Answer:
[tex]|R_n (x)| \leq \dfrac{|x - \dfrac{\pi}{2}|^{n+1}}{(n+1)!}[/tex]
Step-by-step explanation:
From the given question; the objective is to show that :
[tex]\lim_{n \to \infty} R_n (x) = 0[/tex] for all x in the interval of convergence f(x)=cos x, a= π/2
Assuming for the convergence f the taylor's series , f happens to be the derivative on an open interval I with a . Then the Taylor series for the convergence of f , for all x in I , if and only if [tex]\lim_{n \to \infty} R_n (x) = 0[/tex]
where;
[tex]\mathtt{R_n (x) = \dfrac{f^{(n+1)} (c)}{n+1!}(x-a)^{n+1}}[/tex]
is a remainder at x and c happens to be between x and a.
Given that:
a= π/2
Then; the above equation can be written as:
[tex]\mathtt{R_n (x) = \dfrac{f^{(n+1)} (c)}{n+1!}(x-\dfrac{\pi}{2})^{n+1}}[/tex]
so c now happens to be the points between π/2 and x
If we recall; we know that:
[tex]f^{(n+1)}(c) = \pm \ sin \ c \ or \ cos \ c[/tex] (as a result of the value of n)
However, it is true that for all cases that [tex]|f ^{(n+1)} \ (c) | \leq 1[/tex]
Hence, the remainder terms is :
[tex]|R_n (x)| = | \dfrac{f^{(n+1)}(c)}{(n+1!)}(x-\dfrac{\pi}{2})^{n+1}| \leq \dfrac{|x - \dfrac{\pi}{2}|^{n+1}}{(n+1)!}[/tex]
If [tex]\lim_{n \to \infty} R_n (x) = 0[/tex] for all x and x is fixed, Then
[tex]|R_n (x)| \leq \dfrac{|x - \dfrac{\pi}{2}|^{n+1}}{(n+1)!}[/tex]
Allied Corporation is trying to sell its new machines to Ajax. Allied claims that the machine will pay for itself since the time it takes to produce the product using the new machine is significantly less than the production time using the old machine. To test the claim, independent random samples were taken from both machines. You are given the following results.
New Machine Old Machine
Sample Mean 25 23
Sample Variance 27 7.56
Sample Size 45 36
As the statistical advisor to Ajax, would you recommend purchasing Allied's machine? Explain.
Answer:
z(s) is in the acceptance region. We accept H₀ we did not find a significantly difference in the performance of the two machines therefore we suggest not to buy a new machine
Step-by-step explanation:
We must evaluate the differences of the means of the two machines, to do so, we will assume a CI of 95%, and as the interest is to find out if the new machine has better performance ( machine has a bigger efficiency or the new machine produces more units per unit of time than the old one) the test will be a one tail-test (to the left).
New machine
Sample mean x₁ = 25
Sample variance s₁ = 27
Sample size n₁ = 45
Old machine
Sample mean x₂ = 23
Sample variance s₂ = 7,56
Sample size n₂ = 36
Test Hypothesis:
Null hypothesis H₀ x₂ - x₁ = d = 0
Alternative hypothesis Hₐ x₂ - x₁ < 0
CI = 90 % ⇒ α = 10 % α = 0,1 z(c) = - 1,28
To calculate z(s)
z(s) = ( x₂ - x₁ ) / √s₁² / n₁ + s₂² / n₂
s₁ = 27 ⇒ s₁² = 729
n₁ = 45 ⇒ s₁² / n₁ = 16,2
s₂ = 7,56 ⇒ s₂² = 57,15
n₂ = 36 ⇒ s₂² / n₂ = 1,5876
√s₁² / n₁ + s₂² / n₂ = √ 16,2 + 1.5876 = 4,2175
z(s) = (23 - 25 )/4,2175
z(s) = - 0,4742
Comparing z(s) and z(c)
|z(s)| < | z(c)|
z(s) is in the acceptance region. We accept H₀ we did not find a significantly difference in the performance of the two machines therefore we suggest not to buy a new machine
The very hight dispersion of values s₁ = 27 is evidence of frecuent values quite far from the mean
1. The mean performance score on a physical fitness test for Division I student athletes is 947 with a population standard deviation of 205. Select a random sample of 64 of these students. Hint: we have a sample so use the standard error. What is the probability the mean of the sample is below 900
Answer:
0.033316
Step-by-step explanation:
We use the z score formula to solve for this question.
