It extensively proven below that f(n) = 2n + n² - n - 3 is O(n²).
It is shown that f(n) = log₂(n) × n³ is O(n³).
How to prove equations?1. To show that f(n) = 2n + n² - n - 3 is O(nᵃ), find a constant C and a positive integer N such that for all n ≥ N, |f(n)| ≤ C × nᵃ.
First simplify f(n):
f(n) = 2n + n² - n - 3
= n² + n - 3
Next, find a value for C. Choose C as the maximum value of the absolute expression |f(n)| when n is large. Analyze the behavior of f(n) as n approaches infinity.
As n becomes very large, the dominant term in f(n) is n². The other terms (2n, -n, -3) become relatively insignificant compared to n². Therefore, choose C as a constant multiple of the coefficient of n², which is 1.
C = 1
Now, find N. Find a value for N such that for all n ≥ N, |f(n)| ≤ C × nᵃ.
Since f(n) = n² + n - 3, observe that for all n ≥ 3, |f(n)| ≤ n² + n ≤ n² + n² = 2n².
Therefore, if chosen, N = 3:
|f(n)| ≤ 2n² ≤ C × n², for all n ≥ N.
This means that for all n ≥ 3, f(n) is bounded above by a constant multiple of n², satisfying the definition of O(nᵃ).
Thus, it is shown that f(n) = 2n + n² - n - 3 is O(n²).
2. To show that f(n) = log₂(n) × n³ is O(nᵃ), find a constant C and a positive integer N such that for all n ≥ N, |f(n)| ≤ C × nᵃ.
Simplify f(n) first:
f(n) = log₂(n) × n³
As n becomes very large, the logarithmic term log₂(n) grows slowly compared to the polynomial term n³. Therefore, choose C as a constant multiple of the coefficient of n³, which is 1.
C = 1
Now, find N. Find a value for N such that for all n ≥ N, |f(n)| ≤ C × nᵃ.
Since f(n) = log₂(n) × n³, observe that for all n ≥ 1, |f(n)| ≤ n³.
Therefore, if chosen N = 1:
|f(n)| ≤ n³ ≤ C × n³, for all n ≥ N.
This means that for all n ≥ 1, f(n) is bounded above by a constant multiple of n³, satisfying the definition of O(nᵃ).
Thus, it is shown that f(n) = log₂(n) × n³ is O(n³).
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The box part of the box plot contains all the values between whích numbers?
25
30
35
40
45
50
between 27 and 36 and between 37 and 40
between 32 and 36
between 32 and 37
between 27 and 32 and between 37 and 40
Answer:
between 32 and 37
Step-by-step explanation:
that's my answer
triangle abc with vertices at a(−1, −1), b(1, 1), c(0, 1) is dilated to create triangle a′b′c′ with vertices at a′(−3, −3), b′(3, 3), c′(0, 3). determine the scale factor used.
1] The scale factor used to dilate triangle ABC to create triangle A'B'C' is
2] To determine the scale factor used, we can compare the corresponding side lengths of the original triangle ABC and the dilated triangle A'B'C'.
Using the distance formula, we can calculate the lengths of the sides:
Side AB:
For triangle ABC: AB = √[(1 - (-1))^2 + (1 - (-1))^2] = √8 = 2√2
For triangle A'B'C': A'B' = √[(3 - (-3))^2 + (3 - (-3))^2] = √72 = 6√2
Side AC:
For triangle ABC: AC = √[(0 - (-1))^2 + (1 - (-1))^2] = √5
For triangle A'B'C': A'C' = √[(0 - (-3))^2 + (3 - (-3))^2] = √72 = 6√2
Side BC:
For triangle ABC: BC = √[(1 - 0)^2 + (1 - 1)^2] = 1
For triangle A'B'C': B'C' = √[(3 - 0)^2 + (3 - 3)^2] = 3
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What is the fraction for 0.85?, what is the percent for 0.85?. What is the ratio for 0.85
Answer:
Fraction: 17/20
Percent: 85%
Ratio: 85 out of 100
Step-by-step explanation:
Step-by-step explanation:
Fraction = 85/100 = 17 / 20
Percent = 17 / 20 × 100 = 85%
Ratio = 17 : 20
Question 10 of 10
In a unit circle, 0 = 3pi/2 radians. What is the terminal point?
A. (0,1)
B. (1,0)
C. (-1,0)
D. (0, -1)
The terminal point is (0, -1) if in a unit circle the angle is 3π/2 option (D) is correct.
