By using the squeeze theorem, we conclude that, [tex]$$\lim _{x \rightarrow a} f(x)=b .$$[/tex]
Limit of a function, Squeeze Law:
The squeeze theorem states that if f(x) ≤ g(x) ≤ h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them. We can use the theorem to find tricky limits like sin(x)/x at x=0, by "squeezing" sin(x)/x between two nicer functions and using them to find the limit at x=0
We say that a function f(x) approaches a limit L approaches a, or, [tex]$\lim _{x \rightarrow a} f(x)=L$[/tex]
if, for every number [tex]$\epsilon > 0$[/tex], however small, such that [tex]$|f(x)-L| < \epsilon$[/tex],
there always exists a number δ>0 such that
[tex]|x-a| < \delta \text {. }[/tex]
The Squeeze Law:
Suppose that g(x) ≤ f(x) ≤h(x) in some neighborhood of a and also that
[tex]$$\lim _{x \rightarrow a} g(x)=L=\lim _{x \rightarrow a} h(x)$,[/tex]
then,
[tex]$$\lim _{x \rightarrow a} f(x)=L .$$[/tex]
We have to find the limit of f(x) as x approaches to x.
Let [tex]$b-|x-a|=g(x)$[/tex] and [tex]$b+|x-a|=h(x)$[/tex].
Then we have the inequality, g(x) ≤ f(x) ≤h(x)
Let us see what happens to g(x) and h(x) as [tex]$$x \rightarrow a$.[/tex]
[tex]$$\begin{aligned}g(x) & =b-|x-a| \\\\lim _{x \rightarrow a} g(x) & =\lim _{x \rightarrow a} b-|x-a| . \\& =b\end{aligned}$$[/tex]
Similarly,
[tex]$$\begin{aligned}h(x) & =b+|x-a| \\\lim _{x \rightarrow a} h(x) & =\lim _{x \rightarrow a} b+|x-a| . \\& =b\end{aligned}$$[/tex]
We have the conditions required by the Squeeze law,
[tex]$$\lim _{x \rightarrow a} g(x)=b=\lim _{x \rightarrow a} h(x)$$[/tex]
so we conclude that, [tex]$$\lim _{x \rightarrow a} f(x)=b .$$[/tex]
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If C is ¼ of the distance from A to D if A is (3,7) and D is (11,11) where is C?
Answer:
(8.25, 9.75)
Step-by-step explanation:
To find the point that is ¼ of the distance from A to D, we first need to find the coordinates of D. Since A is located at the point (3, 7) and D is located at the point (11, 11), we can use the distance formula to find the distance between these two points. The distance formula is given by:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) and (x2, y2) are the coordinates of the two points. In this case, we have x1 = 3, y1 = 7, x2 = 11, and y2 = 11, so the distance between A and D is:
distance = √((11 - 3)^2 + (11 - 7)^2) = √(8^2 + 4^2) = √(64 + 16) = √80 = 8.944
Now that we know the distance between A and D, we can use this information to find the point that is ¼ of the distance from A to D. Since C is ¼ of the distance from A to D, it is located ¼ of the way along the line between A and D. This means that we can find the coordinates of C by taking the average of the coordinates of A and D, weighted by ¼ and ¾, respectively. In other words, the coordinates of C are given by:
C = ¼ * A + ¾ * D
where A and D are the coordinates of points A and D, respectively. Plugging in the coordinates of A and D, we get:
C = ¼ * (3, 7) + ¾ * (11, 11) = (0.75 * 3 + 2.25 * 11, 0.75 * 7 + 2.25 * 11) = (8.25, 9.75)
Therefore, the point C is located at the coordinates (8.25, 9.75).
Will give brainliestttt PLEASE I NEED HELP :((((
Aurora is planning to participate in an event at her school's field day that requires her to complete tasks at various stations in the fastest time possible. To prepare for the event, she is practicing and keeping track of her time to complete each station.
The x-coordinate is the station number, and the y-coordinate is the time in minutes since the start of the race that she completed the task.
(1, 4)(2, 8)(3, 16)(4, 32)
Part A: Is this data modeling a linear function or an exponential function? Explain your answer.
Part B: Write a function to represent the data. Show your work. (4 points)Part C: Determine the average rate of change between station 2 and station 4. Show your work. (4 points)
Step-by-step explanation:
A
for a linear function the slope ratio has to be the same between every pair of data points.
between (1, 4) and (2, 8) we have
x changes by +1 (from 1 to 2).
y changes by +4 (from 4 to 8).
the slope is
+4/+1 = 4
between e.g. (2, 8) and (3, 16) we have
x changes by +1 (from 2 to 3).
y changes by +8 (from 8 to 16).
the slope here is +8/+1 = 8.
which is not the same as 4.
so, it is not a linear function.
it is an exponential function.