Since we are given the number of samples in the question, our z score formula is given as:
z = (x-μ)/ S.E
where x is the raw score
μ is the sample mean
S.E is the Standard error.
x is the raw score = 900
μ is the sample mean = Population mean = 947
Standard error =
This is calculated as Population standard deviation/ √No of samples
= 205/√64.
= 205/8
= 25.625
We proceed to calculate the z score
z = (x-μ)/ S.E
z = 900 - 947/25.625
= -1.83415
Using the z score table for normal distribution,
P(x≤ z) = P(z ≤ -1.83) = P(x ≤ 900)
P(x<900) = 0.033316
Therefore, the probability the mean of the sample is below 900 is 0.033316
The letters x and y represent rectangular coordinates. Write the given equation using polar coordinates (r,θ) . Select the correct equation in polar coordinates below.
x2+y2−4x=0
a. r=4 sinθ
b. r=4 cosθ
c. r cos2θ=4 sinθ
d. r sin2θ=4 cosθ
Answer:
B. r = 4cosθStep-by-step explanation:
Given the expression in rectangular coordinate as x²+y²−4x=0, in order to write the given expression in polar coordinates, we need to write the value of x and y as a function of (r, θ).
x = rcosθ and y = rsinθ.
Substituting the value of x and y in their polar form into the given expression we have;
x²+y²−4x=0
( rcosθ)²+( rsinθ)²-4( rcosθ) = 0
Expand the expressions in parenthesis
r²cos²θ+r²sin²θ-4rcosθ = 0
r²(cos²θ+sin²θ)-4rcosθ = 0
From trigonometry identity, cos²θ+sin²θ =1
The resulting equation becomes;
r²(1)-4rcosθ = 0
r²-4rcosθ = 0
Add 4rcosθ to both sides of the equation
r²-4rcosθ+4rcosθ = 0+4rcosθ
r² = 4rcosθ
Dividing both sides by r
r²/r = 4rcosθ/r
r = 4cosθ
Hence the correct equation in polar coordinates is r = 4cosθ
simplify -3(2g - 6) +4g
-3-2g+6+4g
3+2g
hope it helps
Answer:
-2g + 18
Step-by-step explanation:
-3(2g - 6) + 4g
First we use distributive property.
-3 × 2g = -6g
-3 × -6 = 18
now we have
-6g + 18 + 4g
Now we combine the like terms
-6g + 4g = -2g
Finally we have
-2g + 18
and they are not like terms so we leave them and the equation is solved.
omplete the following multiplication problems.
a. 0.34 × 6
b. 0.11 × 4
c. 17 × 0.07
d. 28 × 0.003
e. 3.8 × 5
f. 5.931 × 7
g. 14.07 × 13
h. 3.005 × 32
i. 0.8 × 0.3
j. 0.45 × 0.05
k. 0.09 × 0.02
l. 0.074 × 0.08
m. 2.3 × 0.9
n. 7.25 × 0.3
o. 4.53 × .003
p. 53.67 × 0.056
q. 1.1 × 3.7
r. 3.76 × 18.9
s. 4.57 × 6.1
t. 24.13 × 1.48
If 2( a^2 +b^2 ) = ( a+b)^2 , then
a. a+b =0
b. a =b
c. 2a =b
d. ab =0
Answer:
the answer is a=b
Step-by-step explanation:
The grade appeal process at a university requires that a jury be structured by selecting individuals randomly from a pool of students and faculty. (a) What is the probability of selecting a jury of all students? (b) What is the probability of selecting a jury of all faculty? (c) What is the probability of selecting a jury of students and faculty
Correct question is ;
The grade appeal process at a university requires that a jury be structured by selecting eight individuals randomly from a pool of nine students and eleven faculty. (a) What is the probability of selecting a jury of all students? (b) What is the probability of selecting a jury of all faculty? (c) What is the probability of selecting a jury of six students and two faculty?
