Answer:
2x^4−11x^3+7x^2+5x−3
Step-by-step explanation:
The ^ means exponent
It was reported that 23% of U.S. adult cellphone owners called a friend for advice about a purchase while in a store. If a sample of 15 U.S adult cellphone owners is selected, what is the probability that 7 called a friend for advice about a purchase while in a store
Answer:
[tex] P(X=7)[/tex]
And using the probability mass function we got:
[tex]P(X=7)=(15C7)(0.23)^7 (1-0.23)^{15-7}=0.0271[/tex]
Step-by-step explanation:
Let X the random variable of interest, on this case we now that:
[tex]X \sim Binom(n=15, p=0.23)[/tex]
The probability mass function for the Binomial distribution is given as:
[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]
Where (nCx) means combinatory and it's given by this formula:
[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]
And we want to find the following probability:
[tex] P(X=7)[/tex]
And using the probability mass function we got:
[tex]P(X=7)=(15C7)(0.23)^7 (1-0.23)^{15-7}=0.0271[/tex]
Identifico el nombre de la propiedad a la que hacen referencia las siguientes expresiones:
Hacen falta las expresiones para poder responder a tu pregunta, estuve investigando y adjuntaré una imagen que hace referencia a tus preguntas, espero no equivocarme.
Si este es el caso, son 9 expresiones y el nombre de cada propiedad es:
1. Inverso aditivo (Sumar un número por su opuesto el resultado es 0)
2. Ley conmutativa (El orden de los factores no altera el producto)
3. Ley asociativa (Agrupar los términos sin alterar el resultado)
4. Ley de la identidad, (Sumar un número con 0 se obtiene el mismo número)
5. Ley distributiva (La misma respuesta cuando multiplicas un conjunto de números por otro número que cuando se hace cada multiplicación por separado)
6. Ley distributiva
7. Ley distributiva
8. Ley asociativa
9. Ley conmutativa
Professor Sanchez has been teaching Principles of Economics for over 25 years. He uses the following scale for grading. Grade Numerical Score Probability A 4 0.090 B 3 0.240 C 2 0.360 D 1 0.165 F 0 0.145 a. Convert the above probability distribution to a cumulative probability distribution. (Round your answers to 3 decimal places.)
The question is incomplete! Complete question along with answer and step by step explanation is provided below.
Question:
Professor Sanchez has been teaching Principles of Economics for over 25 years. He uses the following scale for grading. Grade Numerical Score Probability A 4 0.090 B 3 0.240 C 2 0.360 D 1 0.165 F 0 0.145
a. Convert the above probability distribution to a cumulative probability distribution. (Round your answers to 3 decimal places.)
b. What is the probability of earning at least a B in Professor Sanchez’s course? (Round your answer to 3 decimal places.)
c. What is the probability of passing Professor Sanchez’s course? (Round your answer to 3 decimal places.)
Answer:
a. Cumulative Probability Distribution
Grade P(X ≤ x)
F 0.145
D 0.310
C 0.670
B 0.910
A 1
b. P(at least B) = 0.330
c. P(pass) = 0.855
Step-by-step explanation:
Professor Sanchez has been teaching Principles of Economics for over 25 years.
He uses the following scale for grading.
Grade Numerical Score Probability
A 4 0.090
B 3 0.240
C 2 0.360
D 1 0.165
F 0 0.145
a. Convert the above probability distribution to a cumulative probability distribution. (Round your answers to 3 decimal places.)
The cumulative probability distribution is given by
Grade = F
P(X ≤ x) = 0.145
Grade = D
P(X ≤ x) = 0.145 + 0.165 = 0.310
Grade = C
P(X ≤ x) = 0.145 + 0.165 + 0.360 = 0.670
Grade = B
P(X ≤ x) = 0.145 + 0.165 + 0.360 + 0.240 = 0.910
Grade = A
P(X ≤ x) = 0.145 + 0.165 + 0.360 + 0.240 + 0.090 = 1
Cumulative Probability Distribution
Grade P(X ≤ x)
F 0.145
D 0.310
C 0.670
B 0.910
A 1
b. What is the probability of earning at least a B in Professor Sanchez’s course? (Round your answer to 3 decimal places.)