What is a circle?It is described as a set of points, where each point is at the same distance from a fixed point (called the center of a circle)
We have:
In a unit circle, θ = 3π/2 radians.
r = 1
Angle θ = 3π/2
The terminal point is (rcosθ, rsinθ)
= (cos3π/2, sin3π/2)
= (0, -1)
Thus, the terminal point is (0, -1) if in a unit circle the angle is 3π/2 option (D) is correct.
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Which of these is an example of a continuous random variable?
O A. Number of breaths in a minute
O B. Number of employees at a company
O C. Scores in a bowling tournament
D. Length of a fish
Answer:
D Length of F I S H
Step-by-step explanation:
I chose the bowling one and got it wrong, but you can have the right answer
Answer: length of a fish
Step-by-step explanation: quizzzz
verify x+(y+z) =(x+y) +z, when x=-4/15, y=-4/5and z =17/8
Answer:
The equality is true
Step-by-step explanation:
x+(y+z)=(x+y)+z
x+(y+z)=(x+y)+zwhen we plug in -4/15 for x,-4/5 for y
x+(y+z)=(x+y)+zwhen we plug in -4/15 for x,-4/5 for yand 17/8 for z,we get
x+(y+z)=(x+y)+zwhen we plug in -4/15 for x,-4/5 for yand 17/8 for z,we get-4/15+(-4/5+17/8)=(-4/15+-4/5)+17/8
x+(y+z)=(x+y)+zwhen we plug in -4/15 for x,-4/5 for yand 17/8 for z,we get-4/15+(-4/5+17/8)=(-4/15+-4/5)+17/817/24=17/24
x+(y+z)=(x+y)+zwhen we plug in -4/15 for x,-4/5 for yand 17/8 for z,we get-4/15+(-4/5+17/8)=(-4/15+-4/5)+17/817/24=17/24so this equality is true
Let f be a given function. A graphical interpretation of the 2-point backward difference formula for approximating f'(x) is the slope of the line joining the points of abscissas xo - h and X, with h > 0. False True
A graphical interpretation of the 2-point backward difference formula for approximating f'(x) is the slope of the line joining the points of abscissas xo - h and X, with h > 0 is False
The 2-point backward difference formula for approximating f'(x) is given by:
f'(x) ≈ (f(x) - f(x - h)) / h
In this formula, the slope is calculated using the values of f(x) and f(x - h) at two points: x and x - h. The graphical interpretation of this formula involves finding the slope of the line passing through these two points.
However, the given statement states that the line is joining the points of abscissas xo - h and X, with h > 0. This implies that the line is connecting a fixed point xo - h to a variable point X. This interpretation does not align with the 2-point backward difference formula.
Therefore, the statement is false.
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Three angles of a quadrilateral measure 45°, 95°, and 111°. What is the measure of the fourth angle of the quadrilateral?
Answer:
109°
Step-by-step explanation:
sum of angles of a quadrilateral=360°
45+95+111+fourth angle=360
fourth angle=360-251=109°
The equations of two lines are:
2x-y=4 and y=-2x+8.
What is the value of x in the solution for this system?
Answer:
x = 3
Step-by-step explanation:
2x - y = 4
y = -2x + 8
-y = -2x + 4
y = -2x + 8
0 = -4x + 12
-4x = -12
x = 3
If the point (-4,-8) is on the graph of the function L(x) and the slope of the line is 15, what is the equation for the function L(x).
The equation of the function L(x) is given by y = 15x + 52.
To find the equation of the function L(x), given that the point (-4, -8) is on the graph of the function L(x) and the slope of the line is 15, we use the point-slope form of the equation of a line.
Let's assume that the equation of the function L(x) is of the form y = L(x).
The slope of the line, m = 15
The point (-4, -8) is on the line, which means that it satisfies the equation of the line: y - y1 = m(x - x1),
where (x1, y1) = (-4, -8)
Substituting m, x1 and y1 in the equation of the line, we get:
y - (-8) = 15(x - (-4))y + 8
= 15(x + 4)y + 8
= 15x + 60y
= 15x + 52
The equation of the function L(x) is y = 15x + 52.
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Find the surface area.
Answer:
600ft^2
Step-by-step explanation:
Step One: There are two identical faces of each, which means we can multiply each face times two to make it faster. The first face will be 2(4*12)=96. I muliplied by two, because, once again, there are two faces of each.