Answer:
A) 7.144 × 10^(-5)
B) 0.00131
C) 0.0367
Step-by-step explanation:
We are given;
Number of students = 9
Number of faculty members = 11
A) Now, the number of ways we can select eight students from 9 =
C(9, 8) = 9!/(8! × 1!) = 9
Also, number of ways of selecting 8 individuals out of the total of 20 = C(20,8) = 20!/(8! × 12!) = 125970
Thus, probability of selecting a jury of all students = 9/125970 = 7.144 × 10^(-5)
B) P(selecting a jury of all faculty) = (number of ways to choose 8 faculty out of 11 faculty)/(Total number of ways to choose 8 individuals out of 20 individuals) = [C(11,8)]/[C(20,8)] = (11!/(8! × 3!))/125970 = 0.00131
C) P(selecting a jury of six students and two faculty) = ((number of ways to choose 6 students out of 9 students) × (number of ways to choose 2 faculty out of 11 faculty))/(Total number of ways to choose 8 individuals out of 20 individuals) = [(C(9,6) × C(11,2)]/125970
This gives;
(84 × 55)/125970 = 0.0367
Choose the inequality that represents the following graph.
Answer:
option a
Step-by-step explanation:
give person above brainliest :)
The radius of a circle measures 5 inches A central angle of the circle measuring 12 radians cuts off a sector
What is the area of the sector?
Enter your answer as a simplified fraction in the box
area =
inches squared
Answer:
25/4 square inches
Step-by-step explanation:
The area of a sector of a circle is given by the formula ...
A = (1/2)r²θ
where r is the radius and θ is the central angle in radians.
For your sector, the area is ...
A = (1/2)(5 in)²(1/2) = 25/4 in²
If A and B are independent events with P( A) = 0.35 and P( B) = 0.55, then P( A| B) is:_________.
a. .19
b. 1.57
c. .64
d. .91
Answer:
P( A| B)= 0.35. None of the options are correctStep-by-step explanation:
Two events A and B are said to be independent if the occurrence of one of the events does not affect the other occurring. For example, the event of tossing two coins is an independent event since they occur simultaneously. Two events are therefore independent if the following are true.
P(A|B) = P(A)
P(B|A) = P(B)
P(A and B) = P(A)P(B)
If A and B are independent events with P( A) = 0.35 and P( B) = 0.55,
then P( A| B) is a probability of A occurring provided that B has occurred. This is known as conditional probability for an independent event.
From the condition above for independent events, P(A|B) = P(A) and since P(A) = 0.35, hence P(A|B) =0.35
Does artistic ability determine which type of operating system a person prefers? Suppose that a market research company randomly selected n=259 adults who used a desktop or laptop outside of the workplace (tablets and smartphones were excluded).
Answer:
Your question lacks some parts attached below is the complete question
Answer : 2.66
Step-by-step explanation:
The expected number ( E ) can be calculated using the formula below
[tex]E = \frac{row total * column total }{gross total}[/tex]
since we are computing the number of subjects that would prefer Linux operating system and are also rated as exceptional
The row total to be used = 53 ( row total of exceptional )
The column total to be used = 13 ( column total of Linux )
The gross total to be used = summation of row total of both exceptional and no-exceptional = 259
BACK TO THE EQUATION
E = [tex]\frac{53*13}{259}[/tex] = 689 / 259
E = 2.6602 ≈ 2.66
I need help please, m bda =
And m bca =
Step-by-step explanation:
Exterior angle BOA = 250°
Interior angle BOA = 360°- 250° = 110°
Now,
(A) BDA = interior angle BOA / 2 = 55°( Property of circles)
(B) From the figure, we observe that AOBC is a cyclic quadrilateral (i.e. sum of opposite angles is 180°).
Therefore, BCA + BOA = 180°
BCA = 180° - 110° = 70°
Which phrase describes the algebraic expression 8f+7?
the product of 8 and 7 more than a number
the quotient of 8 and 7
8 times the sum of a number and 7
8 times a number plus 7
Answer:
8 times a number plus 7
Step-by-step explanation:
Let the number be f.
Simply, 8 f+7 is the expression.
