. let f be a bounded function on [a, b], and let p be an arbitrary partition of [a, b]. first, explain why u(f) ≥ l(f,p). now, prove lemma 7.2.6.
Answers
U(f) ≥ L(f,p) means the upper bound (U) of a bounded function f on a given interval [a, b] is greater than or equal to the lower bound (L) of the same function f on the same interval [a, b] using a particular partition (p) of the interval.
The proof of this statement can be broken down into two parts:
Upper bound (U) of a function f on a given interval [a, b] is greater than or equal to the lower bound (L) of the same function f on the same interval [a, b] using a particular partition (p) of the interval. For any interval \([x_i, x_i+h_i]\) in the partition p, the lower bound (L) of the function f in this interval is the maximum value of f over this interval. Similarly, the upper bound (U) of the function f in this interval is the minimum value of f over this interval.
Now, for any value of the function f in the interval [a, b], we can find a partition p of [a, b] such that the maximum value of the function f in this interval is in one of the subintervals \([x_i, x_i+h_i]\) of the partition, and the minimum value of the function f in this interval is in another subinterval \([y_j, y_j+h_j]\) of the partition.
Therefore, the difference between the upper bound (U) and the lower bound (L) of the function f in the interval [a, b] is the maximum value of the function f minus the minimum value of the function f in this interval, which is less than or equal to 0.
Prove Lemma 7.2.6:
Let f be a bounded function on [a, b] and p be an arbitrary partition of [a, b]. Then, limsupf of n(p) = ±∞, if and only if limsup \([f_n(x_i)]\) = ±∞, for any subsequence f_n(x_i) of f.
By the definition of Riemann sums, we have:
lim\(S_n(p) = lim[[f_n(x_i) * (b_i - a_i)]/b_i]\)
= lim[∑\([f(x_i) * (b_i - a_i)]/b_i]\)
= lim[∑\([f(x_i) * (b_i - a_i) + f(a_i) * (b_i - a_i)]/b_i]\)
The second term in the above equation is 0, since \(f(a_i) = L * (b_i - a_i) =\)0. Therefore, we have: limS of n(p)
= lim[∑\([f(x_i) * (b_i - a_i)]/b_i]\)
= lim[∑\([f(x_i) * (b_i - a_i)]/b_i]\)
= lim[∑\([f(x_i) * (b_i - a_i)]/b_i]\)
= lim[∑\([f_n(x_i) * (b_i - a_i)]/b_i]\)
Therefore, we have proven that if limsupf of n(p) = ±∞, then limsup \([f_n(x_i)]\)= ±∞,
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Related Questions
Single-tickets sell for $20 each and a couple-tickets for $35 each. A total of 160 tickets are
sold, and the total value of the ticket is $4100. How many of each kind of ticket were sold?
Answers
Answer:
100 of the 20$ tickets, and 60 of the 35$ tickets
Step-by-step explanation:
20*x + 35*y = 4100
x + y = 160
That means x = 160 - y, put this into the first equation
20(160 - y) + 35y = 4100
3200 - 20y + 35y = 4100
15y + 3200 = 4100
15y = 900
y = 60
Put y = 60 into the second equation and
x + 60 = 160
x = 100
y = 60, x = 100
Double check that this works for both equations to make sure its right
20 * 100 + 35 * 60= 4100
2000 + 2100 = 4100 /Correct
100 + 60 = 160 /Correct
Find the area and round it to the nearest hundredth if necessary
Answers
Answer:
16.28 squared inches
Step-by-step explanation:
Use the formula A = bh.
b = base
h = height
A = (4.4) (3.7)
A = 16.28
after allowing a 15% discount on the Marked price of a radio becomes rupees 1997 find the Marked price of the radio
Answers
Answer:
after discount sales price is rs 1997 which is 85 % of market price the market price = 1997/85×100 = 2350 ans.
At a certain time of day, the angle of elevation is 40°. To the nearest foot, find the height of a tree whose shadow is 31 feet long.
Answers
Answer: Height of tree = 26 ft
Make a triangle ABC with 40 degrees angle. The shadow is the base of the triangle equals 31 feet.
Base = AC = 31
Perpendicular = CB = ?
Hypotenuse = AB
Using trigonometric equation
tan 40 (deg) = CB / AC
CB = Tan 40 x AC
CB = 0.83909 x 31
CB = 26.011
Thus the height of the tree = 26 ft
Please answer with steps
{0.4), (-4,-6)
Answers
\((0.4), (-4,-6)\)
The LCM of a, b is the smallest multiplier that is divisible by both a and b
\(= 0.4(-4, -6)\)
An atom of rhodium (Rh) has a diameter of about 2.7×10−8cm. If the atom is assumed to be a sphere, what is the volume in m3 of a single Rh atom?