B.
the function is actually
f(x) = y = 2^(x+1)
an exponential function is of the form
y = a^x
in our case we have
4 = a¹ = 2²
8 = a² = a¹×2 = 2² × 2 = 2^(2+1)
16 = a³ = a²×2 = a¹×2×2 = 2²×2×2
we see
f(n) = a1×2^(n-1) = 2²×2^(n-1) = 2^(n+1)
C.
the average rate of change is
(the difference of the function results at the interval ends) / (the difference of the interval ends).
so, in our case
(f(4) - f(2))/(4 - 2) = (32 - 8)/2 = 24/2 = 12
the average rate of change between stations 2 and 4 is 12 (or 12/1).
Answer:
Um in an old question of mine you said "its been forever". That was my elijahmiraclechild account. I don't really remember you but I would like to
Step-by-step explanation:
The median of a boulevard i 8 feet wide and 300 feet long. What i the area of the median?
By using the formula for area of rectangle, it can be calculated that Area of the median is 2400 sq. feet
What is area of rectangle?
The area of a rectangle is calculated by multiplying the length by the width. For example, if a rectangle has a length of 10 feet and a width of 6 feet, the area would be 10 x 6 = 60 square feet. The area of a rectangle is a two-dimensional measurement and is typically expressed in units squared, such as square feet or meters squared. Other shapes, such as a triangle or a circle, also have areas and are calculated differently.
If l is the length of the rectangle and b is the breadth of the rectangle, then area of rectangle is calculated by the formula
Area = [tex]l \times b[/tex]
Length of the median = 300 feet
Width of the median = 8 feet
Area of the median = 300 [tex]\times[/tex] 8 = 2400 sq. feet
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You need to determine where to place the beams so that the chains are fastened to the rollercoaster at a height of 25 feet. Write the equation you would need to solve to find the horizontal distance each beam is from the origin. (10 points).
Using the roller coaster as an example, draw a right triangle.
The equation used to find the horizontal distance: h² = 30² - 25².16.58 feet is the horizontal distance.What are ways to arrive at the equation?25 feet is listed as the height.The answer to the whole question is that the roller coaster measures 30 feet long.Use h to symbolize the horizontal distance.
The following expression can be used to get h given that roller coaster serves as a representation of the hypotenuse side length.
30² = h² + 25².
Assemble similar phrases.
h² = 30² - 25².
horizontal separation
In (a), there is:
h² = 30² - 25².
This results:
h² = 275
Take both sides' roots.
h = 16.58
Consequently, 16.58 feet are the horizontal distance.
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find the value of x.
A rocket will launch in 3. 75 hour and the launch time i 11:20 am. What time i it now??
The time presently is 8:15 a.m, and the rocket will launch in 3.75 hours, at 11:20 a.m.
What is subtraction?Subtraction (represented by the minus sign) is one of four mathematical operations, the others being addition, multiplication, and division. Subtraction is the procedure of removing items from a collection. Subtraction may be thought of as the inverse of addition. Counting forwards helps children learn to add. To subtract, they just count backwards.
Here,
launch time=11:20 am
time left to launch=3.75 hour
=0.75*60
=45 minutes
= 3 hours 45 minutes
=11 hours 20 minutes-3 hours 45 minutes
=8 hours 15 minutes
=8:15 AM
The time right now is 8:15 AM when rocket will launch in 3. 75 hour and the launch time is 11:20 am.
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Solve the equation by subtracting the appropriate number from both sides and enter the value of x below. x + 2551 = 3105
Answer:
554
Step-by-step explanation:
x + 2551 - 2551 = 3105 - 2551
x = 554
Answer:
[tex]x + 2551 = 3105 \\ x + 2551 - 2551 = 3105 - 2551 \\ x = 554[/tex]
The expression (x + 3)(x + 2) is the product of two binomials. Which expression is also
a product of binomials?
The expression which has product of two binomials is (5-n)(n+7).
What is Expression?An expression is combination of variables, numbers and operators.
Binomials are expressions containing the sum (or difference) of two terms.
The expression (x + 3)(x + 2) is the product of two binomials.
(pq)(qp), (5-n)(n+7), 3x(2x+5), 6x-2y
(pq)(qp) is not the product of binomials because pq and qp are products.
