P value of the test statistic = 0.0220
NULL and alternative hypothesis
In statistical hypothesis testing, null and alternative hypotheses are used. A test's null hypothesis always predicts that there will be no effect or relationship between variables, whereas the alternative hypothesis states your research prediction of an effect or relationship.Given that,
p = 0.75
n =200
p^ = 0.68
Testing statistic;
p = 0.75
q = 1 - p
= 1 - 0.75
= 0.25
[tex]Z = \frac{p^ -p }{\sqrt{\frac{p.q}{n} } }[/tex]
= 0.68 - 0.75 / √(0.75 × 0.25)/200
= -2.2861
Z = -2 .29
p value = 2 × (Z ≤ -2.29 )
= 2 × 0.0110
p value = 0.0220
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A survey asked people what alternative transportation modes they use. The results are below:
16% of people answered light rail.
10% of people answered bus.
11% of people answered carpool.
2% of people answered carpool and bus.
6% of people answered light rail and carpool.
3% of people answered light rail and bus.
2% of people answered all three.
Fill in the following Venn Diagram with the percents that each region represents then answer the
questions below.
light rail
IV.
11.
VII.
bus
%
VI.
carpool
III.
What percent of people answered light rail or bus, but not carpool? 15
What percent of people answered carpool and bus?
What percent of people did not answer light rail, carpool, or bus?
y
What percent of people answered exactly one of the options?
What percent of people answered light rail, carpool, or bus?
VIII.
%
%
%
U
%
Fill in the Venn Diagram and answer the questions.
Thank you.
The completed Venn Diagram is given by the image shown at the end of the answer.
The percentages are given as follows:
Light rail or bus, but not carpool: 10%.Carpool and bus: 0%.Did not answer light rail, carpool or bus: 72%.Exactly one: 21%.Light rail, carpool, or bus: 28%.How to fill the Venn Diagram?The Venn diagram starts to be filled from the most restrictive conditions, towards the least restrictive.
2% of people answered all three, hence:
V = 2%.
3% of people answered light rail and bus, hence:
IV + V = 3%
IV = 1%.
6% of people answered light rail and carpool, hence:
II + V = 6%
II = 4%.
2% of people answered carpool and bus, hence:
V + VI = 2%
V = 0%.
11% of people answered carpool, hence:
III + II + V + VI = 11
III + 4 + 2 + 0 = 11
III = 5%.
10% of people answered bus, hence:
IV + V + VI + VII = 10
1 + 2 + 0 + VII = 10
VI = 7%.
16% of people answered light rail, hence:
I + II + IV + V = 16
I + 4 + 2 + 1 = 16
I = 9%.
Then the or percentage is given as follows:
9 + 7 + 5 + 1 + 2 + 4 = 28%.
The percentage that uses other mediums is given as follows:
VIII = 100 - 28 = 72%.
Then the missing percentages are given as follows:
Light rail or bus, but not carpool: I + IV = 9 + 1 = 10%.Carpool and bus: VI = 0%.Exactly one: I + III + VII = 9 + 5 + 7 = 21%.More can be learned about Venn Diagrams at https://brainly.com/question/24713052
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You are refilling your inventory of welcome packets for new customers. For each packet, it takes 5 minutes to print, 2 minutes to organize, and 5 minutes to proofread. The ink in the printer will need to be changed every 5 packets, which takes 3 minutes. You have 3 hours to dedicate to this task. Assume thathe ink in the printer was already changed before beginning and that you have to complete each packet before completing any steps for the next one. How many welcome packets will you be able to fully complete in the time allotted?
Answer:
5x + 2x + 5x + (3/5x) = 180
12x + 0.6x = 180
12.06x = 180
x = 14
14 packetsStep-by-step explanation:
heart and star pls <3 brainliest will be appreciated <3(っ◔◡◔)っ -{ elyna s }-How factorize this : ( + )( – ) + (– + )( – )
Answer:
this very easy to factorize this because (-+)is- (+-)-
(++)+ (--)+ but the sign which is bigger we have to keep that
find a cubic function, in the form below, that has a local maximum value of 4 at -4 and a local minimum value of 0 at 2.f (x)
The cubic function is f(x) = x^3 + 8x^2 - 32x + 32, which has a local maximum value of 4 at x = -4 and a local minimum value of 0 at x = 2.