Step Two: The next is 2(18*12)=432 and lastly, 2(4*18)= 72
Step Three: We just have to add it all up: 72+96+432= 600ft^2
ill give brainliest
Write and solve the equation for the following situation:
Angles 1 and 2 are complementary. The measure of angle 1 is 16° larger than the measure of angle 2.
A. x + 16 = 90 x = 74
B. x + (x - 16) = 90 x = 53
C. 2x + 16 = 90 x = 37
D. x + 16 = 180 x = 164
Find the surface area.
24 in.
40 in.
10 in.
26 in.
NEED HELP ASAP!!
Answer:
200
Step-by-step explanation:
Let T: R3 → R3 be the linear transformation that projects u onto v = (a) Find the rank and nullity of T (b) Find a basis for the kernel of T.
The linear transformation T: R³ → R³ that projects u onto v has Rank(T) = 1 and the basis for the kernel of T is {a}.
To find the rank and nullity of the linear transformation T: R³→ R³, we need to determine the dimensions of the image space (range) and the kernel (null space) of T.
(a) Rank of T:
The rank of T is equal to the dimension of the image space. Since T projects u onto v, the image of T is the span of the vector v. Therefore, the rank of T is 1.
Rank(T) = 1
(b) Basis for the kernel of T:
The kernel of T consists of all vectors u in R³ that are mapped to the zero vector in R³. In other words, it consists of all vectors u perpendicular to the vector v.
To find a basis for the kernel, we need to solve the equation T(u) = 0. Since T projects u onto v, we can express this as u - proj_v(u) = 0.
For any vector u in R³, the projection of u onto v can be computed as:
proj_v(u) = (u · v) / (||v||²) * v
where u · v represents the dot product of u and v, and ||v|| is the norm (length) of v.
In this case, v = (a), so we can rewrite the projection formula as:
proj_v(u) = (u · (a)) / (||a||²) * (a)
Since T(u) = u - proj_v(u) = 0, we have:
u - (u · (a)) / (||a||²) * (a) = 0
This equation can be rearranged as:
u = (u · (a)) / (||a||²) * (a)
Now we can find a basis for the kernel by setting u to be a free variable and expressing it in terms of (a).
Let's denote the scalar (u · (a)) / (||a||²) as k:
u = k * a
Therefore, any vector in the kernel of T can be written as k * a, where k is a scalar.
A basis for the kernel of T is {a}.
So, the basis for the kernel of T is {a}.
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The mean number of years of marriage preceding divorce is 7. The median mber of years is 6. Most divorces occur, however, either at 3 years of marriage 22 years. Which measure of central tendency best describes these data, and y?
The measure of central tendency that best describes the given data is the Mode.
The mode is the value that occurs most frequently in a data set. According to the given data, most divorces occur either at 3 years of marriage or 22 years. Hence, the mode best describes these data.
The mean is the average value of a data set, which is calculated by adding all the values and dividing by the number of values.
The median is the middle value of a data set when arranged in numerical order. In the given data, the mean number of years of marriage preceding divorce is 7, and the median number of years is 6. Since most divorces occur at either 3 years or 22 years, the mode best describes the given data.
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x^2-8x+15
expresión racional
Answer:
x = 3, 5
Step-by-step explanation:
(x - 3)(x - 5) = 0
What is a scatterplot?
a graph showing the relationship between two sets of data
a relationship between two variables
a data point that does not fit the general pattern in the data
a group of data points around a value
Answer:
a graph showing the relationship between two sets of data
Step-by-step explanation:
Help fastttt someone pls and no files will block you
Bryce is testing whether school is more enjoyable when students are making high grades. He asked 100 students if they enjoyed school and whether their GPA was
above or below 3.0. He found that 33 of the 40 students with a GPA above 3.0 reported that they enjoyed school, and 5 of the 60 students with a GPA below 3.0
reported that they enjoyed school. What is the probability that a student with a GPA below 3.0 does not enjoy school?
look at photo for the question and answer choices... NO LINKS OR BLANK ANSWERS
Answer:
B and E
Step-by-step explanation:
Poosjsjdiwjqjsnsjjdd
Answer:
poosjsjdwjqisnsjjdd
Step-by-step explanation:
Answer:
what's the question?
You spin the spinner and flip a coin. Find the probability of the compound event. The probability of spinning a number less than 3 and flipping tails is
Answer:
1/5
Step-by-step explanation
The spinner has the numbers 1-5 and they want the numbers less than 3 there a 2/5 then multiply the denominator by 1/2 thin divide it by 2
2/10 divided by 2 = 1/5
.
2+x5-3=(2+)x(5-3) is the equation true or not show work why
Please help I will give 20 points if right
Answer:
for my own advice you should add those numbers then that Square unit if she's there it's 90° so you should find the area of the polygon which that thing when you come this sides you get the number of the polygon with your plus then by those numbers then he write his equals to 90 degree then you'll get your answer thank you
Carissa’s gerbil has a tail that is the same length as its body length. Its tail is 102 millimeters. How long is her gerbil in centimeters?
Answer:
10.2 cm
Step-by-step explanation:
Divide the tail length by 10
HoPe ThIs HeLpEd youuu! lol :)
Answer: 10.2 cm
Hope this helped! :D
Data Mining. Data Mining cannot automatically find beneficial patterns for a business. True False
False. Data mining can automatically find beneficial patterns for a business by utilizing various techniques and algorithms to extract valuable insights and uncover hidden patterns from large datasets.
Data mining refers to the process of discovering patterns, relationships, and insights from large datasets. It involves using various techniques and algorithms to extract valuable information and knowledge from data. One of the primary goals of data mining is to uncover patterns that can be beneficial for businesses, such as identifying customer preferences, market trends, or predicting future outcomes.
Through automated analysis and pattern recognition, data mining can uncover hidden patterns and relationships that may not be apparent through traditional manual analysis. Therefore, data mining has the potential to automatically find beneficial patterns for businesses, making the statement "Data Mining cannot automatically find beneficial patterns for a business" false.
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7.13 hm³ = _____m³
Convert this!
Options
713 x 10^4
71.3 x 10²
71.3 x10³
713 x 10²
Answer:
71.3 * 10² that's the answer to your question
Step-by-step explanation:
The answer is 713 X 10^4
I hope it helps
Use Cramer's vale to solve the following system of equation: J2X1 - X2-3 = 0 ./) X1 + 3X2-7= 0 3X1 + 2X2-1=0 4X1 + 5X2 = 14
Using Cramer's rule the solutions to the system of equations are:
X1 = 21 / (-J2 - 9)
X2 = -12 / (-J2 - 9)
Using Cramer's rule, we can solve the system of equations:
J2X1 - X2 - 3 = 0
X1 + 3X2 - 7 = 0
3X1 + 2X2 - 1 = 0
4X1 + 5X2 = 14
The values of X1 and X2, we'll calculate the determinants.
Let D be the determinant of the coefficient matrix:
D = |J2 -1 0| = J2(-1) - 3(3) = -J2 - 9
D1 is the determinant obtained by replacing the first column of the coefficient matrix with the constants:
D1 = |0 -1 0| = 0(-1) - (-7)(3) = 21
D2 is the determinant obtained by replacing the second column of the coefficient matrix with the constants:
D2 = |J2 0 0| = J2(0) - 3(4) = -12
Now, we can calculate the values of X1 and X2 using the determinants:
X1 = D1 / D = 21 / (-J2 - 9)
X2 = D2 / D = -12 / (-J2 - 9)
Therefore, the solutions to the system of equations are:
X1 = 21 / (-J2 - 9)
X2 = -12 / (-J2 - 9)
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the annual
The present population of
of a town is 66550. If the annual population growth
rate is 10% what was
the Population growth of town 3 years ago ?
Answer:
Population (t=-3)= 50,000
Step-by-step explanation:
Giving the following information:
Annual growth rate (g)= 10%
Number of periods (n)= 3 years
Present Value (PV)= 66,550 people
To calculate the population three years ago, we need to use the following formula:
Population (t=-3)= PV / (1 + g)^n
Population (t=-3)= 66,550 / (1.1^3)
Population (t=-3)= 50,000
When light arrives and leaves a reflecting surface the angle with the line perpendicular to the surface is what?
Answer:
the angle of reflection is the angle between the reflected wave and the "normal" (the perpendicular line to the reflecting surface).
Step-by-step explanation:
Answer:
The angle which the light leaves or bounces off a surface is angle of reflection.
Step-by-step explanation:
has two types of light rays. The incoming ray and the outgoing, or reflected, ray. The angle between the incident ray and an imaginary perpendicular line drawn to the surface of the mirror is called the angle of incidence.