a. For the function f(x) = (x + x)(log(x) + 3x) is O(x) with the witnesses C = 7 and k = 1.
b. It is proved that f(x) = (x + x)(log(x) + 3x) is (x) with the witnesses C = 1 and k = 1.
c. The obtained witnesses C = 1, C' = 10, and k =[tex]10^C.[/tex] such that g(x)log(x) < Cg(x) whenever x > k. And g(x) = 1.
a) To show that f(x) = (x + x)(log(x) + 3x) is O(x),
find witnesses C and k such that f(x) ≤ C × x for all x > k.
Let's simplify the expression for f(x):
f(x) = 2x × (log(x) + 3x)
= 2x × log(x) + 6x²
Now, find a witness C and a value k such that f(x) ≤ C × x for all x > k.
Let's choose C = 7 and k = 1.
This means show that f(x) ≤ 7x for all x > 1.
For x > 1,
f(x) = 2x × log(x) + 6x²
< 2x × log(x) + 6x² + 7x
= 2x × log(x) + 6x² + 7x
= x(2log(x) + 6x + 7)
≤ x(2log(x) + 13x)
≤ x × 7
= 7x
This implies,
f(x) = (x + x)(log(x) + 3x) is O(x) with the witnesses C = 7 and k = 1.
b) To show that f(x) = (x + x)(log(x) + 3x) is (x),
find witnesses C and k such that f(x) ≥ C × x for all x > k.
Let us simplify the expression for f(x):
f(x) = 2x × (log(x) + 3x)
= 2x × log(x) + 6x²
Now, find a witness C and a value k such that f(x) ≥ C × x for all x > k.
Let us choose C = 1 and k = 1.
This means show that f(x) ≥ x for all x > 1.
For x > 1,
f(x) = 2x × log(x) + 6x²
> x × log(x) + 6x²
= x(log(x) + 6x)
≥ x(log(x) + x)
≥ x × log(x)
≥ x
Therefore, we have shown that f(x) = (x + x)(log(x) + 3x) is (x) with the witnesses C = 1 and k = 1.
c) To find the obtained witnesses C, C', and k .
such that g(x)log(x) < Cg(x) whenever x > k,
Examine the expression f(x) = (x + x)(log(x) + 3x) and determine the function g(x).
Let us simplify the expression for f(x),
f(x) = 2x × (log(x) + 3x)
= 2x × log(x) + 6x²
From the given condition, we have g(x)log(x) < Cg(x) rewrite this as,
log(x) < C
Since log(x) is an increasing function, if log(x) < C, it means x < [tex]10^C.[/tex]Therefore, the witness k is [tex]10^C.[/tex]
Now let us determine g(x).
Since g(x)log(x) appears in the inequality, we can take g(x) = 1.
C = 1, C' = 10, and k = [tex]10^C.[/tex] for g(x)log(x) < Cg(x) whenever x > k. And g(x) = 1.
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i dont have a question for this. look at the photo
Answer:
first is no second is yes third is no fourth is yes
Step-by-step explanation:
I'm not sure btw
3. A piece of wire 16 feet long is strung from
the top of the side of a house to the
ground. The distance from the base of the
house to where the wire meets the ground
is 9 feet, as shown in the picturt.
st
19 lei
(The figure is not drawn to scale.)
How tall, in feet, is the house?
B. 20
C. 175
D. 337
Answer:
its c because its not a b or d and its alround 13 feet
Step-by-step explanation:
Plz help I’ll give brainliest
Answer:
2) 4/5
3) 3
Step-by-step explanation:
2) 4/8-3=4/5
3)5x-4=3x+2
2x=6
x=3
Answer:
2) 4/5
3)x=3
Step-by-step explanation:
1) 4/8-3= 4/5
2) 5x-4=3x+2
bring 3x to left side and bring -4 to right side (remember signs will be reversed that is 3x becomes -3x and -4 will be 4
5x-3x=4+2
2x=6
x=6/2
x=3
put 3 in place of x
5*3-4=3*3+2
15-4=9+2
11=11
f (x) = -3x + 1
g(x) = x2 + 2x – 5
Find g (f (x))
Answer: f
= x = − 3 x + 1 = x = 1 4 g = x 2 + 2 x − 5f( x ) = f ( x )
Step-by-step explanation:
Answer fast plsssssss
Answer:
i cat see it
Step-by-step explanation:
Beth filled 48 jars with paint. If each jar holds 1 pint of paint, how many gallons of paint did Beth use?