3x(2x+5) is not the product of binomials because 3x is not a sum or difference.
6x-2y is also not because it is one binomial, not two multiplied together.
(5-n)(n+7) is the product of two binomials.
Hence, (5-n)(n+7) is the expression which has product of two binomials.
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sqrt{ 45 Multiplyed bysqrt 10
Answer:
Step-by-step explanation:
SOMEONE PLEASE HELP MEEEEEEEEEEEEEEEEEEEEEEEEEE
Answer:
y=-2/3x+4
Step-by-step explanation:
Rise/run (from the y-intercept to the next point you go down 2 and then to the right 3)
Y-intercept (where the line touches the y-axis) 4
when the spring is stretched and the distance from point a to point b is 5.3 feet, what is the value of θ to the nearest tenth of a degree?
a. 60.0
b. 35.2
c. 45.1
d. 55.5
When the spring is stretched and the distance from point a to point b is 5.3 feet, the value of θ is 53.13 degrees
The distance between point a to point b = 5.3 feet
The length of the top side = 3 feet
Therefore, it will form a right triangle
Here we have to use trigonometric function
Here adjacent side and the hypotenuse of the triangle is given
The trigonometric function that suitable for the given conditions is
cos θ = Adjacent side / Hypotenuse
Substitute the values in the equation
cos θ = 3 / 5
θ = cos^-1(3 / 5)
θ = cos^-1(0.6)
θ = 53.13 degrees
Therefore, the value of θ is 53.13 degrees
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And bottom says solve for X someone help me
Answer:
2(x+8)=126
Step-by-step explanation:
GD is the bisector of CGE and 126 and the angle CGE are equal
help plss and if u could plot the point as well thats better
TY
The linear functions defined in this problem are given as follows:
1. y = -x + 3.
2. y = x - 3.
3. y = 3x + 1.
4. y = -2x - 2.
How to define the linear functions?The slope-intercept definition of a linear function is given as follows:
y = mx + b.
The coefficients and their meaning are given as follows:
m is the slope, representing the rate of change of the output variable y relative to the input variable x.b is the y-intercept, representing the value of y when the graph of the function touches or crosses the y-axis.From the second bullet point above, the intercepts for each function in this problem are given as follows:
1. b = 3.2. b = -3.3. b = 1.4. b = -2.The slopes are given as follows:
1: m = -1, when x increases by 1, y decays by 1.2: m = 1, when x increases by 1, y increases by 1.3. m = 3, when x increases by 1, y increases by 3.4: m = -2, when x increases by 1, y decays by 2.Missing Information
The problem is incomplete, hence we just define the linear functions.
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Determine the period.
Enter
Answer:
Your answer should be [tex]\frac{\pi }{7}[/tex]
Step-by-step explanation:
Looking at the x-axis we can say that is our b.
Period Formula: 2π/b.
x-axis is from 0 to 20.
b = 14 because of the rise to rise.
2π/14 = π/7
One ector of a circle ha an area of 121 quare centimeter and an arc meaure of 36'. What i the area of the circle to the nearet
quare centimeter
The Area of the circle will be = is 144.87 [tex]cm^{2}[/tex]
Given in the question,
The sector of the circle has an area of = 121 [tex]cm^{2}[/tex]
The measure of an arc = is 36 inches.
According to the attached figure,
Let us consider the arc length = L
Let radius = r
As per the question L = 36'
Let the radius be
To calculate the length of the arc we need to use the formula,
L = r x θ
36 = r x θ
θ = [tex]\frac{36}{r}[/tex] degrees ------- equation 1
Now according to the area of the sector of the circle,
A = [tex]r^{2}[/tex] x θ / 2
121 = [tex]r^{2}[/tex] x θ / 2
[tex]r^{2}[/tex] = 2 x 121 / ( [tex]\frac{36}{r}[/tex] )
[tex]r^{2}[/tex] = [tex]\frac{2 * 121 * r}{36}[/tex] ----- equation 2
r = [tex]\frac{2 * 121 }{36}[/tex]
r = 6.72 cm.
Now if the radius is = 6.72 cm
Diameter = 2 x radius
= 2 x 6.72
= 13.44 cm
Area of the circle = [tex]\pi r^{2}[/tex]
= [tex]\pi[/tex] x 6.72 x 6.72
= 144.87 [tex]cm^{2}[/tex]
Therefore, the area of the circle = is 144.87 [tex]cm^{2}[/tex]
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The diagram below is used to sort out numbers.
Type each number in the correct location on the diagram.
3
30
333
3001
two-digit
numbers
multiples
of 3
three-digit
numbers
The solution is given as,
Two-digit numbers Multiples of 3 Three-digit numbers
30 3,30, 333, 333
The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
here,
Place the given number in the appropriate place.
Numbers are 3, 30, 333, 3001,
Two digit number among the numbers = 30
Multiples of 3 among numbers = 30, 333, 3
Three-digit numbers among the numbers = 333
Thus, It is shown that the numbers are placed in their appropriate positions.
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This week, a pack of tomato plants costs $25. Next week, the plants are on sale for 15% off.
Larissa wants to know the price of the tomato plants if she buys them next week.
How much is the discount?
Price This Week
%
$
$
Price Next Week
%
25
$
Discount
15
?
%
The discount is $3.75 and the amount next week will be $21.25
How to illustrate the percentage?This has to do with a percentage. A percentage is a value or ratio that may be stated as a fraction of 100. If we need to calculate a percentage of a number, we should divide it's entirety and then multiply it by 100.
In this case, the pack of tomato plants costs $25. Next week, the plants are on sale for 15% off. Larissa wants to know the price of the tomato plants if she buys them next week.
The discount will be:
= 15% × $25
= $3.75
The amount next week will be:
= $25 - $3.75
= $21.25
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g there are 10 clubs on campus. how many ways are there if you want to choose 3 student clubs out of 10 to join?
there are 10 clubs on campus. how many ways are there if you want to choose 3 student clubs out of 10 to join is 120
A fractional factorial experiment utilizes subset of combinations that are gotten from the full factorial experiment. They are applicable when there are too many inputs to screen practically or cost or time would be excessive.
Also, using the fractional factorial experiment makes running the experiments faster and cheaper.
10C3 = 10! / (7!*3!)
= 10*9*8/6
= 120
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the difference between the sample statistic and the population parameter when sampling is done correctly is referred to as what?
A sampling error is the difference between a population parameter and a sample statistic.
Parameters are numbers that describe the properties of entire populations. Statistics are numbers that describe the properties of samples. For example, the average income for the United States is a population parameter. Conversely, the average income for a sample drawn from the U.S. is a sample statistic. The difference between a parameter vs a statistic is that a parameter is a fixed measure describing the whole population, while a statistic is a characteristic of a sample, a portion of the target population.
Thus, the difference is defined as sampling error.
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PLS HELP WILL GIVE BRAINLIEST!
1) the slope of the line x=8 is ______
2) the domain and range of a linear function are _________
Answer:
I tried as I can to solve it
Identify the line(s) of symmetry for the quadratic function below.
Answer:
y=-3
Step-by-step explanation:
A line of symmetry is the line where a function can be folded in half evenly. Here it occurs at y=-3
line of symmetry, the line at which point the graph begins to mirror itself, Check the picture below.
Line a passes through the points (0, 4)
and (4, 0). Line b passes through the
points (0, 7) and (7, 0). Are the lines
parallel? Explain.
Answer:
Yes, the lines are parallel. This is because the lines have the same slope. The slope of a line is a measure of how steep the line is, and it is calculated by taking the difference of the y-coordinates of two points on the line and dividing it by the difference of the x-coordinates of the same two points. In this case, the slope of line a is -1, and the slope of line b is also -1. Because the lines have the same slope, they are parallel to each other.
Addilyn mow two lawn. The firt lawn i 6 meter long and 4 meter wide. The area of the econd lawn i 6 time big but ha the ame width a the firt. What i the length of the econd lawn?
The length of second lawn is found as the 36 sq. m.
Explain the term area of the rectangle?The area a rectangle occupies is the space it takes up inside the limitations of its four sides. The dimensions of a rectangle determine its area.In essence, the area of a rectangle is equal to the sum of its length and breadth.For the stated question-
First lawn:
Length = 6 mWidth = 4 mArea = length x width
A = 6 x 4 = 24 sq. m.
Second lawn:
Length = LWidth = 4 mArea A2 = 6 x Area of first lawnA2 = length x width
6 x A = L x 4
6 x 24 = L x 4
L = 6 x 6
L = 36 sq. m
Thus, the length of the second lawn is found as the 36 sq. m.
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Answer:
Step-by-step explanation:
36 is the answer
if there are 720 different classes in one quarter, each in its own room, how many different rooms will be required?
if there are 720 different classes in one quarter at time, each in its own room, different rooms will be required 720 rooms.