Answers
Therefore, the volume of a single Rh atom is approximately 1.011 × 10^(-29) m^3.
The diameter of a single Rh atom is given as 2.7 × 10^(-8) cm. The radius of the atom, which is half of the diameter, is therefore:
r = 2.7 × 10^(-8) cm / 2 = 1.35 × 10^(-8) cm
To find the volume of the sphere, we can use the formula:
V = (4/3)πr^3
where π is the constant pi.
Substituting the values, we get:
V = (4/3)π(1.35 × 10^(-8) cm)^3
= 1.011 × 10^(-23) cm^3
However, the answer is required in m^3, so we need to convert cm^3 to m^3. We know that:
1 cm = 0.01 m
Therefore:
(1 cm)^3 = (0.01 m)^3
1 cm^3 = 1 × 10^(-6) m^3
Substituting this conversion factor, we get:
V = 1.011 × 10^(-23) cm^3 × 1 × 10^(-6) m^3/cm^3
= 1.011 × 10^(-29) m^3
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Replace the values of mand n to show the solutions of this equation.
X^2 + 6x - 5=0
X=m + n
_
Answers
Lana has 4 bags of apples. Each bag has 9 apples. She puts an equal number of apples on each of 6 plates.
Let n be the number of apples on each plate.
What equation can be used to find n?
Answers
4x9 to get the total number of apples
Then divide by 6 to see how many apples for each plate (since there’s 6)
TRUE/FALSE. The term degrees of freedom refers to how many values are free to vary in a set of values if we know the summary statistic and the number of values in the set.
Answers
True. The term "degrees of freedom" refers to the number of values that are free to vary in a set of values when the summary statistic and the number of values in the set are known.
Degrees of freedom is a statistical concept that pertains to the number of independent pieces of information available to estimate a parameter or to make inferences about a population. In the context of a set of values, it represents the number of values that are free to vary once certain constraints are imposed by known summary statistics and the total number of values in the set.
To understand this concept, let's consider an example. Suppose we have a set of n values and we know the mean of the set. If we are to calculate the individual values from this information, we would have n-1 degrees of freedom. This is because once the mean is known, there is a constraint on the values in the set, leaving only n-1 values free to vary independently.
Degrees of freedom are also relevant in hypothesis testing and statistical analysis. When conducting a t-test or an analysis of variance (ANOVA), degrees of freedom are used to determine the critical values and calculate the p-values associated with the test statistics. It provides an indication of the reliability and precision of the statistical estimates.
In summary, degrees of freedom represent the number of values that can vary independently in a set of values given the knowledge of the summary statistic and the total number of values in the set. It is a crucial concept in statistics that has implications in various areas of data analysis.
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The whittendale family purchases a new refrigerator on a no interest for one year plan. The cost is $1,385. There is no down payment If they make a monthly payment of x dollars until the last month, express their last month's payment algebraically.
Answers
1.2x-0.6y=3
0.8x-1.4y=3
Answers
Answer:
1.2−0.6=30.8−1.4=3
The median of a quanitive data set is always one of the infiviual data values. True or false
Answers
The given statement "The median of a quantitative data set is always one of the individual data values" is false.
What is a median?When a given data set is arranged in ascending order the observation which at the between of the data is the median of that data set. It is a value in the set whose left and right both have the same number of observations.
The given statement is not true, this is because the median is the mid-value of any data set.
If the number of data observations is odd then the median of a quantitative data set is always one of the individual data values.But if the number of data observations is even then the median is the average of the two numbers present in between the data observations.Hence, the given statement "The median of a quantitative data set is always one of the individual data values" is false.
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please please!! help!!
Answers
9514 1404 393
Answer:
2
Step-by-step explanation:
Each orange vertex is twice as far from the origin as the corresponding black vertex. Each orange line segment is twice as long as the corresponding black line segment. The scale factor is 2.
given f(x)= 1/x-2. Find x if f(x)= -1
Answers
Answer:
2,5 and 7
Step-by-step explanation:
factors of 1/x-2 are the use of integers that
we can split
help pls thx guys photo attached
Answers
Answer:
x = 18
A = 110
Step-by-step explanation:
Angles A and B are supplementary angles so they add to 180 degrees
A + B = 180
5x+20 + 9x -92 = 180
Combine like terms
14x -72 = 180
Add 72 to each side
14x-72+72 = 180+72
14x = 252
Divide each side by 14
14x/14 = 252/14
x =18
A = 5x+20
= 5*18 +20
=90 +20
= 110
Henry will buy grass seed for his lawn. Each bag of grass seed covers an area of 925
square feet. Henry's lawn is 128 feet long and 72 feet wide. He estimates that he will
need 8 bags of grass seed. Is this estimate reasonable?
Show all your work!