At least B means equal to B or greater than B grade.
P(at least B) = P(B) + P(A)
P(at least B) = 0.240 + 0.090
P(at least B) = 0.330
c. What is the probability of passing Professor Sanchez’s course? (Round your answer to 3 decimal places.)
Passing the course means getting a grade of A, B, C or D
P(pass) = P(A) + P(B) + P(C) + P(D)
P(pass) = 0.090 + 0.240 + 0.360 + 0.165
P(pass) = 0.855
Alternatively,
P(pass) = 1 - P(F)
P(pass) = 1 - 0.145
P(pass) = 0.855
Determine what type of study is described. Explain. Researchers wanted to determine whether there was an association between high blood pressure and the suppression of emotions. The researchers looked at 1800 adults enrolled in a Health Initiative Observational Study. Each person was interviewed and asked about their response to emotions. In particular they were asked whether their tendency was to express or to hold in anger and other emotions. The degree of suppression of emotions was rated on a scale of 1 to 10. Each person's blood pressure was also measured. The researchers analyzed the results to determine whether there was an association between high blood pressure and the suppression of emotions.
Answer:
Experimental Study
Step-by-step explanation:
In an experimental study, the researchers involve always produce and intervention (in this case they were asked whether their tendency was to express or to hold in anger and other emotions. The degree of suppression of emotions was rated on a scale of 1 to 10) and study the effects taking measurements.
These studies are usually randomized ie subjects are group by chance.
As opposed to observation studies, where the researchers only measures what was observed, seen or hear without any intervention on their parts.
A map's scale is 1 inch : 3.5 miles.
If the distance on the map is
8 inches, then the actual distance
in real life is __miles.
Answer:
28 miles
Step-by-step explanation:
to fin the actual distance you must multiply the didtance on the map by the map scale
3.5*8=28
Evaluate. Write your answer as a fraction or whole number without exponents. 7^–1 =
Answer:
1/7 = 0.142857... repeating
Step-by-step explanation:
7^(-1) = 1/(7^1) =1/7 = 0.142857... repeating
Answer:
[tex] \frac{1}{7} [/tex]Solution,
[tex] {7}^{ - 1} \\ = \frac{1}{ {7}^{1} } \\ = \frac{1}{7} [/tex]
Laws of indices:Law of zero index:[tex] {x}^{0} = 1[/tex]
Product law of indices:[tex] {x}^{m} \times {x}^{n} = {x}^{m + n} [/tex]
( powers are added in multiplication of same base)
Power law of indices:[tex] {( {x}^{m} )}^{n} = {x}^{m \times n} [/tex]
law of negative index:[tex] {x}^{ - m} = \frac{1}{ {x}^{m} } [/tex]
Root law of indices:[tex] {x}^{ \frac{p}{q} } = \sqrt[q]{ {x}^{p} } [/tex]
[tex]( \frac{x}{y} ) ^{n} = \frac{ {x}^{n} }{ {y}^{n} } [/tex] [tex] {(xy)}^{m} = {x}^{m} {y}^{m} [/tex][tex] \sqrt[n]{x} = x \frac{1}{n} [/tex]Hope this helps ....
Good luck on your assignment...
Help me please! I need an answer!
Answer: [tex]\bold{\dfrac{b_1}{b_2}=\dfrac{3}{2}}[/tex]
Step-by-step explanation:
Inversely proportional means a x b = k --> b = k/a
Given that a₁ = 2 --> b₁ = k/2
Given that a₂ = 3 --> b₂ = k/3
[tex]\dfrac{b_1}{b_2}=\dfrac{\frac{k}{2}}{\frac{k}{3}}=\large\boxed{\dfrac{3}{2}}[/tex]
A math teacher asks Nico and Katya to solve the following word problem. A car travels 98 miles in 1.7 hours on a freeway where the speed limit is 55 mph. Was the car speeding? Nico and Katya both agree that they should use their calculators to divide the miles by the hours to find the speed of the car, and then compare the answer to 55 mph. However, Nico says it's okay to round what his calculator says to the nearest whole number. Katya says that because the calculator displays eight numbers after the decimal point, they shouldn't round. She says they should write down exactly what the calculator shows. Do you agree with Nico or with Katya? In a short paragraph, explain who you agree with and provide the reasons why.
