Approximate the area under the
function between a and b using a
left-hand sum with the given
number of intervals.
f(x) = x² + 2
a = 0
b= 6
6 intervals
Answers
Answer:
67 square units
Step-by-step explanation:
The area using the left-hand sum is the sum of products of the function value at the left side of the interval and the width of the interval.
AreaThe attachment shows a table of the x-value at the left side of each interval, and the corresponding function value there. The interval width is 1 unit in every case, so the desired area is simply the sum of the function values.
The approximate area is 67 square units.
Split up the interval [0, 6] into 6 equally spaced subintervals of length [tex]\Delta x = \frac{6-0}6 = 1[/tex]. So we have the partition
[0, 1] U [1, 2] U [2, 3] U [3, 4] U [4, 5] U [5, 6]
where the left endpoint of the [tex]i[/tex]-th interval is
[tex]\ell_i = i - 1[/tex]
with [tex]i\in\{1,2,3,4,5,6\}[/tex].
The area under [tex]f(x)=x^2+2[/tex] on the interval [0, 6] is then given by the definite integral and approximated by the Riemann sum,
[tex]\displaystyle \int_0^6 f(x) \, dx \approx \sum_{i=1}^6 f(\ell_i) \Delta x \\\\ ~~~~~~~~ = \sum_{i=1}^6 \bigg((i-1)^2 + 2\bigg) \\\\ ~~~~~~~~ = \sum_{i=1}^6 \bigg(i^2 - 2i + 3\bigg) \\\\ ~~~~~~~~ = \frac{6\cdot7\cdot13}6 - 6\cdot7 + 3\cdot6 = \boxed{67}[/tex]
where we use the well-known sums,
[tex]\displaystyle \sum_{i=1}^n 1 = \underbrace{1 + 1 + \cdots + 1}_{n\,\rm times} = n[/tex]
[tex]\displaystyle \sum_{i=1}^n i = 1 + 2 + \cdots + n = \frac{n(n+1)}2[/tex]
[tex]\displaystyle \sum_{i=1}^n i^2 = 1 + 4 + \cdots + n^2 = \frac{n(n+1)(2n+1)}6[/tex]
Related Questions
The periodic time, t, of a pendulum varies directly as the square root of its length, l. t = 6 when l = 9. find t when l = 25.
Answers
The periodic time, t, of a pendulum range directly as the square root of its length, l. t = 6 when l = 9. If l = 25 then the periodic time, T exists at 10.
What is periodic time?The period, or periodic time, of a periodic variation of a quantity, exists described as the time interval between two consecutive repetitions.
Given: The periodic time, t, of a pendulum goes directly as the square root of its length, l if t = 6 when l = 9.
If [tex]T \alpha\ l^2[/tex] then [tex]T^2[/tex] α [tex]\sqrt{l}[/tex]
[tex]T^2 = l[/tex]
Let, [tex]T^2 = kl,[/tex] where k exists constant
t = 6 when l = 9.
So, [tex]6^2 = k*9[/tex]
[tex]k = 6^2/9 = 4[/tex]
[tex]T^2 = 4*l[/tex]
If l = 25
[tex]T^2 = 4 * 25 = 100[/tex]
[tex]T = \sqrt{100}[/tex]
T = 10
Therefore, the periodic time, T exists 10.
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200 σ j=1 2j( j 3) describe the steps to evaluate the summation. what is the sum?
Answers
The sum of the equation is = 5494000.
What does summation mean in math?The outcome of adding numbers or quantities mathematically is a summation, often known as a sum. A summation always has an even number of terms in it. There may be just two terms, or there may be 100, 1000, or even a million. Some summations include an infinite number of terms.
Briefing:Distribute 2j to (j+3).
Rewrite the summation as the sum of two individual summations.
Evaluate each summation using properties or formulas from the lesson.
The lower index is 1, so any properties can be used.
The sum is 5,494,000.
Calculation according to the statement:[tex]\sum_{j=1}^{200} 2 j(j+3)[/tex]
simplifying them we get:
[tex]\sum_{j=1}^{200} 2 j^{2}+6 j[/tex]
Split the summation into smaller summations that fit the summation rules.