Answers
Henry needs to buy the 9.96 bags of grass seed to cover the lawn area:
What is the definition of the area of a geometric figure?The total space occupied by a flat (2-D) surface or the shape of an object is defined as its area. It is a two-dimensional figure. The area of the shape on the paper is referred to as its Area. Consider your square to be made up of smaller unit squares.
What is an example of a geometrical figure?Geometrical shapes are two-dimensional shapes and closed figures such as circles, squares, rectangles, rhombuses, and so on. The three-dimensional shapes in solid geometry are the cube, cuboid, cone, sphere, and cylinder. All of these shapes can be found in various forms all over the place.
According to the given data:The length of the lawn L = 128.
the width of the lawn B = 72.
bag of grass seed covers an area of 925 ft².
He anticipates that he will require 8 bags of grass seed.
Now first we find the area of the lawn:
area of lawn = L x B
= 128 * 72
= 9,216 ft²
The grass seed required to covers an area of lawn = 9,216 ft²
one bag of grass seed covers an area of 925 ft².
= 9216/925
= 9.96
henry needs to buy the 9.96 bags of grass seed to cover the lawn area:
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Suppose () = 1/8 for 0 ≤ ≤ 4 for x being a continuous random variable Is () a probability density function? Prove or disprove.
Answers
Answer:
The expected value of x ; E(x) = 1
Step-by-step explanation:
F(x) = 1/8 for 0 ≤ x ≤ 4
To prove that it is a probability density function we will find E(x )
attached below is the required prove
It is proven that F(x) = 1/8 for 0 ≤ x ≤ 4 is probability density function
The expected value of X = 1
Consider the equation -3*e^5w=-88
solve for the equation w. express the solution as log in base e
Answers
Answer: The solution for w, expressed as a natural logarithm, is:
w = ln(88/3) / 5
Step-by-step explanation: Starting with the given equation:
-3e^(5w) = -88
Dividing both sides by -3:
e^(5w) = 88/3
Taking the natural logarithm of both sides:
ln(e^(5w)) = ln(88/3)
Using the property that ln(e^x) = x:
5w = ln(88/3)
Finally, solving for w by dividing both sides by 5:
w = (1/5)ln(88/3)
So the solution for w, expressed as a natural logarithm, is:
w = ln(88/3) / 5
CAN YOU ANSWER BOTH OF THESE QUESTIONS? ILL GIVE YOU 3X THE BRAINLIEST :D
Johnny gets a deal on new Time Warp skateboard. The board normally sells for $60, but they are now discounted down to $24. What is the discount percent? *
Ekaterina wants to buy and awesome pair of boots that normally sells for $75. The boots are on sale for at a 32% discount. She has $50 in her pocket. Does she have enough to buy the boots? *
Answers
Answer:
75.9550
Step-by-step explanation:
Answer:
40% and yes she can afford the boots
Step-by-step explanation:
just multyply 60 by 40% and you'll get $24
Same for the second question $75*32%=24
100 points! :) How do you find the equation of a line given two points? Please explain step-by-step!
So for example: Find the equation of the line containing the points (-2, 1) and (2,9).
Answers
Answer:
1,9
Step-by-step explanation:
Answer:
-2+4=2, 1+8=9
Hope I helped
What does 6x − 9 = 45 equal?
Answers
Answer:
9
Step-by-step explanation:
Add nine to both sides:
\(6x = 54\)
Divide by six:
\( \frac{6x = 54}{6} \)
\(x = 9\)
Answer:
x = 9
Step-by-step explanation:
Add 9 to each side to begin simplifying. It should now look like this: 6x = 54Now, divide each side by 6 to find the value of x. It should look like this: x = 9I hope this helps!
If f(x) = 2x - 3, find the following values: a) f(0)= b) f(-2) = c) f(5)=
Answers
\(f(0) = 2 \cdot 0 - 3 = 0 - 3 = -3\\f(-2) = 2 \cdot (-2) - 3 = -4 - 3 = -7\\f(5) = 2 \cdot 5 - 3 = 10 - 3 = 7\)
Pls urgent help needed Bcz exam tmrw
Answers
The simplification of the expression 4a + 3b - a + 3b + 6 is 3(a + 2b + 2)
The expression im terms of x for the perimeter of the triangle is 31 = (3x - 5) + (2x - 1) + (x + 1)The value of x is 6How to find the perimeter of a triangle?Simplify 4a + 3b - a + 3b + 6
collect like terms
= 4a - a + 3b + 3b + 6
= 3a + 6b + 6
factorize
= 3(a + 2b + 2)
side a = 3x - 5
side b = 2x - 1
side c = x + 1
Perimeter = 31 cm
The perimeter of a triangle = side a + side b + side c
31 = (3x - 5) + (2x - 1) + (x + 1)
31 = 3x - 5 + 2x - 1 + x + 1
31 = 6x - 5
Add 5 to both sides
31 + 5 = 6x
36 = 6x
divide both sides by 6
x = 36/6
x = 6
Therefore, the value of x from the perimeter of the triangle is 6
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answer this and ill give brainliest
Answers
Answer:
-4.65< -4 and 2/5
Step-by-step explanation:
To solve, this we can convert the fraction -4 and 2/5 into a decimal so that the values can be more easily compared. When we do this, we get the result that -4 and 2/5 equals -4.4. Now we can clearly see which value is greater because they are written in the same form:
-4.65<-4.4
-4.65<-4 2/5
Answer:
<
Step-by-step explanation:
You have to convert the mixed number by multiplying the whole number by the denominator, adding the numerator, and dividing the sum by the denominator.