Answer:
- Was the car speeding?
Yes, the car was speeding as its current speed of 57.65 mph was more than the speed limit of that freeway.
- Do you agree with Nico or with Katya?
I agree somewhat with both Nico and Katya, but, I agree more with Nico.
- Explain your reasoning.
Like I said, I agree more with Nico's method of rounding the speed to the nearest whole number. This is because in this question, the standard speed we want to compare the calculated speed with is given as a whole number. Hence, it is more proper to estimate the calculated speed to its nearest whole number too.
Step-by-step explanation:
Speed during a travel is given as distance travelled divided by time taken
Speed = (Distance/time)
Distance = 98 miles
Time = 1.7 hours
Speed = (98/1.7) = 57.6470588235 = 57.65 mph = 58 mph
- Was the car speeding?
The speed limit for the road is 55 mph and the current speed of the car = 57.65 mph
Since 57.65 > 55
The car was overspeeding.
- Nico says it's okay to round what his calculator says to the nearest whole number. Katya says that because the calculator displays eight numbers after the decimal point, they shouldn't round. She says they should write down exactly what the calculator shows. Do you agree with Nico or with Katya?
I agree somewhat with both Nico and Katya as the both methods of recording the speed ate right, depending on what the speed is required for.
Although, I agree more with Nico's method as it seems like a better fit for the situation described in the question.
- explain who you agree with and provide the reasons why.
Like I said earlier, I agree more with Nico's method of rounding the speed to the nearest whole number. This is because in this question, the standard speed we want to compare the calculated speed with is given as a whole number. Hence, it is more proper to estimate the calculated speed to its nearest whole number too.
Katya's method of writing the calculated speed as is will be correct in cases where extreme accuracy is required, not an estimate. For this question, the estimate will do.
Hope this Helps!!!
Answer:
Yes, the car was speeding as its current speed of 57.65 mph was more than the speed limit of that freeway.
Step-by-step explanation:
Nico and Katya i agree with.
The number of degrees of freedom for the appropriate chi-square distribution in a test of independence is a. k – 1. b. A chi-square distribution is not used. c. number of rows minus 1 times number of columns minus 1. d. n – 1.
Answer:
Option C
Step-by-step explanation:
The chi square test of independence is used to determine if there is a significant association between two categorical variables from a population.
It tests the claim that the row and column variables are independent of each other.
The degrees of freedom for the chi-square are calculated using the following formula: df = (r-1) (c-1) where r is the number of rows and c is the number of columns.
There are 7 students in a class: 5 boys and 2 girls.
If the teacher picks a group of 4 at random, what is the probability that everyone in the group is a boy?
Answer:
4/7
Step-by-step explanation:
5+2=7
7 children
4 boys Out of 7 children
Answer:1/7
Step-by-step explanation:
Khan academy
The completion times for a job task range from 11.1 minutes to 19.2 minutes and are thought to be uniformly distributed. What is the probability that it will require between 14.8 and 16.5 minutes to perform the task?
