show that 3x4 1 is o(x4∕2) and x4∕2 is o(3x4 1).
Answers
3\(x^4\) + 1 is equivalent to \(x^{(4/2)}\) in terms of asymptotic growth rate. The value of 3\(x^4\) + 1 is O(\(x^{(4/2)}\)) and \(x^{(4/2)}\) is O(3\(x^4\) + 1).
To show that 3\(x^4\) + 1 is O(\(x^{(4/2)}\)), we need to find a constant C and a value of x such that:
|3\(x^4\) + 1| <= C|\(x^{(4/2)}\)|
For x >= 1, we can say that:
3\(x^4\) + 1 <= 4\(x^4\)
Taking the square root of both sides, we get:
sqrt(3\(x^4\) + 1) <= 2x²
Thus, we can choose C = 2 and x0 = 1, and we have:
|3\(x^4\) + 1| <= 2|\(x^{(4/2)}\)| for all x >= 1
Therefore, 3\(x^4\) + 1 is O(\(x^{(4/2)}\)).
To show that \(x^{(4/2)}\) is O(3\(x^4\) + 1), we need to find a constant C and a value of x such that:
|\(x^{(4/2)}\)| <= C|3\(x^4\) + 1|
For x >= 1, we can say that:
\(x^{(4/2)}\) <= x²
Thus, we can choose C = 1 and x0 = 1, and we have:
|\(x^{(4/2)}\)| <= |3\(x^4\) + 1| for all x >= 1
Therefore, \(x^{(4/2)}\) is O(3\(x^4\) + 1).
Since both 3\(x^4\) + 1 is O(\(x^{(4/2)}\)) and \(x^{(4/2)}\) is O(3\(x^4\) + 1).
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Related Questions
Drew is making a dart board, the dart board is 50 centimeters what is the circumefrence
Answers
The circumference of the dartboard, given that 50 centimeters is the diameter, would be 157. 14 cm
How to find the circumference ?The circumference of a circular dartboard can be calculated using the formula:
Circumference = 2 x π x r
In this case, the dartboard has a diameter of 50 centimeters. The radius is half the diameter, so the radius of the dartboard is:
Radius = Diameter / 2
= 50 cm / 2
= 25 cm
The circumference is therefore ;
= 2 x π x 25 cm
= 2 x 3. 14159 x 25 cm
= 6. 28318 x 25 cm
= 157. 14 cm
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a frog is positioned at the origin of the coordinate plane. from the point (x, y), the frog can jump to any of the points (x 1, y), (x 2, y), (x, y 1), or (x, y 2). find the number of distinct sequences of jumps in which the frog begins at (0, 0) and ends at (4, 4).
Answers
The minimum number of jumps required for the frog to go from (0,0) to (4,4) and each distance is 5 units is 3.
Graph Theory, in discrete mathematics, is the study of the graph. A graph is determined as a mathematical structure that represents a particular function by connecting a set of points. It is used to create a pairwise relationship between objects.
The graph is made up of vertices (nodes) that are connected by the edges (lines). The applications of the linear graph are used not only in Maths but also in other fields such as Computer Science, Physics and Chemistry, Linguistics, Biology, etc. In real-life also the best example of graph structure is GPS, where you can track the path or know the direction of the road.
We have, the initial position of frog =(0,0)
The endpoint of the jump of frog is (4,4)
The minimum number of jumps required for the frog to go from (0,0) to (4,4) and each distance is 5 units is 3
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Additive Inverses are numbers that add to make-101/11
Answers
The aditive inverse of a number is given by itself multiplied by -1. Then if we have any number x its additive inverse is -x and their sum is:
\(x-x=0\text{ for all values of x}\)Then theanswer is the second option, 0.
Stuck on this……………….
Answers
Answer:
the value of x that makes the equation true is -9. Makes the equation turn into 27/-9 = -3
for value of x that makes the equation false, substitute any number,
like 27, for example, which would make the equation = 1.
Complete the table for the equation of the line y = 3x - 1 X
f(x)
-1
0
2
Answers
Answer:
Step-by-step explanation:
Given the equation of a line expressed as y = 3x-1, we are to find y = f(x) for the following values of x in the table.
If y = 3x-1, then f(x) = 3
When x = -1
f(-1) = 3(-1)-1
f(-1) = -3-1
f(-1) = -4
Hence when x = -1, f(x) = -4
When x = 0
f(0) = 3(0)-1
f(0) = 0-1
f(0) = -1
Hence when x = 0, f(x) = -1
When x = 2
f(2) = 3(2)-1
f(2) = 6-1
f(2) = 5
Hence when x = 2, f(x) = 5
15 out of 20 students own hand-held games; 105 out of 160 students own hand held games. PLEASE HELP ME ASAP
Answers
Answer: No the ratios are NOT related
Step-by-step explanation:
the ________ is a line graph that plots the cumulative relative frequency distribution.
Answers
The ogive is a line graph that plots the cumulative relative frequency distribution.
An ogive, also known as a cumulative frequency polygon, is a line graph that shows the cumulative frequency distribution of a data set. The cumulative frequency is calculated by adding up the frequencies of each value up to a certain point in the data set.
The cumulative relative frequency is calculated by dividing the cumulative frequency by the total number of observations in the data set. The ogive plots these cumulative relative frequencies against the corresponding values in the data set, usually on the x-axis.
By plotting the cumulative relative frequencies, the ogive shows how the data is distributed over the entire range of values. It can be used to identify patterns in the data, such as whether it is skewed or symmetrical. It is also useful for determining percentiles, as the percentile for a given value can be read directly from the ogive.
Overall, the ogive is a helpful tool for summarizing and visualizing the distribution of a data set, particularly when dealing with large data sets or complex distributions.
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Joey’s family wants to save $5000 to finance a vacation trip to a popular amusement park. If they save $240 at the beginning of each month and the fund is invested to earn 5% compounded monthly, how long will it take them to save enough money to take the trip?
Answers
It will take Joey's family approximately 21 months to save enough money, compounded monthly at a 5% interest rate, to finance their vacation trip to the amusement park.
To determine the time it will take to save enough money, we can use the formula for the future value of an ordinary annuity:
FV = P * [(1 + r)^n - 1] / r,
where FV is the future value, P is the monthly payment, r is the monthly interest rate, and n is the number of periods.
Given:
Desired future value (FV) = $5000,
Monthly payment (P) = $240,
Monthly interest rate (r) = 5% / 12 = 0.05 / 12 = 0.004167.
We need to solve for the number of periods (n) in the formula. Rearranging the formula, we have:
n = log(1 + (FV * r) / P) / log(1 + r),
where log denotes the logarithm with base 10.
Plugging in the given values, we get:
n = log(1 + ($5000 * 0.004167) / $240) / log(1 + 0.004167).
Using a calculator, we find that n is approximately 20.998, which rounds up to 21.
Therefore, it will take Joey's family approximately 21 months to save enough money, compounded monthly at a 5% interest rate, to finance their vacation trip to the amusement park.
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A tank contains 9,000 L of brine with 12 kg of dissolved salt. Pure water enters the tank at a rate of 90 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. (a) How much salt is in the tank after t minutes? y = kg (b) How much salt is in the tank after 20 minutes? (Round your answer to one decimal place.) y = kg
Answers
Therefore, After 20 minutes, there are approximately 11.9 kg (rounded to one decimal place) of salt in the tank.
To solve this problem, we need to consider the rate of change of the amount of salt in the tank over time.
(a) Let's denote the amount of salt in the tank after t minutes as y (in kg). We can set up a differential equation to represent the rate of change of salt:
dy/dt = (rate of salt in) - (rate of salt out)
The rate of salt in is given by the concentration of salt in the incoming water (0 kg/L) multiplied by the rate at which water enters the tank (90 L/min). Therefore, the rate of salt in is 0 kg/L * 90 L/min = 0 kg/min.
The rate of salt out is given by the concentration of salt in the tank (y kg/9000 L) multiplied by the rate at which water leaves the tank (90 L/min). Therefore, the rate of salt out is (y/9000) kg/min.
Setting up the differential equation:
dy/dt = 0 - (y/9000)
dy/dt + (1/9000)y = 0
This is a first-order linear homogeneous differential equation. We can solve it by separation of variables:
dy/y = -(1/9000)dt
Integrating both sides:
ln|y| = -(1/9000)t + C
Solving for y:
y = Ce^(-t/9000)
To find the particular solution, we need an initial condition. We know that at t = 0, y = 12 kg (the initial amount of salt in the tank). Substituting these values into the equation:
12 = Ce^(0/9000)
12 = Ce^0
12 = C
Therefore, the particular solution is:
y = 12e^(-t/9000)
(b) To find the amount of salt in the tank after 20 minutes, we substitute t = 20 into the particular solution:
y = 12e^(-20/9000)
y ≈ 11.8767 kg (rounded to one decimal place)
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The length of a rectangle is 3 more than five times its width. If the perimeter of
the rectangle is 30 meters, find the dimensions of the rectangle.
Answers
Answer:
Image result for The length of a rectangle is 3 more than five times its width. If the perimeter of the rectangle is 30 meters, find the dimensions of the rectangle.
To find the width, multiply the length that you have been given by 2, and subtract the result from the perimeter. You now have the total length for the remaining 2 sides. This number divided by 2 is the width.
Consider the three infinite series below. (-1)-1 (n+1)(,2−1) (1) 5n 4n³ - 2n + 1 n=1 n=1 (a) Which of these series is (are) alternating? (b) Which one of these series diverges, and why? (c) One of
Answers
(a) Among the three infinite series given, the first series (-1)-1 (n+1)(,2−1) (1) is alternating.
(b) The series 5n 4n³ - 2n + 1 diverges.
In summary, the first series is alternating, and the series 5n 4n³ - 2n + 1 diverges.
(a) To determine if a series is alternating, we need to check if the signs of consecutive terms alternate. In the first series, we have (-1)-1 (n+1)(,2−1) (1), where the negative sign alternates between terms. Therefore, it is an alternating series.
(b) To determine if a series diverges, we examine its behavior as n approaches infinity. In the series 5n 4n³ - 2n + 1, we can observe that as n increases, the dominant term is 4n³, which grows faster than any other term. The other terms become relatively insignificant compared to 4n³ as n becomes large. Since the series does not converge to a finite value as n approaches infinity, it diverges.
In conclusion, the first series is alternating, and the series 5n 4n³ - 2n + 1 diverges because its terms do not approach a finite value as n increases.
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Please help me..............
Answers
Answer:
it is A
Step-by-step explanation:
The table shows the height of several buildings in Chicago. Use the table to determine whether the height of the building is a linear function of the number of stories.
Stories Height (ft)
Harris Bank III: 35 510
One Financial: 40 515
Federal build: 45 545
Mid Plaza: 50 582
North tower: 55 552
Answers
Answer:
no (mark brainlest plz)
Step-by-step explanation:
because North tower dips down when its the highest. Its also not linear because it +5 then +30 all the sudden
In June, Ryan earned $20 mowing lawns. He made $35 mowing lawns in
July. What is the percent increase in his earnings?
Answers
Answer:
75% increase
Step-by-step explanation:
20 divided by 4 = 5 ( 25%)
5 x 3 = 15 ( 75%)
Hope this Helps!
An assembly line has 16 hours to make 1.000 units. What is the required cycle time? (slide 23) 72sec 216sec 57.65sec 14,4sec
Answers
The required cycle-time is approximately 57.6 seconds.
To find the required cycle time, we need to divide the total available time by the number of units to be produced.
Total available time: 16 hours = 16 * 60 minutes = 960 minutes = 960 * 60 seconds = 57,600 seconds
Number of units to be produced: 1,000 units
Required cycle time: Total available time / Number of units
Cycle time = 57,600 seconds / 1,000 units
Cycle time ≈ 57.6 seconds
Therefore, the required cycle time is approximately 57.6 seconds.
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you have four pieces of a chain necklace, which you wish to join into a single circle of 12 links. it costs $2 to open a link, and 3$ to close a link you have 15 dollars how can you do this
Answers
The four pieces of chain necklaces into a single circle of 12 links, you need to open six links (1.5 links for each necklace) and then close two links, which will cost $18.