[tex]\sum_{j=1}^{200} 2 j^{2}+6 j=2 \sum_{j=1}^{200} j^{2}+6 \sum_{j=1}^{200} j[/tex]
[tex]\text { Evaluate } 2 \sum_{j=1}^{200} j^{2}[/tex]
The formula for the summation of a polynomial with degree 2
is:
[tex]\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}[/tex]
Substitute the values into the formula and make sure to multiply by the front term.
[tex](2)$$\left(\frac{200(200+1)(2 \cdot 200+1)}{6}\right)$$[/tex]
we get: 5373400
Evaluating same as above : [tex]6 \sum_{j=1}^{200} j[/tex]
we get: 120600
Add the results of the summations.
5373400 + 120600
= 5494000
The sum of the equation is = 5494000.
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A variety of two types of snack packs are delivered to a store. The box plots compare the number of calories in each snack pack of crackers to the number of calories in each snack pack of trail mix.
A number line goes from 65 to 115. Crackers's whiskers range from 70 to 100, and the box ranges from 75 to 85. A line divides the box at 80. Cookies's whiskers range from 70 to 115, and the box ranges from 90 to 105. A line divides the box at 100.
Which statement is true about the box plots?
The interquartile range of the trail mix data is greater than the range of the cracker data.
The value 70 is an outlier in the trail mix data.
The upper quartile of the trail mix data is equal to the maximum value of the cracker data.
The number of calories in the packs of trail mix have a greater variation than the number of calories in the packs of crackers.
Answers
The statement that is true about the box plots is: D. The number of calories in the packs of trail mix have a greater variation than the number of calories in the packs of crackers.
What is a Measure of Variation?For a data distribution that is displayed by a box plot, the measure of variation that can be used to describe the data distribution is the interquartile range.
What is the Interquartile Range of a Data Distribution?Interquartile range, which is a measure of variation can be determined from a box plot by finding the value of the third quartile and first quartile, the finding their difference. The difference is the interquartile range.
Interquartile range = third quartile - first quartile.
Interquartile range for crackers = 85 - 75
Interquartile range for crackers = 10
Interquartile range for trail mix = 105 - 90
Interquartile range for trail mix = 15.
The bigger the value of the interquartile range, the more variation there is that exist in a data distribution.
Interquartile range for trial mix is the largest, therefore, the statement that is true about the box plots is: D. The number of calories in the packs of trail mix have a greater variation than the number of calories in the packs of crackers.
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pls pls help meee i dont understand the fundamental theorem of algebra
Answers
Answer:
Hey! I'm gonna go ahead and give ya the "meaning" of the fundamental theorem of algebra but not the answer to the A. B. and C.
Step-by-step explanation:
fundamental theorem of algebra, theorem of equations proved by Carl Friedrich Gauss in 1799. It states that every polynomial equation of degree n with complex number coefficients has n roots, or solutions, in the complex numbers. The roots can have a multiplicity greater than zero.
10 by 28 + - 32 by 49 rational number
pls answer it
thank you
Answers
By simplifying the fraction 10/28 ÷ 32/49 exists 35/64.
How to simplify fractions?The first stage of dividing fractions exists to estimate the reciprocal (reverse the numerator and denominator) of the second fraction. Subsequently, multiply the two numerators. Then, multiply the two denominators. Finally, simplify the fractions if required.
To find the reciprocal of the divisor
Reciprocal of 32/49 = 49/32
Now, multiply it by the dividend, then we get
So, 1028 ÷ 3249 = 10/28 × 49/32
= (10 × 49)/(28 × 32) = 490/896
After decreasing the fraction, the answer exists at 35/64.
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In the graph above, which vertical line (V) and horizontal line (H) can be used to graph point D?
Question 5 options:
A)
V: x = –4; H: y = 4
B)
V: y = –4; H: x = –4
C)
V: y = 4; H: x = –4
D)
V: x = 4; H: y = 4
Answers
Answer: A. V: x = –4; H: y = 4
Step-by-step explanation:
30 metres is percentage of 100 metres
Answers
Answer:
30%
Step-by-step explanation:
1) Based on the given conditions, formulate:
30 meters/100 meters
2) Calculate 30 meters/100 meters:
3/10
3) Convert to percentage:
- 3/10 x 10/10
- 3x10/10x10
30/100
30/100 = 30%
You’re done!