Answer:
[tex] P(14.8< X<16.5)= \frac{16.5-11.1}{19.2-11.1} -\frac{14.8-11.1}{19.2-11.1}= 0.667-0.457= 0.210[/tex]
The probability that it will require between 14.8 and 16.5 minutes to perform the task is 0.210
Step-by-step explanation:
Let X the random variable "completion times for a job task" , and we know that the distribution for X is given by:
[tex] X \sim Unif (a= 11.1, b= 19.2)[/tex]
And for this case we wantto find the following probability:
[tex] P(14.8< X<16.5)[/tex]
And for this case we can use the cumulative distribution given by:
[tex] F(x) =\frac{x-a}{b-a} , a\leq X \leq b[/tex]
And using this formula we got:
[tex] P(14.8< X<16.5)= \frac{16.5-11.1}{19.2-11.1} -\frac{14.8-11.1}{19.2-11.1}= 0.667-0.457= 0.210[/tex]
The probability that it will require between 14.8 and 16.5 minutes to perform the task is 0.210
A graph is given to the right. a. Explain why the graph has at least one Euler path. b. Use trial and error or Fleury's Algorithm to find one such path starting at Upper A, with Upper D as the fourth and seventh vertex, and with Upper B as the fifth vertex. A C B D E A graph has 5 vertices labeled A through E and 7 edges. The edges are as follows: Upper A Upper C, Upper A Upper B, Upper A Upper D, Upper C Upper D, Upper C Upper E, Upper B Upper D, Upper D Upper E. a. Choose the correct explanation below. A. It has exactly two odd vertices. Your answer is correct.B. It has exactly two even vertices. C. It has more than two odd vertices. D. All graphs have at least one Euler path. b. Drag the letters representing the vertices given above to form the Euler path.
Answer:
a. It has exactly two odd vertices
b. A C E D B A D C
Step-by-step explanation:
(a) There will not be an Euler path if the number of odd vertices is not 0 or 2. Here, the graph has exactly two odd vertices: A and C.
__
(b) We are required to produce a path of the form {A, _, _, D, B, _, D, _}.
Starting at A, there is only one way to get to node D as the 4th node on the path: via C and E. Node B must follow. From B, there is exactly one way to cover the remaining three edges that have not been traversed so far.
The Euler path meeting the requirements is ...
A C E D B A D C
It is shown by the arrows on the edges in the graph of the attachment.
I NEED HELP PLEASE, THANKS! :)
Answer:
Option D
Step-by-step explanation:
x is given to be 4 in this case, so all we would have to is plug it into the following function -
[tex]f ( x ) = \left \{ {{x - 2, x < 4 } \atop {x + 2, x \geq 4 }} \right[/tex]
Through substitution, you would receive the following function -
[tex]f ( x ) = \left \{ {{2, 4 < 4 } \atop 6, 4 \geq 4 }} \right[/tex]
Now the graph proves that this function is closer to 4, and thus proves that the y - coordinate is about 2 at the same time. However, the graph is cut off, so the limit doesn't exists.
Which equation represents the statement below?
Thirteen less than a number is forty-two.
Select one:
a. n – 13 = 42
b. 42 – 13 = n
c. 13 – n = 42
d. 13 – 42 = n
The answer is option A
Step-by-step explanation:
Thirteen less than a number is written as
n - 13
Equate it to 42
We have
n - 13 = 42
Hope this helps you
PLSSS PEOPLE I NEED HELP
Answer:C
Step-by-step explanation:
The vertical line test
Among 20 golden hamster litters recorded, there was a sample mean of =7.72 baby hamsters, with a sample standard deviation of s=2.5 hamsters per liter. Create a 98% confidence interval for the mean number of baby hamsters per liter.
Answer:
[tex] 7.72 -2.539 \frac{2.5}{\sqrt{20}} =6.30[/tex]
[tex] 7.72 +2.539 \frac{2.5}{\sqrt{20}} =9.14[/tex]
Step-by-step explanation:
For this case we have the following info given:
[tex]\bar X= 7.72[/tex] represent the sample mean
[tex]s= 2.5[/tex] represent the sample deviation
[tex] n=20[/tex] represent the sample size
The confidence interval is given by:
[tex] \bar X \pm t_{\alpha/2} \frac{s}{\sqrt{n}}[/tex]
The confidence interval is 98% and the significance level is [tex]\alpha=0.02[/tex] the degrees of freedom are given by:
[tex] df= n-1 = 20-1=19[/tex]
And the critical value would be:
[tex] t_{\alpha/2}= 2.539[/tex]
And replacing we got:
[tex] 7.72 -2.539 \frac{2.5}{\sqrt{20}} =6.30[/tex]
[tex] 7.72 +2.539 \frac{2.5}{\sqrt{20}} =9.14[/tex]
The 98% confidence interval is between 6.42 hamsters per liter to 9.02 hamsters per liter
Mean (μ) = 7.72, standard deviation (σ) = 2.5, sample size (n) = 20, Confidence (C) = 98% = 0.98
α = 1 - C = 0.02
α/2 = 0.01
The z score of α/2 is the same as the z score of 0.49 (0.5 - 0.01) which is equal to 2.326.