Thank you!
y - 4= -2(x + 3)
Complete the missing value in
the solution to the equation.
(-3, _ )
Answer:
4
Step-by-step explanation:
i distributed the -2 to what's in the parentheses. that equal 0. I then moved the 4 to the zero so that it becomes positive. I just assumed that you were ask for Y
Step-by-step explanation:
y-4=-2(x+3)....eq(1)
y- 4= -2x-6
y=-2x-2...eq(2)
subtituting equation 2 in equation 1
(-2x-2)-4=-2x-6
-2x-6=-2x-6
=0
in a village in hawaii, about 80% of the residents are of hawaiian ancestry. Let n be the number of people you meet until you encounter the 1st person of hawaiian ancestry in the village. write a formula for the probability distribution
Answer:
The formula for the probability distribution is:
P(X = n) = q^(n - 1)p
= [0.2^(n - 1)]0.8
Step-by-step explanation:
This is a geometric probability distribution.
The probability of success p = 80% = 0.8
The probability of failure is q = 1 - p = 0.2
The formula is:
P(X = n) = q^(n - 1)p
= [0.2^(n - 1)]0.8
Please answer this correctly without making mistakes
Answer:
7/10 mi
Step-by-step explanation:
The total distance is 3 miles = 30/10 miles.
The other distances added gives 7/10+7/10+9/10 = 23/10
Therefore the last hop from Kingwood to Silvergrove is 30/10 - 23/10 = 7/10
how many solutions are there to this non-linear systems/graph a. one solution,b.two solutions,c.no solutions
Find a polar equation r for the conic with its focus at the pole and the given eccentricity and directrix. (For convenience, the equation for the directrix is given in rectangular form.)
Conic: Parabola Eccentricity: e = 1 Directrix: y = 4
Answer:
The equation is [tex]r = \frac{4 }{ 1 + cos (\theta )}[/tex]
Step-by-step explanation:
From the equation we are told that
The Eccentricity: e = 1
The Directrix is y = 4
Generally the polar equation for e = 1 and y = + c is mathematically represented as
[tex]r = \frac{e * c }{ 1 + ecos (\theta )}[/tex]
So
[tex]r = \frac{1 * 4 }{ 1 + 1 * cos (\theta )}[/tex]
[tex]r = \frac{4 }{ 1 + cos (\theta )}[/tex]
Algebra Review
Write an algebraic expression for each verbal expression.
1. the sum of one-third of a number and 27
2. the product of a number squared and 4
3. Write a verbal expression for 5n^3 +9.
Answer:
Step-by-step explanation:
1. The sum of one-third of a number and 27
= [tex]\frac{1}{3}\times x +27\\= 1/3x +27[/tex]
2. The product of a number squared and 4
[tex]Let\:the\:unknown\: number\: be \:x\\\\x^2\times4\\\\= 4x^2[/tex]
3.Write a verbal expression for 5n^3 +9.
The sum of the product and of 5 and a cubed number and 9
Here is some information about the goals scored in some hockey games. Each game has four quarters. Please give the answer asap with full explanation and working out.
Answer:
8 home games and 10 away games
Step-by-step explanation:
Total home goals
= 8+5+9+8
= 30
Number of home games
= 30/3.75
= 8
Total away game goals
= 7+8+4+5
= 24
Number of away games
= 24/2.4
= 10
Answer:
i think it is 8 home and 10 away matches
Step-by-step explanation:
Select the best with the least expensive corn per ounce The choices are in the image
Answer:
B
Step-by-step explanation:
Option A:
1.50÷18≈0.0833
3.00÷36≈0.0833
4.50÷54≈0.0833
Option B:
0.75÷15=0.05
Option C:
2.20÷15=0.146
How hot does it get in Death Valley? The following data are taken from a study conducted by the National Park System, of which Death Valley is a unit. The ground temperatures (°F) were taken from May to November in the vicinity of Furnace Creek. Compute the mean, median, and mode for these ground temperatures. (Enter your answers to one decimal place.) 147 153 170 172 185 181 182 185 181 170 181 167 153 145
Answer:
Mean: 169.4
Median: 171
Mode: 181
Step-by-step explanation:
I first sorted the numbers by value, least to greatest.