The gallons of paint that Beth used is 6 gallons.
How many gallons did Beth use?The first step is to convert pint to gallon:
1 pint = 0.125 gallon
The second step is to multiply 0.125 by 48 : 0.125 x 48 = 6 gallons
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The product of 3x^3-5x² + 3 and 2x² + 5x – 4 in Z7[x] / < x² +1 > is_________a. 2x + 3 b.2x+2 c. 2x d. 2x + 1
The product of 3x³-5x² + 3 and 2x² + 5x – 4 in Z7[x] / < x² +1 > is 2x + 1.
The product of the polynomials 3x³-5x² + 3 and 2x² + 5x – 4 in Z7[x] / < x² +1 > can be determined using polynomial long division.
To perform the division, we divide 3x³-5x² + 3 by x² + 1. The result of this division gives us the quotient and the remainder. In this case, the quotient is 2x and the remainder is 2x + 1.
The quotient represents the coefficient of x in the resulting product, while the remainder represents the constant term. Therefore, the resulting product is d. 2x + 1.
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what is the solution
a: (6,5)
b: (5,6)
c: (1,2)
d: (2,1)
Answer:
(5,6)
Step-by-step explanation:
I'm guessing you mean the coordinates for the point where the lines intersect
Answer:
I believe it is b: (5,6)
Explanation:
The point (5,6) is the only point on the graph that both lines go through.
A circle with a shaded sector has a circumference of 12 pi millimeters. What is the area of the shaded sector? Express the answer in terms of Pi.
Answer:
The answer is C. 8 mm
Step-by-step explanation:
For me, i was able to divide the 80 by 12 and get 6.667 thus i was able to round up and conclude the answer in the answer choices
Answer:
c
Step-by-step explanation:
Let f be the function given by f(x) = 2 cos x +1. What is the approximation for f(1.5) found by using 1 the tangent line to the graph off at x = ? 2 (A) -2 (B) 1 (C) 1-2 (D) 4-2
To approximate the value of f(1.5) using the tangent line to the graph of f at x = 1, we first find the derivative of f(x) to determine the slope of the tangent line at x = 1. The derivative is f'(x) = -2 sin x. Evaluating f'(x) at x = 1, we find the slope to be approximately -1.6829.
Next, we use the point-slope form of a line to find the equation of the tangent line. We have the point (1, f(1)) ≈ (1, 1.5839) and the slope -1.6829. Plugging these values into the point-slope form, we obtain the equation of the tangent line as y ≈ -1.6829x + 3.2668.
Finally, we substitute x = 1.5 into the equation of the tangent line to approximate f(1.5). After calculation, we find that f(1.5) is approximately 0.0009.
Therefore, the approximation for f(1.5) using the tangent line to the graph at x = 1 is approximately 0.0009.
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4. Evaluate the expression for the given value of each variable.
5a2 -7 -b3
when a = -2 and b = -3
please help!
NO LINKS!!!
Answer:
-18
Step-by-step explanation:
5a2-7-b3
[tex]5*a*2-7-b*3[/tex]
[tex]5\cdot \left(-2\right)\cdot \left(2\right)-7-\left(-3\right)\cdot 3[/tex]
Therefore, -18 is your answer.
Hope this helped! :)
A teacher would like to estimate the mean number of steps students take during the school day. To do so, she selects a random sample of 50 students and gives each one a pedometer at the beginning of the school day. They wear the pedometers all day and then return them to her at the end of the school day. From this, she computes the 98% confidence interval for the true mean number of steps students take during the school day to be 8,500 to 10,200. If the teacher had used a 90% confidence interval rather than a 98% confidence interval, what would happen to the width of the interval?
It would increase by 8%.
It would decrease by 8%.
It would increase, but not necessarily by 8%.
It would decrease, but not necessarily by 8%.