Also, if you could label this brainliest that would be a great help!
Thanks xx
-Dante
The daily high temperature of Elk Creek drops by 1.2°C every day. What is the net change in the daily high temperature after two calendar weeks?
Answers
[tex]1.2 \: degrees \: drop \: a \: day \\ 14 \: days \: in \: two \: weeks \\ net \: change = 14 \times 1.2 = 16.8 \: deg[/tex]
Which has no real roots?
Answers
[tex] \sqrt[4]{ - 8} [/tex]
Explanation:The input of a radical should be strictly positive or equal to zero when the nth root is even.The limits of the class intervals are
Answers
The lower and the upper class limit are as follows
interval lower limit upper limit
10-14 10 14
15-19 15 19
20-24 20 24
25-29 25 29
This is further explained below.
What is the class limit?Generally, The lower class limit and the upper-class limits are just the minima and maximum values that are allowed in each class, respectively: The gap that exists between the upper-class limit and the lower class limit is referred to as the class interval.
In conclusion, The lower and the upper class limit are as follows
interval lower limit upper limit
10-14 10 14
15-19 15 19
20-24 20 24
25-29 25 29
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The graph of the step function g(x) = –⌊x⌋ + 3 is shown.
On a coordinate plane, a step graph has horizontal segments that are each 1 unit long. The left end of each segment is a closed circle. The right end of each segment is an open circle. The left-most segment goes from (negative 2, 5) to (negative 1, 5). Each segment is 1 unit lower and 1 unit farther to the right than the previous segment. The right-most segment goes from (4, negative 1) to (5, negative 1).
What is the domain of g(x)?
{x| x is a real number}
{x| x is an integer}
{x| –2 ≤ x < 5}
{x| –1 ≤ x ≤ 5
Answers
Using it's concept, the domain of the function is given as follows:
{x| –2 ≤ x <= 5}
What is the domain of a function?The domain of a function is the set that contains all possible input values for the function.
The function described is defined for x between -2 and 5, hence the domain is given by the following interval:
{x| –2 ≤ x <= 5}
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Answer:
{x| –2 ≤ x < 5}
Step-by-step explanation:
Mike wants to find the confidence interval for a set of data. he knows the sample size and the sample proportion. which other piece of information does he need to determine the confidence interval?
Answers
Using the formula for a z-distribution confidence interval of proportions, he also needs to know the confidence level.
What is a confidence interval of proportions?A confidence interval of proportions is given by:
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which:
[tex]\pi[/tex] is the sample proportion.z is the critical value.n is the sample size.Of these 3 itens, the problem states that he knows the sample size and the sample proportion, hence he needs to know z to determine the confidence interval. z depends on the confidence level, which is the remaining parameter.
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Change
3
-
4
to a percent
Answers
Answer:
75%
Step-by-step explanation:
3/4 ie 75/100. .........edge
which of the following tools is used to construct an angle bisector
Answers
Answer:
Tools used to construct a bisector are 1》pencil ,2》Eraser ,3》protector ,4》Compass ,5》Sharpner etc ....
At the break-even point of 2000 units, variable costs are $45000, and fixed costs are $35000. how much is the selling price per unit? $40. 00 $17. 50 $22. 50 $5. 0
Answers
Answer:
$40/unit?
Step-by-step explanation:
I'll assume the the fixed and variable costs are for 2000 units.
Fixed Costs = $35,000
Variable Costs = $45,000
Tota Costs = $80,000
The selling price can be anything. Free and we lose $80,000; $40 each, net profit of $0; $100 each for a net profit of $120,000.
In the absence of any infoprmation regarding selling price (e.g., 50% profit margin), I'd select the $40/unit selling price since that is breakeven.
Which must be true by the corresponding angles
theorem?
041 = 27
42 = 26
23 45
25= 27
Answers
The corresponding angles are ∠2 and ∠6 i.e., ∠2≅∠6 according to the corresponding angles theorem. So, option 2 is correct.
What is the corresponding angle theorem state?The theorem states that,
The lines 'AB' and 'CD' are parallel and cut by a transverse 'EF' if and only if their corresponding angles are congruent.
Corresponding angles:The corresponding angles are those which lie on the same side of the transverse one is interior and the other is an exterior angle.
From the diagram,
Lines 'AB' and 'CD' are parallel and line 'EF' cuts these parallel lines.
So, the corresponding angles formed are:
(∠1, ∠5), (∠2, ∠6), (∠3, ∠7), and (∠4, ∠8)
According to the theorem, they are congruent. I.e.,
∠1 ≅ ∠5
∠2 ≅ ∠6
∠3 ≅ ∠7
∠4 ≅ ∠8
So, option 2 is correct.
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Disclaimer: The given question is incorrect. Here is the correct question.
Question: Lines 'c' and 'd' are parallel lines cut by the transversal 'p'. Which must be true by the corresponding angles theorem?
1) ∠1 = ∠7
2) ∠2 = ∠6
3) ∠3 = ∠5
4) ∠5 = ∠7
A mini-lottery involves selecting 2 numbers between 1 and 10. What is the probability of correctly picking the two numbers?
a) 2/10
b) 1/30
c) 2/90
d) 2/100
Answers
The probability of correctly picking the two numbers is 2/10.
What is the probability?Probability determines the chances that an event would happen. The probability the event occurs is 1 and the probability that the event does not occur is 0.
The more likely the event is to happen, the closer the probability value would be to 1. The less likely it is for the event not to happen, the closer the probability value would be to zero.
The probability of correctly picking the two numbers = (numbers that have to be selected to win / total numbers)
2/10
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HELP HELP ASAP
Figure BBB is a scaled copy of Figure AAA.
What is the scale factor from Figure AAA to Figure BBB?
Answers
Answer:
figure A = 3 units
figure B = 9 x 3 = 27 units
scale factor =
[tex] \frac{3}{27 } = \frac{1}{9} [/tex]
so the scale factor = 9
a recently televised broadcast of a popular televison show had a 15 share, meaning that among 5000 monitored households with TV sets is use, less than 20% were turned in to the show. find the p-value
a) 1.9998
b) 0.9999
c) 0.0002
d) 0.0001
Answers
The p-value from the information about the recently televised broadcast is B. 0.9999.
How to get the p-value?From the information given, n = 5000 and the level of significance is 0.01.
The computed z value is -8.84.
From the z table, the probability value is the value corresponding to the row 8.8 and column 0.04.
This will give a value of 0.9999.
In conclusion, the correct option is B.
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The total cost charged for a day of fun at Aussie World consists of an entry fee of $15 and a rate of $3 per ride. Write the rule for the total cost, C, where r rides are taken.
Answers
C = 3r + 15 is the rule for the total cost, C, where r rides are taken given that the total cost charged for a day of fun at Aussie World consists of an entry fee of $15 and a rate of $3 per ride. This can be obtained by converting the statements to algebraic equation with variables and constants.
Find the rule for the total cost:Here in the question it is given that,
The total cost charged for a day of fun at Aussie World consists of
an entry fee of $15 a rate of one ride is $3We have to find a rule for the total cost, C, where r rides are taken.
For one ride the rate is 3 ⇒ therefore for r rides the rate is 3r
We can convert the statements to algebraic equation so that we get the required rule,
⇒ The total cost consists of entry fee and rate of rides
⇒ The total cost = entry fee + rate of rides
⇒ C = 15 + 3r (from the given information)
⇒ C = 3r + 15
Hence C = 3r + 15 is the rule for the total cost, C, where r rides are taken given that the total cost charged for a day of fun at Aussie World consists of an entry fee of $15 and a rate of $3 per ride.
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What is the last digit of the product of all the numbers between 11 and 29?
Answers
we can conclude that the last digit of the product of all the numbers between 11 and 29 is 0.
What is the last digit of the product of all the numbers between 11 and 29?
Here we want to find the last digit of the product between all the whole numbers larger than 11 and smaller than 29.
Then we have the product:
P = 12*13*14*15*16*17*18*19*20*21*22*23*24*25*26*27*28
Now, notice that there is a 20 there.
Any number times 20 will end with a zero, then:
P = 20*(12*13*14*15*16*17*18*19*21*22*23*24*25*26*27*28)
Only with that, we can conclude that the last digit of the product of all the numbers between 11 and 29 is 0.
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Clark earns $31 per hour. Calculate his weekly wage if he works 38 hours per week.