The margin of error E is:
[tex]E = Z_\frac{\alpha }{2} *\frac{\sigma}{\sqrt{n} } \\\\E=2.326*\frac{2.5}{\sqrt{20} } =1.3[/tex]
The confidence interval = (μ ± E) = (7.72 ± 1.3) = (6.42, 9.02)
Hence the 98% confidence interval is between 6.42 hamsters per liter to 9.02 hamsters per liter
Find out more at: https://brainly.com/question/24131141
The three-dimensional figure below is a cylinder with a hole in the shape of a rectangular prism going through the center of it.
The radius is 10 yards. Find the volume of the solid in cubic yards, rounded to the nearest ten. Use 3.14 for pie.
A. 1,980
B. 1,788
C. 1,034
D. 1,884
Answer:
B. 1788
Step-by-step explanation:
The volume of solid shaped is expressed in cubic yards. The sides of the shape are multiplied or powered as 3 for the volume determination. Volume is the total space covered by the object. It includes height, length, width. The three dimensional objects volume is found by
length * height * width
The volume for current object is :
12 * 28 * 5
= 1788 cubic yards.
Answer: 1778
Step-by-step explanation:
because Ik I had the question
Which number is a solution of the inequality: B > 2.1
A: -8
B: -12
C:5
D: 1
Answer:
C. 5 is solution of the inequality: B>2.1
given that f(x) = x² + 6x and g(x) = x + 9 calculate
a) f•g (4) =
B) g•f (4) =
Answer:
247
49
Step-by-step explanation:
a) f•g (4) =
f•g (x) = f(g(x)) = (x + 9)^2 + 6(x + 9)
f•g (4) = (4 + 9)^2 + 6(4 + 9)
= 13^2 + 6(13)
= 247
B) g•f (4) =
g•f (x) = g(f(x)) = x^2 + 6x + 9
g•f (4) = 4^2 + 6(4) + 9
= 16 + 24 + 9
= 49
For the triangle show, what are the values of x and y (urgent help needed)
we just have to use the Pythagoras theorem and then calculate the value of x and y.
Find the area of a triangle whose two sides are 12 inches and 14 inches long, and has a perimeter of 34 inches.
Answer:
[tex]\huge\boxed{A=3\sqrt{255}\ in^2\approx47.91\ in^2}[/tex]
Step-by-step explanation:
We have two sides
[tex]a=12in;\ b=14in[/tex]
and the preimeter
[tex]P=34in[/tex]
We can calculate the length of the third side:
[tex]c=P-a-b[/tex]
substitute
[tex]c=34-12-14=8\ (in)[/tex]
Use the Heron's formula:
[tex]A=\sqrt{p(p-a)(p-b)(p-c)[/tex]
where
[tex]p=\dfrac{P}{2}[/tex]
substitute:
[tex]p=\dfrac{34}{2}=17\ (in)\\\\A=\sqrt{17(17-12)(17-14)(17-8)}=\sqrt{(17)(5)(3)(9)}\\\\=\sqrt{9}\cdot\sqrt{(17)(5)(3)}=3\sqrt{255}\ (in^2)\approx47.91\ (in^2)[/tex]
Betty tabulated the miles-per-gallon values for her car as 26.5, 28, 30.2, 29.6, 32.3, and 24.7. She wants to construct the 95% two-sided confidence interval. Which value should Betty use for the value of t* to construct the confidence interval?