Answer:
It would decrease, but not necessarily by 8%
If the teacher had used a 90% confidence interval rather than a 98% confidence interval, the width of the interval would decrease, but not necessarily by 8%.
Option D is the correct answer.
What is a confidence interval?A confidence interval is a range of values that is likely to contain the true value of an unknown parameter, such as a population means or proportion, based on a sample of data from that population.
It is a statistical measure of the degree of uncertainty or precision associated with a statistical estimate.
We have,
The width of a confidence interval is determined by the level of confidence and the sample size.
A higher confidence level requires a wider interval, and a larger sample size typically results in a narrower interval.
Since the sample size is fixed in this case, changing the confidence level will result in a change in the width of the interval.
As the confidence level increases, the width of the interval also increases. This is because a higher confidence level requires a larger margin of error, which is added to the point estimate to create the interval.
Therefore,
If the teacher had used a 90% confidence interval rather than a 98% confidence interval, the width of the interval would decrease, but not necessarily by 8%.
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The null hypothesis is that the laptop produced by HP can run on an average 120 minutes without recharge and the standard deviation is 25 minutes. In a sample of 50 laptops, the sample mean is 130 minutes. Test this hypothesis with the alternative hypothesis that average time is not equal to 120 minutes. What is the p- value?
The p-value for testing the hypothesis that the average runtime of HP laptops is not equal to 120 minutes, based on a sample mean of 130 minutes from a sample of 50 laptops, is approximately 0.0006 (rounded to four decimal places).
To calculate the p-value, we use the t-test. Given the null hypothesis that the average runtime is 120 minutes, the alternative hypothesis is that it is not equal to 120 minutes. We compare the sample mean to the hypothesized population mean using the t-distribution.
Using the formula for the t-statistic:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
t = (130 - 120) / (25 / sqrt(50))
t = 10 / (25 / 7.0711)
t = 2.8284
The degrees of freedom for the t-distribution are (sample size - 1) = (50 - 1) = 49.
Using the t-distribution table or statistical software, we find that the two-tailed p-value for a t-value of 2.8284 with 49 degrees of freedom is approximately 0.0006.
Therefore, the p-value for this hypothesis test is approximately 0.0006.
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If n e z, use modulo to show that -13n2 +5n+23 is odd.
The given expression -13n^{2}+5n+23[/tex] is odd using modulo.
We are given to show that -13n2 + 5n + 23 is odd by using modulo.
Let's try to understand what modulo means:Modulo is a mathematical operation that finds the remainder when one number is divided by another. It is represented by the percentage symbol (%).
Now we can use the concept of modulo to show that -13n2 + 5n + 23 is odd.Note:An integer is odd if it is not even. In other words, odd numbers can't be divided evenly by 2. They leave a remainder of 1 when divided by 2.To show that -13n2 + 5n + 23 is odd, we need to take the given expression modulo 2.$$-13n^2 + 5n + 23 \; \mathrm{mod} \; 2$$We know that every odd integer can be written in the form of 2n + 1 where n is an integer. Now we just have to show that the expression -13n2 + 5n + 23 can be written in the form 2n + 1.$$-13n^2 + 5n + 23 \equiv 1 \; \mathrm{mod} \; 2$$Let's assume k is an integer such that $$-13n^2 + 5n + 23 = 2k + 1$$Rearranging the above expression, we get$$-13n^2 + 5n + 22 = 2k$$Now, we can use the concept of odd and even numbers. We know that an even number can be represented in the form 2n, where n is an integer. Now we just have to show that the expression -13n2 + 5n + 22 can be represented in the form 2n.$$-13n^2 + 5n + 22 = -13n^2 + 13n - 8n + 22$$$$= 13(n-1)(n-2) - 8$$We know that the product of two odd numbers is odd, and the subtraction of an even number from an odd number is odd. Here, (n-1) and (n-2) are consecutive integers, which means one of them is even and the other is odd. Therefore, the product (n-1)(n-2) is even. Hence, the above expression is odd.Since -13n2 + 5n + 23 is equivalent to 1 modulo 2, which means it can be represented in the form of 2n+1. Therefore, -13n2 + 5n + 23 is odd.