The cubic function f(x) is of the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants. In this case, a = 1, b = 8, c = -32, and d = 32. The local maximum value of 4 occurs when x = -4 and the local minimum value of 0 occurs when x = 2.To find the equation, we must use the given information to solve for the constants. The function must equal 4 when x = -4, so f(-4) = 4. Plugging in the values for the constants we have 4 = -4^3 + 8(-4)^2 - 32(-4) + 32. Solving this equation, we find that d = 32, so the equation becomes f(-4) = -64 + 256 + 128 + 32 = 4. Similarly, we set f(2) = 0, so 0 = 8 + 16 - 64 + 32. Solving this equation, we find that c = -32, so the equation becomes f(2) = 8 + 16 - 32 + 32 = 0. Finally, we have the equation f(x) = x
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find a cubic function, in the form below, that has a local maximum value of 4 at -4 and a local minimum value of 0 at 2.f (x) = x^3 + 8x^2 - 32x + 32
Find the composite function
Please help me! I'm so confused and awful at math
Step-by-step explanation:
it is really a simple concept : instead of the isolated variable x you grab the whole function of g(x) and put that into the place of any x appearance in f(x) : g(x) becomes the input argument for f(x).
so,
f(g(x)) = sqrt(36/(x + 3)) = 6×sqrt(1/(x + 3))
If g(x) = 5x, what is the value of x when
g(x) = -60?
A) 12
C) -55
B) -300
D) -12
Answer:
-12
Step-by-step explanation:
If g(x) = 5x, then plugging in -60 for g(x) gives -60 = 5x. Dividing both sides of this equation by 5 gives -12 = x. Therefore, the correct answer is -12.
Problem 2 (D.16 Exercise 5, p 511). Forty rats are placed at random in a house having 4 rooms. There is one door between rooms 1 and 2, one door between rooms 1 and 4, one door between rooms 2 and 4, one door between rooms 2 and 3, and two doors between rooms 3 and 4 see the book for a picture). There are no doors between rooms 1 and 3. After each minute, a rat may change rooms, or stay still. All possibilities are equally likely. So, here is the transition matrix: [1/3 1/4 0 1/5 ] 1/3 1/4 1/4 1/5 0 1/4 1/4 2/5 1/3 1/4 1/2 1/5 Predict the long-term distribution of rats. What is the long-term probability that a given marked rat is in room 4? You must use the method of eigenvalues and eigenvectors.
The long-term probability that a given marked rat is in room 4 is 1/3
What is meant by probability?
In statistics, probability is defined as the measure of the likelihood of an event to happen and it measures the certainty of the event.
Here we have given that Forty rats are placed at random in a house having 4 rooms. There is one door between rooms 1 and 2, one door between rooms 1 and 4, one door between rooms 2 and 4, one door between rooms 2 and 3, and two doors between rooms 3 and 4 see the book for a picture). There are no doors between rooms 1 and 3. After each minute, a rat may change rooms, or stay still.
And we need to find the the long-term probability that a given marked rat is in room 4.
By using the eigenvalues method, we know that
Room 4 has the 3 exits that is from Room 1, Room 2 and Room 3.
So, here we have 3 possibilities for the exit so, the rat can choose any one these three ways,
So, the probability can be written as,
=> 1/3
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Dec 09, 1:25:07 PM Given the following exponential function, identify whe growth or decay, and determine the percentage rate of Growth y = 2400(1.78)* % increase Submit Answer
The exponential function is y = 2400(1.78)x
Since 1.78 < 1, the function is an exponential decay function.
The function can also be written as 2400(1 - 0.04)x.