145 147 153 153 167 170 170 172 181 181 181 182 185 185
We can see that 181 occurs the most, 3 times, so it's the mode.
The median of this set will be the middle number(s).
When we take away 6 numbers from both sides we are left with 170 and 172, and the mean of these two numbers is 171. So the median is 171.
We can add all the numbers and divide by 14 to get the mean.
[tex]147+153+170+172+185+181+182+185+181+170+181+167+153+145=2372\\\\2372\div14\approx169.4[/tex]
Hope this helped!
Fill in the blanks and explain the pattern
0,1,1,2,3,5,__,__,21,34,55
Answer:
8,13
Step-by-step explanation:
Look at the pattern :
0,1,1,2,3,5,...,...,21,34,55.
As you see the number in the pattern was made by the sum of 2 numbers behind it. Then, the blanks must be filled by :
3 + 5 = 88 + 5 = 13So, the blanks must be filled by 8 and 13
Answer:
In the two blanks would be 8, 13.
The pattern is practically the Fibonacci Code.
Step-by-step explanation:
The Fibonacci Code is a mathematical sequencing in which you start with two numbers and add them together to make the third number, then you add the third number and the second number together. Practically you keep adding each new sum and the number before it in the sequence to find the next new sum.
After 55 in this pattern, the pattern would go 89, 144, 233, 377, 610, 987,...
A researcher at the University of Washington medical school believes that energy drink consumption may increase heart rate. Suppose it is known that heart rate (in beats per minute) is normally distributed with an average of 70 bpm for adults. A random sample of 25 adults was selected and it was found that their average heartbeat was 73 bpm after energy drink consumption, with a standard deviation of 7 bpm. In order to test belief at the 10% significance level, determine P-value for the test.
Answer:
Step-by-step explanation:
Given that:
mean μ = 70
sample size = 25
sample mean = 73
standard deviation = 7
level of significance = 0.10
The null hypothesis and the alternative hypothesis can be computed as follows:
[tex]\mathtt{H_o : \mu = 70} \\ \\ \mathtt{H_1 : \mu > 70 }[/tex]
The z score for this statistics can be calculated by using the formula:
[tex]z = \dfrac{X- \mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
[tex]z = \dfrac{73- 70}{\dfrac{7}{\sqrt{25}}}[/tex]
[tex]z = \dfrac{3}{\dfrac{7}{5}}[/tex]
[tex]z = \dfrac{3 \times 5}{{7}{}}[/tex]
z = 2.143
At level of significance of 0.10
degree of freedom = n -1
degree of freedom = 25 - 1
degree of freedom = 24
The p - value from the z score at level of significance of 0.10 and degree of freedom of 24 is:
P - value = 1 - (Z < 2.143)
P - value = 1 - 0.9839
P - value = 0.0161
Decision Rule: since P value is lesser than the level of significance, we reject the null hypothesis.
Conclusion: We conclude that energy drink consumption increases heart rate.
15+9=? (5+3) What number is missing from the expression?
Answer:
[tex] \boxed{ \boxed{ \bold{ \mathsf{3}}}}[/tex]Step-by-step explanation:
Let the missing number be 'x'
⇒[tex] \mathsf{15 + 9 = x(5 + 3)}[/tex]
Distribute x through the parentheses
⇒[tex] \mathsf{15 + 9 = 5x + 3x}[/tex]
Swap the sides of the equation
⇒[tex] \mathsf{5x + 3x = 15 + 9}[/tex]
Add the numbers
⇒[tex] \mathsf{5x + 3x = 24}[/tex]
Collect like terms
⇒[tex] \mathsf{8x = 24}[/tex]
Divide both sides of the equation by 8
⇒[tex] \mathsf{ \frac{8x}{8} = \frac{24}{8} }[/tex]
Calculate
⇒[tex] \mathsf{x = 3}[/tex]
Hope I helped!
Best regards!
Find the minimum sample size n needed to estimate for the given values of c, , and E. c, , and E Assume that a preliminary sample has at least 30 members.
Answer:
hello your question is incomplete below is the complete question
Find the minimum sample size n needed to estimate μ For the given values of c, σ, and E. c=0.98, σ=6.5, and E=22 Assume that a preliminary sample has at least 30 members.