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A tree casts a shadow that is 231 cm long. Janet is 170 cm tall and she casts a shadow that is 102 cm long. How tall is the tree?
Answer:
139 cm
Step-by-step explanation:
ect
Use the number line interactive to evaluate the
following expressions.
8 + 3 =
+
16
(-8) + (-5) =
<
(-8) + 2 =
Answer:
1. 11
2. -13
3. -6
Step-by-step explanation:
Hope this helps. Have a great day! (I got this correct on edge :})
Write an equation for two angles A and B that are supplementary.
A. a + b = 180
B. a + b = 100
C. 2ab = 180
D. a + b = 90
Answer:
A. a + b = 180 is the correct option.
Leo flips a paper cup 50 times and records how the cup landed each time. The table below shows the results. RESULTS OF FLIPPING PAPER CUP Outcome Right-side UP Upside Down On its Side Frequency 10 18 22 Based on the results, how many times can he expect the cup to land on its side if it is flipped 1,000 times? 333 440 550 786
Answer:
The answer is 440
Step-by-step explanation:
I just took the test and got that question right
A scientist used 60 grams of sodium nitrate during an experiment. One ounce is approximately equal to 28.3 grams. Which measurement is closest to the number of ounces of sodium nitrate the scientist used?
Answer:
2
Step-by-step explanation:
60 divided by 28.3 which gives u 2.120141342756184
so round it which gives u 2
PLEASE HELP ME!!! I REALLY NEED TO KNOW THIS...
So my math teacher hates me and i did this for a answer:
5x³ + x² + 16 , this is a different question, and i got the points.
But for a answer that was right and she mared wrong i did this:
2x²+4
(I know it is right because my sister and me have the same math. and she got a 10/10 on this)
My sister's answer : 2x^2 + 4.
Are they the same or not?!
Answer:
The exact same
Step-by-step explanation:
^ just means put it up like 3^2=3²
Question is in picture
You earn $10 per hour working as a manager at a grocery store. You are required to work at the grocery store at least 8 hours per week. You also teach music lessons for $15 per hour. You need to earn at least $120 per week, but you do not want to work more than 20 hours per week. a. Write a system of linear inequalities that represents the situation. Use x for hours worked at the grocery store and y for hours teaching music lessons. i need the inequility for total work. please fast
Answer:
without numbers= 10x + 15y = 120
Step-by-step explanation:
grocery store= 10x (x=8 in this case)
music lessons= 15y
What are the solutions to the quadratic equation 4(x + + 2)2 = 36
O x= -11 and x = 7
Ox= -7 and x = 11
O x= -5 and x = 1
Ox= -1 and x = 5
Answer:
x= -1 and 5
hope that helps
The solutions to the quadratic equation [tex]4(x+2)^2=36[/tex] are x = -5 and x = 1.
Given the following data:
[tex]4(x+2)^2=36[/tex]How to solve a quadratic equation.In this exercise, you're required to determine the value of x by solving for the factors (roots) of the given quadratic equation.
In Mathematics, the standard form of a quadratic equation is given by;
[tex]ax^2 + bx + c =0[/tex]
Where:
a = 4.b = 16.c = -20.Dividing both sides by 4, we have:
[tex]4(x+2)^2=36\\\\(x+2)^2=9[/tex]
Taking the square root of both sides, we have:
[tex]x+2=\pm3\\\\x=\pm 3-2[/tex]
x = -5 or 1.
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Kallen was invited to a birthday party for his best friend. He wanted to buy a gift that was $43.87 and wrapping paper that was $2.94. If he earns $8.25 for each lawn he mows and his mom agrees to match all his earnings how many lawns will he need to complete to buy the gift and wrapping
Answer: 6
Step-by-step explanation:
The total amount needed by Kallen for the gift and the wrapping paper will be:
= $43.87 + $2.94
= $46.81
Since he earns $8.25 for each lawn he mows, the number of lawns they he will need to complete to buy the gift and wrapping will be:
= $46.81 / $8.25
= 5.67
= 6 lawns approximately
He'll need to mow 6 lawns
Find the midpoint of the segment with the following endpoints.