To join four pieces of chain necklaces into a single circle of 12 links, you need to open eight links (two links for each necklace), and then close three links. Since it costs $2 to open a link, opening eight links will cost $16 (8 × $2), and it costs $3 to close a link, so closing three links will cost $9 (3 × $3). Therefore, the total cost will be $16 + $9 = $25.
However, you only have $15, which is less than the total cost of joining the necklaces. Therefore, you need to find another solution to join the necklaces. You can open six links (1.5 links for each necklace) and then close two links. Opening six links will cost $12 (6 × $2), and closing two links will cost $6 (2 × $3).
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A car driving at 10.0 m/s southward accelerates at 4.11 m/s2 southward for 7.25 s. what is the car's speed after this amount of time?
Answers
The car's speed after the given amout of time is 19.8 m/s²
How would you define acceleration?The rate at which an object's velocity with respect to time changes is referred to as acceleration in mechanics. They are vector quantities, accelerations. The direction of the net force acting on an object determines the direction of its acceleration. A point or object going straight ahead is accelerated when it accelerates or decelerates. Any alteration in an object's velocity could take the form of a change in direction of motion or a speed increase or reduction. The moon orbiting the earth, an apple falling to the ground, and a car stopped at a stop sign are a few instances of acceleration.
Average Acceleration =Change in Velocity/ Time Taken
So Change in Velocity is 10-x m/s
Time taken is 7.25 s
Given accelaration 4.11 m/s²
Putting the values we get,
4.11 = (10-x)/7.25
⇒29.8 = 10-x
⇒x=19.8 m/s²
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30 ft
10 ft ;
20 ft
10 ft
10 ft
a. How much carpet does he need?
b. The carpet costs $3 per square
foot. How much will it cost to
carpet the bedroom?
Answers
Answer:30+15
Step-by-step explanation:
GRADE 8
The following figure shows triangle XYZ. The length of XY is 19 units, and the length of YZ is 181 units.
Y
19
X
181
What is the value of
Z
Note: Figure not drawn to scale.
? Enter your answer as a fraction in the spaces provided.
YZ
Answers
The value of \(\frac{XZ}{YZ}\) include the following: C. \(\frac{180}{181}\)
What is Pythagorean theorem?In Mathematics and Geometry, Pythagorean's theorem is represented by the following mathematical equation:
a² + b² = c²
Where:
a, b, and c represent the side lengths of a right-angled triangle.
In order to determine the length of XZ in right-angled triangle XYZ, we would have to apply Pythagorean's theorem.
YZ² = XY² + XZ²
181² = 19² + XZ²
XZ² = 32,761 - 361
XZ = √32,400
XZ = 180 units
For the ratio, we have:
Ratio = XZ/YZ
Ratio = 180/181
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
which of the following could be the value of |x-3|+4 for some value of x? A) -2 B) 1 C) 3 D) 5
Answers
Answer:
5
Step-by-step explanation:
|x-3|+4
The minimum value that an absolute value can take is zero
0+4
4
The smallest value is 4
The answer must be 4 or greater
The only answer that is possible is 5
Prove that 1+3+9+27+…+3^n=3^n+1−1/2 Let n be a positive integer,
Answers
Using mathematical induction, we can prove that the equation 1 + 3 + 9 + 27 + ... + 3^n = (3^(n+1) - 1) / 2 holds true for all positive integers n.
To prove the equation 1 + 3 + 9 + 27 + ... + 3^n = (3^(n+1) - 1) / 2, we can use mathematical induction.
1. Base Case:
For n = 1, we have 1 = (3^(1+1) - 1) / 2.
1 = (3^2 - 1) / 2.
1 = (9 - 1) / 2.
1 = 8 / 2.
1 = 4.
The base case holds true.
2. Inductive Step:
Assume that the equation holds true for some positive integer k, i.e., 1 + 3 + 9 + 27 + ... + 3^k = (3^(k+1) - 1) / 2.
We need to prove that it also holds true for k + 1, i.e., 1 + 3 + 9 + 27 + ... + 3^k + 3^(k+1) = (3^((k+1)+1) - 1) / 2.