4x5=20
20+2=22
22÷5=4.4
-4.65 < -4.40
The real-life scenarios below represent the importance of using basic arithmetic and algebra to solve real-world quantitative problems.
Choosing a gym membership that fits my budget and lifestyle.
Gym A has a monthly payment of $50.
Gym B has a cost of $10 per visit.
Explain how you would calculate the result. You can infer and add any additional details you feel you might need to create the calculation.
Answers
To determine which gym membership is more cost-effective, you would need to compare the total cost for a given period. Let's consider a time frame of one month.
For Gym A, the monthly payment is $50. This means that regardless of how many times you visit the gym, you will pay a fixed amount of $50 each month.
For Gym B, the cost is $10 per visit. To calculate the total cost for a month, you need to estimate the number of times you plan to visit the gym in that period. Let's assume you plan to visit the gym 10 times in a month.
The total cost for Gym B would then be calculated as:
Total Cost for Gym B = Number of Visits * Cost per Visit
Total Cost for Gym B = 10 visits * $10 per visit
Total Cost for Gym B = $100
Comparing the total costs, we find that for Gym A, the monthly payment is $50, and for Gym B, the estimated cost for 10 visits is $100. In this case, Gym A would be the more cost-effective option if you plan to visit the gym at least 10 times in a month. However, if you anticipate visiting the gym less than 10 times in a month, Gym B might be the more affordable choice.
By using basic arithmetic to calculate the total costs based on the given pricing structures, you can make an informed decision regarding the gym membership that fits your budget and lifestyle.
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a circular pool is surrounded by a brick walkway 3 m wide. find the ra- dius of the pool if the area of the walk- way is 198 m*.
Answers
The radius of the pool is 9.01 m.
Given,
In the question:
A circular pool is surrounded by a brick walkway 3 m wide.
The area of the walk- way is 198 m^2.
To find the Radius of the pool.
Now, According to the question:
"Area of the circle bounded by the outside edge of the walkway" minus "area of the pool" = "area of the walkway".
Let R = Radius of the pool
Area of the circle bounded by the outside edge of the walkway is:
\(\pi\)(R +3)^2
Area of the pool is:
\(\pi R^2\)
Now, Our equation is:;
\(\pi\)(R +3)^2 - \(\pi R^2\) = 198
\(\pi\)((R+3)^2 - \(R^2\)) = 198
Open the inner bracket :
\(\pi\)(\(R^2+6R+9-R^2\)) = 198
\(\pi\)(6R +9) = 198
6R+9 = 198/\(\pi\)
6R = 198/\(\pi\) - 9
R = (198/\(\pi\) - 9)/6
R = (198/(3.14) - 9)/6
R = (63.057 - 9)/6
R = 54.057/6
R = 9.01 meters
Hence, The radius of the pool is 9.01 m.
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Given u = 144i − 17j, what are the magnitude and direction of −3u? Round to the nearest whole number.
Answers
The magnitude of the vector is 435 units.
The direction of the vector is 173.3⁰ .
What is the magnitude of the vector?
The magnitude of the vector -3u is calculated as follows;
The new vector - 3u is determined as;
u = 144i - 17j
-3u = -3(144i - 17j )
= -432i + 51j
The magnitude of the vector is calculated as;
|u| = √ (-432² + 51²)
|u| = 435 units
The direction of the vector is calculated as follows;
θ = tan⁻¹ (uy / ux)
θ = tan⁻¹ (-51/432)
θ = -6.7⁰ = 173.3⁰
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7. You are on a field trip. You have to travel 73 miles to reach
your destination. You have traveled 29 miles so far. How
much further do you have to travel?
10 marbles.
Answers
Answer:
44
Step-by-step explanation:
Take the miles you already traveled and subtract it from the total.
73-29=44
Solve 6 + 1/4a = 3 for a.
Answers
Answer:
a= -12
Step-by-step explanation:
the area of a rectangle with length 2p cm and width 6 cm is 24 cm²
Answers
Answer:
A = l × b
Step-by-step explanation: