
100+121=c^2
c^2=221
c=14.87(rounded to nearest hundredths )
Find any vertical or slant asymptotes. Consider the following.
f(x) = x^3/ (x² - 49)
The function [tex]f(x) = x^3 / (x^2 - 49)[/tex]has a vertical asymptote at x = 7 and x = -7. There are no slant asymptotes. To find the vertical asymptotes of the function f(x), we need to identify the values of x for which the denominator becomes zero, as division by zero is undefined.
In this case, the denominator is (x² - 49). To determine when it equals zero, we set it equal to zero and solve for x:
x² - 49 = 0
This equation can be factored as the difference of squares:
(x - 7)(x + 7) = 0
Setting each factor equal to zero, we find that x = 7 and x = -7. Therefore, the function has vertical asymptotes at x = 7 and x = -7.
On the other hand, to determine if there are any slant asymptotes, we need to check if the degree of the numerator (x³) is greater than the degree of the denominator [tex](x^2 - 49)[/tex]. In this case, the degree of the numerator is 3, while the degree of the denominator is 2. Since the degree of the numerator is greater, there are no slant asymptotes for this function.
In summary, the function [tex]f(x) = x^3 / (x^2 - 49)[/tex] has vertical asymptotes at x = 7 and x = -7. There are no slant asymptotes.
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What is the measure of the other acute angle ?
Answer:
58
Step-by-step explanation:
Answer: 58°
Step-by-step explanation:
We know it's a right triangle, one of them must be 90°. The sum of the three interior angles are 180°, so we know the sum of the other two angles is 180 - 90 = 90°.
If one of the angles is 32°, another angle is 90 - 32 = 58°
3. For each function, determine whether it is even, odd, or neither. Explain
a&b in photo
c. The function given by = 3 − 4
The function is not symmetric about the y-axis or the origin, which is consistent with the conclusion that it is neither odd nor even. A function can be categorized as odd, even or neither depending on its symmetry about the y-axis and the origin of the graph.
To determine whether a function is odd, even or neither, we need to check whether the function satisfies the following conditions:If f(-x) = f(x), then the function is evenIf f(-x) = -f(x), then the function is oddIf f(-x) ≠ f(x) and f(-x) ≠ -f(x), then the function is neither.
Now let's evaluate whether the given function f(x) = 3 − 4|x| is even, odd or neither. Evaluating f(-x) and f(x) :f(-x) = 3 - 4|(-x)|f(-x) = 3 + 4|x|Comparing the value of f(-x) and f(x) :f(-x) ≠ f(x) and f(-x) ≠ -f(x).
Therefore, the given function is neither odd nor even.Here is the graph of the function. We can see that the function is not symmetric about the y-axis or the origin, which is consistent with the conclusion that it is neither odd nor even.
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how resolve it
plssss
Answer:
The answer is a
Step-by-step explanation:
A person walks 1/5 mile in 1/15 of an hour.
They are going what miles an hour?
Answer:
(1/5 mi) / (1/15 hr) = 1/5 * 15/1 = 3 mi/hr
hope this helps
Step-by-step explanation:
Find the Laurent series of the function cos z, centered at z = 플 1
The Laurent series of cos z centered at z = 1 is: cos(z - 1) = ∑((-1)^n * ((2n C k) * z^(2n - k)))/(2n)!
To obtain the Laurent series of the function cos z centered at z = 1, we can use the known Maclaurin series expansion of the cosine function and then adjust it for the center of expansion.
The Maclaurin series expansion of cos z is given by:
cos z = ∑((-1)^n * z^(2n))/(2n)!
To center the expansion at z = 1, we can substitute z - 1 for z in the series:
cos(z - 1) = ∑((-1)^n * (z - 1)^(2n))/(2n)!
Expanding this expression using the binomial theorem, we have:
cos(z - 1) = ∑((-1)^n * ((-1)^n * (2n C k) * z^(2n - k)))/(2n)!
Simplifying further, we obtain:
cos(z - 1) = ∑((-1)^n * ((2n C k) * z^(2n - k)))/(2n)!
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Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = 5x^3, y = 5x, x ≥ 0; about the x-axis
V=?