The rate of decrease is 4% (that's 0.04 expressed as a percent)
What is exponential function?The exponential function is a mathematical function denoted by f(x)=\exp or e^{x}. Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics".To learn more about algebras refer to:
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X 1 2 3 4 5 6 7 8
Y 0,5 0,8 1,2 1,9 3 4,8 7,5 11,9
regresi linear
The equation of the linear regression is y = 1.49x- 2.76
How to determine the linear regression equationFrom the question, we have the following parameters that can be used in our computation:
X 1 2 3 4 5 6 7 8
Y 0,5 0,8 1,2 1,9 3 4,8 7,5 11,9
Rewrite as
X 1 2 3 4 5 6 7 8
Y 0.5 0.8 1.2 1.9 3 4.8 7.5 11.9
When the points are entered in a graphing calculator, we have
Sum of X = 36Sum of Y = 31.6Mean X = 4.5Mean Y = 3.95Sum of squares (SSX) = 42Sum of products (SP) = 62.6The regression equation is represented as
Regression Equation
ŷ = bX + a
Substitute the known values in the above equation, so, we have the following representation
b = SP/SSX = 62.6/42 = 1.49048
a = MY - bMX = 3.95 - (1.49*4.5) = -2.75714
So, we have
y = 1.49048X - 2.75714
Approximate
y = 1.49x- 2.76
Hence, the regression equation is y = 1.49x- 2.76
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Complete question
Given that
X 1 2 3 4 5 6 7 8
Y 0,5 0,8 1,2 1,9 3 4,8 7,5 11,9
Calculate the regression linear equation
a complete description of simple harmonic motion must take into account several physical quantities and various mathematical relations among them. this information is needed to solve oscillation problems of this type. the position of a 60 gg oscillating mass is given by x(t)
a complete description of simple harmonic motion must take into account several physical quantities and various mathematical relations then the position of a 60 gg oscillating mass is given by x(t) -20 sin 10t
this information is needed to solve oscillation problems of this type
the position of a 60 gg oscillating mass is given by x(t)
x = 2 cos wt = 2 cos 10t ; w = 10
velocity = dx/dt = -2 x 10 sin 10 t.=- 20 sin 10t
t = .4
velocity = -20 sin 10 x .4 = -20 sin 4 = -20 x -0.7568 = 15.136 cm /s
w = √ k / m = 10 = √ k / .05
k = 15.136 N/m
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A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. Conduct a hypothesis test to determine if the population mean time on death row could likely be 15 years. Is this a test of one mean or proportion? State the null and alternative hypotheses. H_o: H_a: Is this a right-tailed, left-tailed, or two-tailed test? What symbol represents the random variable for this test? In words, define the random variable for this test. Is the population standard deviation known and, if so. what is it? Calculate the following: Which test should be used? State the distribution to use for the hypothesis test. Find the p -value. Al a pre -conceived a = 0.05, what is your Decision: Reason for the decision: Conclusion (write out in a complete sentence):
At the null hypothesis, it is tested if the mean is of 15 years, that is:
[tex]H_0: \mu = 15[/tex]
At the alternative hypothesis, it is tested if the mean is different of 15 years, that is:
[tex]H_1: \mu \neq 15[/tex]
What is the test statistic?The equation that gives the test statistic is defined as follows:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
In which:
[tex]\overline{x}[/tex] is the sample mean.[tex]\mu[/tex] is the value tested at the null hypothesis.s is the standard deviation of the sample.n is the sample size.The values of these parameters in this problem are given as follows:
[tex]\overline{x} = 17.4, \mu = 15, s = 6.3, n = 75[/tex]
Hence the test statistic is of:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
[tex]t = \frac{17.4 - 15}{\frac{6.3}{\sqrt{75}}}[/tex]
t = 3.30.
What is the p-value?Considering a two-tailed test, as we are testing if the mean is different of a value, with t = 3.30 and 75 - 1 = 74 df, the p-value of the test is of:
0.0015.
Since the p-value is less than the significance level of 0.05, the null hypothesis is rejected, meaning that there is enough evidence to conclude that prisoners spend a time different of 15 years in the death row.
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Solve the equation x+5/4=20/16
Answer:
x = 0
Step-by-step explanation:
x + 5/4 = 20/16
x + 5/4 = 5/4
x = 0
What is the position of E on the number line below?
Write your answer as a fraction or a mixed number.
E
The position of E as a mixed number is [tex]2\frac{4}{5}[/tex].
What is number line?
A number line is a diagram of a graduated straight line used to represent real numbers in elementary mathematics. It is assumed that every point on a number line corresponds to a real number, and that every real number corresponds to a point.
Let on a number line the position of E is given.
We have to find E as a mixed number.
As we can see there are 5 parts between 2 and 3.