Answer:
Betty should use T = 2.571 to construct the confidence interval
Step-by-step explanation:
The first step to solve this problem is finding how many degrees of freedom, we have. This is the sample size subtracted by 1. So
df = 6 - 1 = 5
95% confidence interval
Now, we have to find a value of T, which is found looking at the t table, with 5 degrees of freedom(y-axis) and a confidence level of [tex]1 - \frac{1 - 0.95}{2} = 0.975[/tex]. So we have T = 2.571
Betty should use T = 2.571 to construct the confidence interval
At noon, ship A is 120 km west of ship B. Ship A is sailing east at 20 km/h and ship B is sailing north at 15 km/h. How fast is the distance between the ships changing at 4:00 PM?
Answer:
1.39 km/h
Step-by-step explanation:
Let the initial position of ship B represent the origin of our coordinate system. Then the position of ship A as a function of time t is ...
A = -120 +20t . . . (east of the origin)
and the position of B is ...
B = 15t . . . (north of the origin)
Then the distance between them is ...
d = √(A² +B²) = √((-120 +20t)² +(15t)²) = √(625t² -4800t +1440)
And the rate of change is ...
d' = (625t -2400)/√(625t² -4800t +14400)
At t = 4, the rate of change is ...
d' = (625·4 -2400)/√(625·16 -4800·4 +14400) = 100/√5200 = 1.39 . . . km/h
The distance between the ships is increasing at about 1.39 km/h.
Is the area of this shape approximately 24 cm* ? If not give the correct area.
311
101
True
False
Answer:
19.2 feet square
Step-by-step explanation:
We khow that the area of an octagon is :
A= 1/2 * h * P where h is the apothem and p the perimeter
A= (1/2)*1.6*(3*8) = 19.2 feet squareFor the following data at the near-ground level, which location will residents likely see dew on their lawns in the morning? Group of answer choices City C: Dew Point Temperature = 25°F, expected low Temperature = 20°F City A: Dew Point Temperature = 65°F, expected low Temperature = 60°F City B: Dew Point Temperature = 45°F, expected low Temperature = 50°F
Answer: CITY B: Dew Point Temperature = 45°F, expected low Temperature = 50°F
Step-by-step explanation:
CITY C: Dew Point Temperature = 25°F, expected low Temperature = 20°F
CITY A: Dew Point Temperature = 65°F, expected low Temperature = 60°F
CITY B: Dew Point Temperature = 45°F, expected low Temperature = 50°F
city B is going to have dew on their lawn in the morning as the dew point temperature is less than the lowest temperature.
When surface temperature drops, eventually reaching the dew point, atmospheric water vapor condenses to form small droplets on the surface. Thus dew will be formed as the conditions are suitable only for city B.
Solve the algebraic expressio (0.4)(8)−2
Answer: -6.4
Step-by-step explanation:
(0.4)(8)(-2)
3.2*-2
-6.4
If a 1/5 of a gallon of paint is needed to cover 1/4 of a wall, how much paint is needed to cover the entire wall
Answer:
4/5 gallon per wall
Step-by-step explanation:
We can find the unit rate
1/5 gallon
------------------
1/4 wall
1/5 ÷ 1/4
Copy dot flip
1/5 * 4/1
4/5 gallon per wall
Answer:
4/5 gallon of paint
Step-by-step explanation:
1/5 gallon of paint is needed to cover 1/4 of the wall.
To cover the whole wall:
1/4 × 4 = 1 (whole)
1/5 × 4 = 4/5
graph y=8 sec1/5 Ø the answers are graphs I am just unsure of how to answer
Answer:
Use a graphing calc.
Step-by-step explanation:
Pls help with this area question
Answer:
1
Step-by-step explanation:
The lateral area of a cylinder is ...