Answers
Answer:
1178 because 31 dollars an hour for 38 hours you would multiply or times 31 and 38 and get 1178 for total money earned
15 The tennis team is selling tickets to a car wash for $6. When they do not
sell very many tickets, the team decreases the price 25%. What is the
new cost of a ticket?
Show your work.
Answers
Answer: $4.50
Step-by-step explanation:
Original Price: $6
Discount: 25%
Discounted Price: 6*25%= $1.5
$6-$1.5= $4.5
Y=2x^2+2x-4 what is the x intercepts for the parabola
Answers
Answer:
Step-by-step explanation:
To find the x intercepts, substitute in 0 for y and solve for x
So,
2x^2+2x-4 = 0
2(x^2+x-4) = 0
2/2(x^2+x-4) = 0/2
x^2+x-4 = 0
(x-1) (x+2) = 0
now,
x-1 = 0
x = 1
or
x+2 = 0
x = -2
The x intercepts for the parabola (1,0), (-2,0)
Jacob is cutting a tile in the shape of a parallelogram. two opposite angles have measures of (6n − 70)° and (2n 10)°. what are the two different angle measures of the parallelogram-shaped tile?
Answers
The measures of the two different angles are 70° and 110°.
What is angle in parallelogram ?
A parallelogram is a flat 2d shape which has four angles. The opposite interior angles are equal. The angles on the same side of the transversal are supplementary, that means they add up to 180 degrees. Hence, the sum of the interior angles of a parallelogram is 360 degrees.Thus from the given question, we have:
(6n − 70)° + (2n + 10)° = 180°
8n - 60 = 180° + 60
8n = 240
n = 240/8
n = 30°
So that,
i. (6n − 70)° = (6[30] − 70)°
= 180 - 70
= 110°
ii. (2n + 10)° = (2[30] + 10)°
= 60 + 10
= 70°
Therefore, the measures of the two different angles are 70° and 110°.
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5. Enter the answer as a decimal
Calculate the rate of interest, when simple interest = $5,616;
loan amount = $7,200 and time = 12 years.
Answers
Answer:
6.5%
Step-by-step explanation:
I = prt
5616 = 7200 × r × 12
r = 5616/(7200 × 12)
r = 0.065
r = 6.5%
Sandy jogs 19.7 miles in 4.5 hours. How many miles does she jog per hour
Answers
Answer:
4.378 MPH
Step-by-step explanation:
Distance in Miles
MPH= -------------------------
Time in Hours
19.7/4.5 = 4.378 MPH
Fifteen students were divided into two classrooms in a ratio of 1:2.
How many students were in each classroom?
Answers
Explanation:
x + 2x = 3x
3x = 15
x = 5, Classroom 1 = 5, Classroom 2 = 5 x 2 = 10
I hope that this helps! :)
Round to the nearest whole number
763.9
Answers
Hi there,
please see below for solution steps :
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
First let us revise the rounding rules before getting down to the process of rounding :
if the number you're rounding to is followed by a number from 0 to 4, then you drop that number.if the number you're rounding to is followed by a number that's 5 or more, you add 1 to that number.∴[tex]\sf{763.9 \ to \ the \ nearest \ whole \ number :}\\\hfill\stackrel{\small\text{add 1 }}{3^{+1}}.\\\hfill\stackrel{\small\text{drop it}}{9}}[/tex]
∴[tex]\sf{763.9\approx764}[/tex]
‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗‗
There are 20 points on a line, each spaced two inches apart. How far, in inches, is it from the first point to the last one?
Answers
Answer: 40 inches
Step-by-step explanation: If the 20 points on the line were spaced 1 inch apart, the result would be 20 inches. Each point you're going to is 1 inch in spacing and there is only going to be 20 inches of spacing (1 inch per point.)
For 2 inches spaced apart, the result would be simply 40 inches. You just need to add the distance of 20 more to the already distance of 20 points, since if 2 were to be divided, it would be 1, which is 20 inches in distance as stated. Another 20 points would be another inch in distance, and adding them gives you 2 inches in distance or 40 inches all together.
I apologize if I wasn't clear enough, I couldn't find a way to explain it as good as I thought it was in my mind.
Hope it helped though!