Sketch the region.
Sketch the solid, and a typical disk or washer.
The volume (V) of the solid obtained by rotating the region bounded by the curves y = 5x^3, y = 5x, x ≥ 0 about the x-axis is V = ∫[0,1] 2πx((5x) - (5x^3))dx.
To find the volume of the solid, we can use the method of cylindrical shells. We will integrate the volume of each shell as it rotates around the x-axis.
First, let's sketch the region bounded by the curves. The curve y = 5x^3 intersects with the line y = 5x at two points: (0, 0) and (1, 5). The region between these curves is bounded by the x-axis and the curves. It looks like a "bowl" shape, opening upwards, with the bottom touching the x-axis.
Next, let's visualize a typical cylindrical shell. Imagine taking a thin strip of thickness Δx at a distance x from the x-axis. When this strip rotates around the x-axis, it forms a cylindrical shell. The height of this shell is the difference between the two curves: (5x) - (5x^3). The circumference of the shell is 2πx since it is the distance around the axis of rotation. The thickness of the shell is Δx.
The volume of the shell is given by V_shell = 2πx((5x) - (5x^3))Δx. To find the total volume, we need to sum up all these shells by integrating with respect to x.
Integrating V_shell from x = 0 to x = 1, we get V = ∫[0,1] 2πx((5x) - (5x^3))dx.
Evaluating this integral will give us the volume of the solid obtained by rotating the region about the x-axis.
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Exercise 1: If tossing 4 coins identical and distinct. Find the number of macrostates and
microstates (explain the distribution in a table).
Exercise 2: Two particles distinct are to be distributed in three cells. Find the number of
macrostates and microstates ( explain the distrubition in a table)
Exercise 1: When tossing 4 identical and distinct coins, the number of macrostates and microstates are given below:MoleculesMacrostatesMicrostates4 coins16 states2^4=16Microstates: The number of ways in which the particles can be distributed among different energy levels is referred to as microstates. Macrostates: The number of ways in which the total energy of the system can be divided into different energy levels is referred to as macrostates. The distribution is represented in the following table: Distribution Microstates (W) Macrostates (Ω)TTTT1111HHHHT4C4,216HHHH3C4,715
Exercise 2:When distributing two distinct particles among three cells, the number of macrostates and microstates are as follows: Molecules Macrostates Microstates 2 particles10 states3^2=9Microstates: The number of ways in which the particles can be distributed among different energy levels is referred to as microstates. Macrostates: The number of ways in which the total energy of the system can be divided into different energy levels is referred to as macrostates. The distribution is represented in the following table: Distribution Microstates (W) Macrostates (Ω)2 in 11C21,23 in 11C31,33 in 11C32,310 in total 9.
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Which of the following are characteristics of a tortoise in a desert environment that distinguish the tortoise as a living thing?
Choose the THREE that apply.
A. It needs food.
B. It reproduces.
C. It uses energy.
D. It takes up space.
E. It is made up of atoms.
Answer:
A.
B.
E.
Step-by-step explanation:
hope this helps ❤️
Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).
Consider the quadratic expression below.
Answer:
x = -3 and x = 2
Step-by-step explanation:
The expression for the quadratic equation is as follows:
(x+3)(x-2)
For (x+3)(x-2) to equal 0, either (x+3) or (x-2) must be equal to 0. So,
(x+3) = 0 and (x-2) = 0
It implies,
x = -3 and x = 2
So, the values of x are -3 and 2.
1.13 UNIT TEST GRAPH OF SINUSOIDAL FUNCTION PART 1
The true statements from the sinusoidal graph are:
The maximum height of the Ferris wheel is 64 ftThe radius of the Ferris wheel is 30 ftFrom the table, we have the following parameters:
Maximum height = 64 ftMinimum height = 4 ftSo, the radius (r) of the wheel is:
[tex]r = \frac{64 - 4}{2}[/tex]
This gives
[tex]r = \frac{60}{2}[/tex]
[tex]r = 30[/tex]
Also, from the table, the lowest point of the wheel is 4 ft;
The Ferris wheel is at this lowest point at 7.5 seconds and 17.5 seconds
Hence, the true statement is (c) and (d)
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Approximately 9% of all people are left-handed. Consider 27 randomly selected people. a) State the random variable. rv X = the number of 27 randomly selected people that are left-handed b) List the given numeric values with the correct symbols. nV = 27 p = 0.09 c) Compute the mean. Round final answer to 2 decimal places. Which of the following is the correct interpretation of the mean? Out of every 27 people, 2.43 of them on average are left-handed d) Compute the standard deviation. Round final answer to 2 decimal places.