So each part is 0.2 cm.
The position of E is in fourth part.
So, 4 x 0.2 = 0.8 cm
2 + 0.8 = 2.8 is the position of E.
2.8 can be written as.
28/10 = 2 8/10 = 2 4/5
Hence, the position of E as a mixed number is [tex]2\frac{4}{5}[/tex].
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Match each expression with an equivalent radical form. One of the radical form
options will not have a match.
7/1/2
7
13
3/7/7/
72
3/7/7
1. √72
2. √√37
3. √√3
4. 3/7
5. √7
Answer:
see the attachment
Step-by-step explanation:
You want to match several expressions with fractional exponents to their equivalent radical forms.
Fractional ExponentThe relationship between the two forms is ...
[tex]a^{\frac{m}{n}}=\sqrt[n]{a^m}[/tex]
If n=2, or m=1, these are omitted from the radical form.
The attachment shows the matching values.
Consider the following partially completed two-way ANOVA table. Suppose there are 6 levels of Factor A and 4 levels of Factor B. The number of replications per cell is 6. Use the 0.05 significance level. (Hint: estimate the values from the F table.)1. Complete an ANOVA table. (Round MS and F to 2 decimal places.)
Source SS df MS F
Factor A 75 Factor B 25 Interaction 300 Error 600 Total 1,000Find the critical values to test for equal means. (Round your answer to 2 decimal places.)
Critical values to test for equal means are Factor A , Factor B
and Interaction respectively
The DF value we get after is 143 and the integration value we get is 1.76
What is a value ?
Value in ethics and social sciences refers to the degree of significance of something or an activity, with the goal of deciding the best courses of action to take or the best ways to live, or to define the significance of certain activities.
The value describes the value of each digit in relation to its position in the number. The place value and face value of the digit are multiplied to arrive at the answer. Value equals Place Value minus Face Value. Consider this: If we take 45 into consideration. The tens column is where the fourth digit is located in this instance.
The things that someone values are known as their values. In other terms, values are what a person or a group of people see as "important."
(a)
DF for Factor A = 6 - 1 = 5
DF for Factor B = 4 - 1 = 3
DF for Interaction of Factor A and B = 5*3 = 15
Number of replications per cell = 6
Total no of observations = 6*(N_Factor A)*(N_Factor B) = 6 * 6 * 4 = 144
DF Residuals = 144 - 5 - 3 - 15 - 1 = 120
Total DF = N - 1 = 143
(b) Critical F values
Factor A = F(5,120) = 2.30
Factor B = F(3,120) = 2.69
Interaction = F(15,120) = 1.76
(c) As F statistics for Factor A, Factor B as well as Interaction is >> Fc, Hence we can Reject the Null Hypothesis and warrant the rejection of the claim for equality of means
Hence there is a significant difference in Factor A and there is a significant difference in Factor B.
(d) As the F statistics for Interaction = 4.2 >> Fc (1.76), Hence we can conclude that there is a significant Interaction impact of Factor A and Factor B.
Hence there is a significant difference for Interaction term.
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(2/15x - 5) - (-0.7x + 2)
The expression (2/15x - 5) - (-0.7x + 2) is simplified to 5/12x - 3
What is an algebraic expression?An algebraic expression can be defined as an expression made up of variables, terms, coefficients, factors and constants.
These expressions are also made up of arithmetic operations.
From the information given, we have;
2/15x - 5) - (-0.7x + 2)
expand the bracket
2/15x - 5 + 7/10x - 2
collect like terms
2/15x + 7/10x - 5 + 2
Add the like terms
4x + 21x /30 - 3
Add the numerators
25x/60 - 3
Divide the common terms, we have;
5/12x - 3
Hence, the expression is 5/12x - 3
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You have at most $23 to spend at a market. Apples cost $0.50 each, and pears cost $0.75 each. Let a be numbers of apples and p be numbers of pears. Write an inequality that represents the numbers of apples and pears you can afford
Do not include the dollar sign ($) in your inequality.
Answer:
0.5a + 0.75p ≤ 23
Step-by-step explanation:
Cost of a apples: 0.5a
Cost of p pears: 0.75p
The total cost must be $23 or less.
0.5a + 0.75p ≤ 23
. DIG DEEPER What is Descartes's number?
• It is less than 300.
• It is greater than 200.
• The ones digit is 2 more than the hundreds digit.
• The tens digit is 2 more than the ones digit.
●
Descartes's number is the 46.989, which is less than 47 and higher than 46.98.
Define Descartes's number.A Descartes number is an odd number with the formula n = m p, where m and p are coprime numbers and 2n = (m) (p + 1). P is regarded as a "spoof" prime in this formula. The only known example is the one that was provided. The invention of Cartesian or analytic geometry, which employs algebra to describe geometry, is one of Descartes's most enduring contributions. In equations, Descartes "introduced the convention of denoting unknowns by x, y, and z, and knowns by a, b, and c." A Descartes number is an odd integer that, if one of its composite elements were prime, would have been an odd perfect number.
Given,
Descartes's number is,
46.989, which is less than 47 and higher than 46.98.
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what is function for 3x-5
A rocket is shot off from a launcher. The accompanying table represents the height of the rocket at given times, where x is time, in seconds, and y is height, in feet. Write a quadratic regression equation for this set of data, rounding all coefficients to the nearest hundredth. Using this equation, find the height, to the nearest foot, at a time of 3 seconds.
The height of the rocket in given time using regression equation is 586 feet.
The regression equation is, y = 7.70961x² + 176.30826x - 12.44515
Given,
A rocket is shot off from a launcher. The accompanying table represents the height of the rocket at given times, where x is time, in seconds, and y is height, in feet
That is,
x ; 0.6, 1.3, 1.8, 2.5, 3.1
y ; 108, 218, 285, 368, 436
We have to write a quadratic regression equation for the given set;
The quadratic regression equation is;
y = 7.70961x² + 176.30826x - 12.44515
Now,
We have to find the height at a time of 3 seconds using the equation;
So,
y = 7.70961 x 3² + 176.30826 x 3 - 12.44515
y = 69.38649 + 528.92478 - 12.44515
y = 585.86612
y = 586
Height of the rocket= 586 feet
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In each of 10 and 11, show that the given system has no periodic solutions other than constant solutions. dx dt = 2x 3y xy2, dy dt = y + x3 x2 y
The system does not have a periodic solution in each of 10 and 11.
A solution of the period depends on the independent variable t is called Periodic Solution. For a periodic solution x(t)( in the case of a system, x is a vector), there is a number T≠0 such that
x(t+T)=x(t) for t∈R.
The given equations are:
[tex]\frac{dx}{dt} =-2x-3y-xy^2\\\\\frac{dx}{dt} =y+x^3+x^2y[/tex]
F is the derivative of dx/dt and Gis the derivative of dy/dt:
[tex]F(x,y)=-2x-3y-xy^2\\G(x,y)=y+x^3+x^2y[/tex]
When Fx+Gx has the same sign on the entire domain D, then there are no periodic solutions on the domain D.
Fx+Gx=-2-y^2+1-x^2
=-1-y^2-x^2
<for all real numbers x,y[tex]\neq[/tex]0
Fx+Gx have the same sign for all pairs of real numbers with x[tex]\neq[/tex]0 and y[tex]\neq[/tex]0
Thus, there are no periodic solutions in the system(excluding contact solutions)
Hence proved.
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Ben bought 20 shares of stock at $10.00 per share and sold them for $12.00 per share.
What was his ROI (Return on Investment)?
)) A football team has $630 to spend on hats for their fans. Each hat costs $5. How many
hats can the team buy?
NEED HELP ASAP , THANKS! :)
I don’t quite understand , please fill the rest in
From the first two rows we can see that:
P = 435,n = 1,r = 0.039.Fill in the missing values.
Row 2[tex]435*1.039^3 = 487.91[/tex]Row 3[tex]435*1.039^t[/tex] and [tex]435*1.039^3 = 487.91[/tex]Row 4[tex]435*e^{0.039t}[/tex] and [tex]435*e^{0.039*3}=435*e^{0.117}=488.99[/tex]
In 2012, James Cameron descended to the bottom of Challenger Deep in the Marianas Trench; the deepest point in the ocean. The vessel he rode in was called DeepSea Challenger. Challenger Deep is 35,814 feet deep at its lowest point. The ascent can be modeled by a different proportional relationship y =kx. What is the value of k in this case?
What the value of k means in the proportional relationship y =kx is; the change in depth per second of the ascent.
How to Interpret Proportional Relationships?We are told that;
James Cameron descended to the bottom of Challenger Deep in the Marianas Trench; the deepest point in the ocean.
The vessel he rode in called DeepSea Challenger was 35,814 feet deep at its lowest point.
Now, if the time spent is denoted as x in seconds and the depth is denoted as y, the the relationship will be;
y = -kx
where -k is the constant of proportionality which is the change in depth per second of descent.
Now, if the ascent can be modeled by a different proportional relationship y = kx, then it means that k here will be the change in depth per second of the ascent.
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Please help, confused which expression is equivalent???
Answer:
D
Step-by-step explanation:
m^ (-3 · -4) = m^12
n ^ (-2 · -4) = n^8
m^12 n^8
Compute the scalar surface integral ∬T(z+y)dS where T is the triangle (including its interior) with vertices (0,0,2), (0,1,0)( and (1,0,0)
the scalar surface integral ∬T(z+y)dS where T is the triangle (including its interior) with vertices (0,0,2), (0,1,0)( and (1,0,0) is 1
We can compute the scalar surface integral ∬T(z+y)dS using the parametric form of the surface integral.
Let S be the surface of the triangle. We can parameterize S as follows:
S(u,v) = (u, v, 2-u-v)
where 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1-u.
Then, the scalar surface integral ∬T(z+y)dS is defined as:
∬T(z+y)dS = ∬S(u,v) (2-u-v + v) dudv
= ∫0→1∫0→1-u (2-u-v + v) dudv
= ∫0→1 [2u + (1-u)2 - (1-u)2u - u2(1-u)] du
= ∫0→1 (2u + 1 - 2u2 - u3) du
= [u2 + u - u4/4]0→1
= 1/4 + 1 - ¼
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Problem 5: (8 marks) Find the Taylor series for the following function, centred at the given a.
(a) f(x) = 7 cos (−x), a = 0.
(b) f(x) = x^4 + x^2 + 1, a = −2.
(c) f(x) = 2^x, a = 1.
(d) f(x) = x tan^−1(x^2), a= 0.
Taylor Series for following parts,
a) f(x) = 7 - (7/2)x² + (7/24)x⁶ +......
b)f(x) = 21 - 18(x+2) + 25(x + 2)² - 8(x+3)³ +......
c)f(x) = 2 + 2log(2)(x -1) +(log2)²(x-1)² + (1/3 (log2)³(x-1)³ + --------
d)f(x) = x² + ----
What is Taylor Series ?The Taylor series giving the expansion of the function f(x) in the neighborhood of the point a. If the function is continuous in the neighborhood, all its derivatives exist, and the series converges, then The form f(x) = f(a)+f′(a)/1!(x−a)+(f′′(a)/2!)(x−a)²+⋯+(fⁿ(a)/n! )(x−a)ⁿ --(1) where fⁿ(a) is the nᵗʰ derivative of f(x) evaluated at a.
a) f ( x) = 7 cos(-x) = 7 cos(x) , a = 0
f(a ) = 7 cos(0°) = 7
f'(x) = - 7 sin(x) , f'(a) = 7 sin(0°) = 0
f"(x) = - 7cos(x) , f"(a) = -7 cos(0°) = -7
f"'(x) = 7 sin(x) , f"'(a) = 7 sin(0° ) = 0
....................................
put all the values in above equation (1) we get,
f(x) = 7 + 0/1! (x -0) - (7/2!)(x-0)² )+ 0 +(7/4!) (x-0)⁴-0 + ----------
f(x) = 7 - (7/2)x² + (7/24)x⁶ +......
b) f(x) = x⁴ + x² + 1, a = -2
f(a= -2) = (-2)⁴ +(-2)² +1 = 21
f'(x) = 4x³ + 2x
f'(a= -2) = 4(-2)³ + 2×(-2) = -36
f"(x) = 12x² + 2
f"( a = -2) = 12(-2)² + 2 = 50
.....
put all the values in above equation (1) we get,
f(x) = 21 - (36/2)(x+2) + (50/2!)(x + 2)² - ( 48/3!)(x+3)³ +......
f(x) = 21 - 18(x+2) + 25(x + 2)² - 8(x+3)³ +......
c) f(x) = 2ˣ ; a = 1
f( a= 1) = 2¹ = 2
f'(x) = 2ˣ log(2)
f'(a= 1) = 2 log(2)
f"(x) = 2ˣ ( log(2) )²
f"( a = 1) = 2 ( log(2) )²
................
put all the values in above formula we get,
f(x) = 2 + 2log(2) +( 2/2!)(log2)² + (2/3!)(log2)³
+--------
f(x) = 2 + 2log(2)(x -1) +(log2)²(x-1)² + (1/3(log2)³(x-1)³ + --------
d) f(x) = x tan⁻¹(x²) ; a= 0
f(a=0) = 0
f'(x) = x (1/(1+x⁴) + tan⁻¹(x²)
f'(a = 0) = 0
f"(x) = ((1+x⁴) - x 4x³)/(1+x⁴)² + (1/(1+x⁴)
= (2/(1+x⁴) - 4x⁴/(1+x⁴)²
f"( a = 0) = 2 - 0 = 2
f"'(x) = - 8x³/(1+x⁴)² - 12x³/ (1+x⁴)² + 16x⁷/(1+x⁴)⁴
f"'(a = 0) = 0
..............
f(x) = 0 +0 + 2/2! (x-0)² + 0 +-------
f(x) = x² + ----
Hence, we get all required Taylor Series for
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what is the arithmetic mean of the altitude lengths of an isosceles triangle with two sides of length 13 cm and one side of length 10 cm? Express your answer as a common fraction.
The arithmetic mean of the three altitudes is equal to 10.1533 or 1523/150
What is the arithmetic mean of a set of observations?The Arithmetic Mean (AM) or called average is the ratio of the sum of all observations to the total number of observations and is given by [tex]A= \frac {1}{n} \sum \limits_{i=1}^n a_i[/tex]
A = arithmetic mean
n = number of values
a₁ = data set values
Given here length of two equal sides is 13 cm and the other side is 10cm
let the Δ be ΔABC with AC=AB= 13 & BC=10cm and the altitudes on sides AC, AB, and BC with the midpoints P, Q, and R be AP, BQ, and CR respectively than by Pythagoras theorem we get AP = √13²-5²
= 12
As the altitude of an isosceles triangle bisects the base in this case BC
Then by area of ΔABC is = 1/2×12×10
= 60 cm²
Again BQ is an altitude therefore area ⇒ 60=1/2×BQ×13
⇒BQ=9.2307cm
Similarly, we can find CR = 9.2307 cm
Thus Arithmetic mean of the altitudes is =( 9.2307+9.2307+12)/3
=10.1533 cm
= 1523/150 cm
Hence, the answer as a common fraction is 1523/150
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the president of blitz sales enterprises sells kitchen products through cable television infomercials. he gathered data from the last 15 weeks of sales to determine the relationship between sales and the number of infomercials. infomercials sales ($000s) 20 3.2 15 2.6 25 3.4 10 1.8 18 2.2 18 2.4 15 2.4 12 1.5 22 2.5 15 2.4 25 3.0 16 2.7 12 2.0 20 2.6 25 2.8
The regression equation is Y=1117.84+73.62X for the data as follows:
(20,3200), (15,3600), (25,3400), (10,1800), (18,2200), (18,2400), (15,2400), (12,1500), (22,2500), (15,2400), (25,3000), (16,2700), (12,2000), (20,2600), (25,2800)
where (x,y) denotes infomercials and sales, respectively.
According to regression equation Y=a+bX,
a = ((Σy) x (Σx^2) - (Σx) x (Σxy)) / (n x (Σx^2) - (Σx)^2)
a = ((36500 x 5126)-(268 x 677000)) / ((15 x 5126)-(268 x 268))
a = 1117.84
and, b = (n(Σxy)-(Σx)(Σy)) / (n(Σx^2)-(Σx)^2)
b = ((15 x 677000)-(268 x 365000)) / ((15 x 5126)-(268 x 268))
b = 73.62
Putting the values of a and b in the regression equation, we get the desired equation Y=1117.84+73.62X.
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