(9,6) and (5,2)
Answer:
[tex]M(7 ; 4)[/tex]
Step-by-step explanation:
midpoint -M
[tex]M(\frac{x_{1}+x_{2}}{2} ; \frac{y_{1}+y_{2}}{2} )[/tex]
[tex]M(\frac{9+5}{2} ; \frac{6+2}{2} )[/tex]
[tex]M(7 ; 4)[/tex]
Answer:
(7, 4 )
Step-by-step explanation:
Given endpoints (x₁, y₁ ) and (x₂, y₂ ) then the midpoint is
( [tex]\frac{x_{1}+x_{2} }{2}[/tex] , [tex]\frac{y_{1}+y_{2} }{2}[/tex] )
Here (x₁, y₁ ) = (9, 6) and (x₂, y₂ ) = (5, 2) , then
midpoint = ( [tex]\frac{9+5}{2}[/tex] , [tex]\frac{6+2}{2}[/tex] ) = ([tex]\frac{14}{2}[/tex] , [tex]\frac{8}{2}[/tex] ) = (7, 4 )
Which equation is not equivalent to 2x+10=-8
Answer:
2x+10=-8
x=9
3x=9
x=3
2x=8
x=4
4x=20
x=5
Step-by-step explanation:
Calculate the volume of a cube with side lengths of 2.8 yards.
Answer:
21.952 [tex]yards^{3}[/tex]
Step-by-step explanation:
The volume of a cube is side length cubed.
So [tex]2.8^{3}[/tex] = 21.952
if an equation indicates addition in its presentation, in order to solve for the unknown you must: a. add b. subtract c. multiply d. divide
If an equation indicates addition in its presentation, in order to solve for the unknown, you must perform the inverse operation, which is subtraction.
Equations typically involve an equality between two expressions, with one or more unknown variables.
To isolate the unknown variable and determine its value, you need to perform operations on both sides of the equation to simplify and solve for the variable.
When an equation shows addition, you can undo that operation by subtracting the same value from both sides of the equation. This ensures that the equation remains balanced and maintains equality.
For example, consider the equation:
x + 5 = 10
To solve for x, you need to eliminate the 5 added to x. To do so, you subtract 5 from both sides of the equation:
x + 5 - 5 = 10 - 5
x = 5
By subtracting 5 from both sides, you isolate the variable x and find that its value is 5.
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consider recurrence (and no such that for every non negative integer ny 2 an = 400-1 - 5an-2 +200-3 suppose recurrence (an) has the following initial value a=1 a2=2 in what follows, let a (n) dehote term an for each nonnegative integern write an explicit formula for a(n)
The given recurrence relation is an = 400 - 5an-2 + 200-3, with initial values a0 = 1 and a1 = 2. We need to find an explicit formula for a(n).
To find an explicit formula for a(n), we will first solve the recurrence relation and then generalize the pattern. Let's expand the relation for a few terms to observe the pattern:
a2 = 400 - 5a0 + 200-3 = 400 - 5(1) + 200-3 = 397
a3 = 400 - 5a1 + 200-3 = 400 - 5(2) + 200-3 = 388
a4 = 400 - 5a2 + 200-3 = 400 - 5(397) + 200-3 = -9603
a5 = 400 - 5a3 + 200-3 = 400 - 5(388) + 200-3 = -1260
From the given examples, we can observe that the recurrence relation alternates between positive and negative values. This suggests that the relation might have a periodic pattern. Since the given recurrence is a second-order relation (in terms of n), it is reasonable to assume that the pattern repeats every two terms.
Based on this observation, we can establish the following explicit formula for a(n):
a(n) = (-1)^(n mod 2) * (200n - 5^(n/2) + 200-3)
In this formula, the term (-1)^(n mod 2) alternates between -1 and 1 depending on the parity of n, ensuring the alternating pattern. The other terms account for the linear and exponential components of the recurrence relation.
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Why is it important to know "first term" and "common ratio" when dealing with geometric sequences?
Answer:
Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.
Step-by-step explanation:
Please give brainliest
please help me
Step-by-step explanation:
mark me brilliant