Starting from the left side of the equation:
1 + 3 + 9 + 27 + ... + 3^k + 3^(k+1) = (3^(k+1) - 1) / 2 + 3^(k+1)
= (3^(k+1) - 1 + 2 * 3^(k+1)) / 2
= (3^(k+1) - 1 + 2 * 3 * 3^k) / 2
= (3^(k+1) + 2 * 3 * 3^k - 1) / 2
= (3^(k+1) + 2 * 3^(k+1) - 1) / 2
= (3 * 3^(k+1) + 3^(k+1) - 1) / 2
= (3^(k+2) + 3^(k+1) - 1) / 2
= (3^(k+2) + 3^(k+1) - 1 * 2/2) / 2
= (3^(k+2) + 3^(k+1) - 2) / 2
= (3^(k+2) + 3^(k+1) - 2) / 2
= (3^(k+2) + 3^(k+1) - 1) / 2 - 1/2
= (3^(k+2+1) - 1) / 2 - 1/2
= (3^((k+1)+1) - 1) / 2 - 1/2
Thus, we have shown that if the equation holds true for k, it also holds true for k + 1.
By the principle of mathematical induction, the equation is true for all positive integers n. Therefore, we have proven that 1 + 3 + 9 + 27 + ... + 3^n = (3^(n+1) - 1) / 2 for any positive integer n.
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Find the area of the shape below please. (:
Answers
Answer:
105
Step-by-step explanation:
15*6=90
6*5=30
30/2=15
90+15=105
Answer:
105 cm^2
Step-by-step explanation:
Area=15*6+(15-9)*(11-6)/2=(90+6*5/2)=90+15=105 cm^2
Write an explicit formula for an, the nth term of the sequence 26, 35, 44, ....
Answers
Here , we need to find an formula for nth term of the sequence 26 , 35 , 44 , .....
Now , here if you notice carefully ,then you can notice that 35 - 26 = 9 , 44 - 35 = 9 , So the given sequence is an AP ( Arithmetic Progression ) with common difference being 9 , first term being 26 , so now as we knows that :
\({\boxed{\bf{a_{n}=a+(n-1)d}}}\)Where , \({\bf{a_n}}\) is nth term of AP , \(\bf a\) being first term while \(\bf d\) is the common difference . So , now putting the values in above formula ;
\({:\implies \quad \sf a_{n}=26+(n-1)9}\)
\({:\implies \quad \sf a_{n}=26+9n-9}\)
\({:\implies \quad \bf \therefore \quad \underline{\underline{a_{n}=9n+17}}}\)
Henceforth , nth term of the sequence is 9n+17
Answer:
\(a_n=9n+17\)
Step-by-step explanation:
From inspection we can see that the sequence is an arithmetic sequence since the common difference between terms is constant.
General form of an arithmetic sequence formula: \(a_n=a+(n-1)d\)
(where a is the first term and d is the common difference).
To find the common difference subtract one term from the next:
\(d = 44-35=9\)
or \(d=35-26=9\)
Given:
\(a=26\)\(d=9\)Therefore, \(a_n=26+(n-1)9\)
Simplified: \(a_n=9n+17\)
Art Club. He then made a scale drawing of his original painting to create a similar rectangular poster. He used a scale factor of 4/5 to create the rectangular poster. If the perimeter of the smaller rectangle is 45 centimeters, what is the perimeter of the larger rectangle?
Answers
The perimeter of the larger rectangle found using the scale factor of 4/5 is 56.25 cm.
What is meant by the perimeter of a figure?
A closed path that covers, encircles, or outlines a one-dimensional length or a two-dimensional shape is called a perimeter. The complete length of a shape's boundary is referred to as the perimeter in geometry. A shape's perimeter is calculated by adding the lengths of all of its sides and edges.
Given,
the scale factor = 4/5
The perimeter of the smaller rectangle = 45 cm
We are asked to find the perimeter of the larger rectangle(x).
Here both the rectangle are similar figures.
So the scale factor is the ratio of their corresponding sides.
But since the perimeter of a rectangle is the sum of the length of its sides.
The scale factor is also the ratio of the perimeter of the smaller rectangle to the larger rectangle.