LA = 2πrh
The total area is that added to the areas of the two circular bases:
A = 2πr² +2πrh
We want the ratio of these to be 1/2:
LA/A = (2πrh)/(2πr² +2πrh) = h/(r+h) = 1/2 . . . . cancel factors of 2πr
Multiplying by 2(r+h) gives ...
2h = r+h
h = r . . . . . subtract h
So, the desired ratio is ...
h/r = h/h = 1
The ratio between height and radius is 1.
Estimate the area under the graph of f(x)=2x^2-12x+22 over the interval [0,2] using four approximating rectangles and right endpoints.
Answer:
The right Riemann sum is 21.5.
The left Riemann sum is 29.5.
Step-by-step explanation:
The right Riemann sum (also known as the right endpoint approximation) uses the right endpoints of a sub-interval:
[tex]\int_{a}^{b}f(x)dx\approx\Delta{x}\left(f(x_1)+f(x_2)+f(x_3)+...+f(x_{n-1})+f(x_{n})\right)[/tex], where [tex]\Delta{x}=\frac{b-a}{n}[/tex].
To find the Riemann sum for [tex]\int_{0}^{2}\left(2 x^{2} - 12 x + 22\right)\ dx[/tex] with 4 rectangles, using right endpoints you must:
We have that a = 0, b = 2, n = 4. Therefore, [tex]\Delta{x}=\frac{2-0}{4}=\frac{1}{2}[/tex].
Divide the interval [0,2] into n = 4 sub-intervals of length [tex]\Delta{x}=\frac{1}{2}[/tex]:
[tex]\left[0, \frac{1}{2}\right], \left[\frac{1}{2}, 1\right], \left[1, \frac{3}{2}\right], \left[\frac{3}{2}, 2\right][/tex]
Now, we just evaluate the function at the right endpoints:
[tex]f\left(x_{1}\right)=f\left(\frac{1}{2}\right)=\frac{33}{2}=16.5\\\\f\left(x_{2}\right)=f\left(1\right)=12\\\\f\left(x_{3}\right)=f\left(\frac{3}{2}\right)=\frac{17}{2}=8.5\\\\f\left(x_{4}\right)=f(b)=f\left(2\right)=6[/tex]
Finally, just sum up the above values and multiply by [tex]\Delta{x}=\frac{1}{2}[/tex]:
[tex]\frac{1}{2}(16.5+12+8.5+6)=21.5[/tex]
The left Riemann sum (also known as the left endpoint approximation) uses the left endpoints of a sub-interval:
[tex]\int_{a}^{b}f(x)dx\approx\Delta{x}\left(f(x_0)+f(x_1)+2f(x_2)+...+f(x_{n-2})+f(x_{n-1})\right)[/tex], where [tex]\Delta{x}=\frac{b-a}{n}[/tex].
To find the Riemann sum for [tex]\int_{0}^{2}\left(2 x^{2} - 12 x + 22\right)\ dx[/tex] with 4 rectangles, using left endpoints you must:
We have that a = 0, b = 2, n = 4. Therefore, [tex]\Delta{x}=\frac{2-0}{4}=\frac{1}{2}[/tex].
Divide the interval [0,2] into n = 4 sub-intervals of length [tex]\Delta{x}=\frac{1}{2}[/tex]:
[tex]\left[0, \frac{1}{2}\right], \left[\frac{1}{2}, 1\right], \left[1, \frac{3}{2}\right], \left[\frac{3}{2}, 2\right][/tex]
Now, we just evaluate the function at the left endpoints:
[tex]f\left(x_{0}\right)=f(a)=f\left(0\right)=22\\\\f\left(x_{1}\right)=f\left(\frac{1}{2}\right)=\frac{33}{2}=16.5\\\\f\left(x_{2}\right)=f\left(1\\\right)=12\\\\f\left(x_{3}\right)=f\left(\frac{3}{2}\right)=\frac{17}{2}=8.5[/tex]
Finally, just sum up the above values and multiply by [tex]\Delta{x}=\frac{1}{2}[/tex]:
[tex]\frac{1}{2}(22+16.5+12+8.5)=29.5[/tex]