a) The random variable is X, which represents the number of 27 randomly selected people that are left-handed.
b) Given values: n = 27 p = 0.09
c) The correct interpretation of the mean is: Out of every 27 people, on average, 2.43 of them are left-handed.
d) The standard deviation is approximately 1.49
a) The random variable is X, which represents the number of 27 randomly selected people that are left-handed.
b) Given values:
n = 27 (sample size)
p = 0.09 (probability of a person being left-handed)
c) To compute the mean, you can multiply the sample size by the probability:
Mean (μ) = n * p
μ = 27 * 0.09 = 2.43
The correct interpretation of the mean is:
Out of every 27 people, on average, 2.43 of them are left-handed.
d) To compute the standard deviation (σ), you can use the formula for the binomial distribution:
Standard Deviation (σ) = √(n * p * (1 - p))
σ = √(27 * 0.09 * (1 - 0.09))
σ = √(2.43 * 0.91)
σ ≈ √2.2153
σ ≈ 1.49 (rounded to 2 decimal places)
Therefore, the standard deviation is approximately 1.49
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Let C be a closed contour and let zo ∈ C be a point not lying on C. The winding number of C about zo is defined by the integral.
n(C,zo) = 1/2πi ∫ 1/(z-zo)dz.
The winding number of a closed contour C about a point zo is defined as the integral of the function 1/(z - zo) over the contour C, divided by 2πi.
The winding number, denoted as n(C, zo), measures how many times the contour C wraps around the point zo in the counterclockwise direction. It is a topological property of the contour and is an integer value.
To calculate the winding number, we evaluate the integral 1/(z - zo)dz along the contour C. The contour must be positively oriented (counterclockwise) and enclose the point zo. The integral measures the net change in the argument of the complex number z - zo as we traverse the contour.
The value of the integral divided by 2πi gives us the winding number, which represents the number of times the contour wraps around the point zo in the counterclockwise direction.
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A bicycle tire is 28 inches in diameter how far does the bicycle move forward each time the wheel goes around use 22/7 as an approximately for π
Answer:
88 inches
Step-by-step explanation:
Well the bike will travel the distance of the circumference of the tire.
Circumference is 2*pi*radius or pi*diameter since the diameter is twice the radius
I guess they want is to use 22/7 for pi, which is weird, but whatever.
C = pi*d
C = 22/7*28
This gets you [tex]C = \frac{616}{7}[/tex] or 88 inches
Carlos rode his bike 1 2/3 miles to Tim's house then 1 2/3 to the park. What number line shows u the total number of miles that Carlos rode from house to the park
Number line wasn't added to the original question
Answer:
Carlos rode his bike 1 2/3 miles to Tim's house then 1 2/3 to the park. What number line shows u the total number of miles that Carlos rode from house to the park
Step-by-step explanation:
Distance biked to Tim's house = 1 2/3 miles
Distance biked to park = 1 2/3 miles
Total number of miles Carlos rode from house to park :
1 2/3 + 1 2/3
5/3 + 5/3
(5 + 5) / 3
10 /3
= 3 1/3 miles
find the missing angle measurement
Answer: 56
Step-by-step explanation: According to AIA(Alternate Interior Angles Congruency) they are equal. So just solve the equation that they're both equal to each other and you get x=-6. Plug this back in to find the angle measure, 56! Hope this helped!
Answer:
x = -6
56 degrees
Step-by-step explanation:
Those two angles that are labeled are equal for a reason that I forgot (but trust me they are). So, -8x+8 = -6x+20
just solve the equation to get x = -6
after plugging in -6 for x, you get that the angles that are labeled are 56 degrees.