Perimeter of small rectangle/Perimeter of large rectangle = 4/5
45/x = 4/5
x = 45 * 5 / 4 = 56.25cm
Therefore the perimeter of the larger rectangle found using the scale factor of 4/5 is 56.25 cm.
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Can someone help me?
I’m not sure if -4/-12 simplified is negative or positive
Answers
Answer:
The answer is 1/3
Step-by-step explanation:
-4/-12 is the same as -4 divided by -12, and a negative divided by a negative will always be a positive; therefore, the answer comes out to be positive.
what's the absolute value of 4.5 ?
Answers
Answer:
the absolute value or 4.5 is 4.5
Step-by-step explanation:
Answer:
4.5
Step-by-step explanation:
absolute value: the distance away from zero, it will always be positive because distance can't be negative.
thus, the absolute value of 4.5 = the distance of 4.5 to zero, which is 4.5
|4.5| = 4.5
What is the gradient of the graph shown?
Give your answer in its simplest form.
Answers
Answer:
-2
Step-by-step explanation:
I think this was the question you were talking about.
If $a>0$ and $b>0$, a new operation $\nabla$ is defined as follows:$$a \nabla b = \dfrac{a + b}{1 + ab}.$$For example,$$3 \nabla 6 = \dfrac{3 + 6}{1 + 3 \times 6} = \dfrac{9}{19}.$$For some values of $x$ and $y$, the value of $x \nabla y$ is equal to $\dfrac{x + y}{17}$. How many possible ordered pairs of positive integers $x$ and $y$ are there for which this is true?
Answers
This happens when
1 + a b = 17 ==> a b = 16
With a and b both positive integers, and 16 = 2^4, we can have
• a = 1 and b = 16
• a = 2 and b = 8
• a = b = 4
and vice versa. So there are 5 possible ordered pairs.
Mrs. Alejandro places four cards on the board. Which does not represent the same distance?
A. 36 yards
B. 432 feet
C. 1296 inches
D. 108 feet
Answers
Answer:
B. 432 feet
Step-by-step explanation:
36 yards * 3 = 108 feet
108 feet * 12 = 1296 inches
Suppose (a.n - - - 31(1-0}10 expresses a in base 10. Prove that. 13 | a. if
and only if 13 i (an ' ' ' (1-1)") + 4H0. (b) Use part. (a) to decide whether
20192018 is divisible by 13.
Answers
We need to check if 13 divides (20192018 - (1-1)") + 4H0. By performing the calculations, we find that 13 does not divide (20192018 - (1-1)") + 4H0. Hence, 20192018 is not divisible by 13.
To prove that 13 divides a number "a" if and only if 13 divides (an - (1-1)") + 4H0, we need to show that both conditions imply each other
First, let's assume that 13 divides "a". This means that "a" can be written as 13k, where k is an integer. We can express this in terms of base 10 as (13k - (1-1)") + 4H0. Simplifying further, we get (13k - 1) + 4H0. Since 13 divides 13k and 13 divides 4H0, it also divides their sum. Therefore, if 13 divides "a", it also divides (an - (1-1)") + 4H0.
Next, let's assume that 13 divides (an - (1-1)") + 4H0. This means that (an - (1-1)") + 4H0 can be written as 13k, where k is an integer. Simplifying the expression, we get an - (1-1)") + 4H0 = 13k. Rearranging, we have an = 13k + (1-1)") + 4H0. Since 13 divides 13k + (1-1)") + 4H0, it also divides their sum. Therefore, if 13 divides (an - (1-1)") + 4H0, it also divides "a".
In conclusion, we have proven that 13 divides "a" if and only if 13 divides (an - (1-1)") + 4H0.
To decide whether 20192018 is divisible by 13, we can apply the result from part (a). Let's express 20192018 as (an - (1-1)") + 4H0. Since 13 divides (an - (1-1)") + 4H0, if 13 divides 20192018, it will also divide (an - (1-1)") + 4H0. Therefore, we need to check if 13 divides (20192018 - (1-1)") + 4H0. By performing the calculations, we find that 13 does not divide (20192018 - (1-1)") + 4H0. Hence, 20192018 is not divisible by 13.
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