Answer : 48
Step-by-step explanation:
Given data:
E = 2.2,
std ( σ ) = 6.5
c ( level of confidence ) = 0.98
To find the minimum sample size
we have to first obtain the value of [tex]Z_{a/2}[/tex]
note : a can be found using this relation :
( 1 - a ) = 0.98 ----- equation 1
a = 1 - 0.98 = 0.02
hence: a/2 = 0.01
This means that P( Z ≤ z ) = 0.99 the value of z can be found using the table of standard normal distribution. from the table the value of z = 2.33
P( Z ≤ 2.33 ) = 0.99
To obtain the sample size n
[tex]n = (\frac{std*z}{E} )^{2}[/tex]
n = [tex](\frac{6.5*2.33}{2.2} )^2[/tex] = (6.88409)^2
Therefore n ≈ 48
x power 8 + x power 4 + 1
factorize
Answer:
[tex]1(x {}^{8} + x {}^{4} + 1)[/tex]
Step-by-step explanation:
[tex]x {}^{8} + {x}^{4} + 1 =1( x {}^{8} + x {}^{2} + 1)[/tex]
Hope this helps ;) ❤❤❤
Let me know if there is an error in my answer.
pls help:Find all the missing elements:
Answer:
B = 48.7° , C = 61.3° , b = 12Step-by-step explanation:
In order to find B we must first angle C
To find angle C we use the sine rule
That's
[tex] \frac{ |a| }{ \sin(A) } = \frac{ |c| }{ \sin(C) } [/tex]
From the question
a = 15
A = 70°
c = 14
So we have
[tex] \frac{15}{ \sin(70) } = \frac{14}{ \sin(C) } [/tex]
[tex] \sin(C) = \frac{14 \sin(7 0 ) }{15} [/tex]
[tex]C = \sin^{ - 1} ( \frac{14 \sin(70) }{15} ) [/tex]
C = 61.288
C = 61.3° to the nearest tenthSince we've found C we can use it to find B.
Angles in a triangle add up to 180°
To find B add A and C and subtract it from 180°
That's
A + B + C = 180
B = 180 - A - C
B = 180 - 70 - 61.3
B = 48.7° to the nearest tenthTo find b we can use the sine rule
That's
[tex] \frac{ |a| }{ \sin(A) } = \frac{ |a| }{ \sin(B) } [/tex]
[tex] \frac{15}{ \sin(70) } = \frac{ |b| }{ \sin(48.7) } [/tex]
[tex] |b| = \frac{15 \sin(48.7) }{ \sin(70) } [/tex]
b = 11.9921
b = 12.0 to the nearest tenthHope this helps you
Fill in the blanks and explain the pattern.
1, 4, 9, 16, 25,__,__,__,__,100
Fill in the blanks and explain the pattern.
1/2, 1/6, 1/18, 1/54,__,__,__
Answer:
1. 36, 49, 64, 81
The pattern is going up by squares.
2. 1/162, 1/486, 1/1458
The pattern is 1/3n.
Step-by-step explanation:
1. Before the blank spaces started we were at 5^2, or 25. Now, we have to find 6^2, 7^2, 8^2, and 9^2. That would be 36, 49, 64, and 81. It goes up by squares.
2. As the pattern continues, each number in the sequence is multiplied by 1/3 create the next number in the sequence.
What is the value of the product (3 – 2i)(3 + 2i)?
Answer:
13
Step-by-step explanation:
(3 - 2i)(3 + 2i)
Expand
(9 + 6i - 6i - 4i^2)
Add
(9 - 4i^2)
Convert i^2
i^2 = ([tex]\sqrt{-1}[/tex])^2 = -1
(9 - 4(-1))
Add
(9 + 4)
= 13
Answer:
13.
Step-by-step explanation:
(3 - 2i)(3 + 2i)
= (3 * 3) + (-2i * 3) + (2i * 3) + (-2i * 2i)
= 9 - 6i + 6i - 4[tex]\sqrt{-1} ^{2}[/tex]
= 9 - 4(-1)
= 9 + 4
= 13
Hope this helps!