To answer this question, we need to understand that each class needs its own room. Therefore, the number of rooms required will be equal to the number of classes. In this case, there are 720 different classes in one quarter, so 720 different rooms will be required. Each room can accommodate one class at a time, so the total number of rooms needed is 720.
Number of Classes = 720
Number of Rooms Required = Number of Classes = 720
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The basketball team scored a total of 109 points last game. They made a total of 47 shots, including 2-point shots and 3-point shots. how many 2-point shots did they make? How many 3-points did they make?
[please include steps]
Based on the above, the number of 3-points that they make is 25.
What is the Simultaneous equation about?Simultaneous equation is a form of two equations that have to be solved at the same time. Thus, the number of 2 points made is 17, while that of 3 points is 25.
A set of equations that has to be solved simultaneously so as to determine the values of the unknown variables is termed simultaneous equation.
The given question would result to a simultaneous equation as follows;
let the number of 2 points made by represented by m, and that of 3 points by n.So that;
m + n = 42 ..................
i2m + 3n = 109 ...............
ii multiply equation 1 by 3 and equation 2 by 1 to have;3m + 3n = 126 ............
iii 2m + 3n = 109 .............
iv subtract eqaution iv from iii
m = 17
substitute the value of m in equation 1 to have:
m + n = 4217 + n = 42n = 42 - 17 = 25m = 17, n = 25
Thus the number of 2 points made was 17, while that of 3 points made was 25.
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A water coller filll 150 glae in 30 minute how many glae of water can the coller filll in one minute
Five water glasses each minute, one minute to fill the cooler.
What is unitary method?The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units. what kinds of values and units
Let's say you go to the store to buy six apples.
You are informed by the shopkeeper that he is offering 10 apples for Rs 100. In this instance, the value and the units are the price of the apples. Recognizing the units and values is crucial when using the unitary technique to a problem.
Always write the items that need to be computed on the right side and the things that are known on the left side to simplify things.
We are aware of the quantity of apples and the amount of money in the aforesaid problem.
According to our question-
divide 150 by 30 equal 5
= 5
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Seth is comparing scenarios to determine which starting point he should get dropped off at to get him to school faster. Seth's house is 17 miles from school. In which scenario does seth start closer to school? in which scenario does seth travel at a greater speed, and what was the rate?.
In linear equation, He travels at a greater speed from his friend's house, at a rate of 0.6667 miles per minute, that is, 2 miles in 3 minutes.
What in mathematics is a linear equation?
A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept.
Sometimes, the aforementioned is referred to as a "linear equation of two variables," where x and y are the variables.
Using linear function concepts, it is found that:
Seth starts closer to the school in the bus.
He travels at a greater speed from his friend's house, at a rate of 0.6667 miles per minute, that is, 2 miles in 3 minutes.
From the graph of the bus, when he takes the bus, he is 13 miles from the school, as he starts at position 4 and ends at 17.
At his friend house, he is 13 miles from the school only after 3 minutes, which means that the bus will be closer.
The scenario with the greater speed is the one with the higher slope, that is, higher rate of change, which is the change in distance divided by the change in time.
In the bus, change of 13 miles in 24 minutes, thus
mᵇ = 13/24 = 0.5417
From the friend's house, change of 2 miles in 3 minutes, thus
mf = 2/3 = 0.6667
Thus, he travels at a greater speed from his friend's house, at a rate of 0.6667 miles per minute, that is, 2 miles in 3 minutes.
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I need to know the question to a problem 10”=14,521
Answer:
The value of n in this equation is 5.
Step-by-step explanation:
This can be determined by using the property of exponents that states that x^n * x^m = x^(n+m). In this case, 10^n * 10^5 = 14,521, so 10^(n+5) = 14,521. Solving for n, we get n = 5.
The domain of a certain exponential function is all real numbers. The range of the function is all real
numbers greater than -3. The graph of the function has a horizontal asymptote at y = -3.
Select a graph or graphs to fit the description of the function.
One exponential function with the given features is y = 2^x - 3, and is graphed at the end of the answer.
How to define the exponential function?An exponential function is defined as follows:
y = a(b)^x + c.
In which:
a is the initial value.b is the rate of change.c is the horizontal asymptote.The graph of the function has a horizontal asymptote at y = -3, hence:
c = -3.
For a and b, they can assume any non-negative values, hence we attribute these following values:
a = 1, b = 2.
The graph is given at the end of the answer.
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Question 4
The weights for 12-month-old baby boys are normally distributed with a mean of 22.5 pounds and a standard deviation of 2.2 pounds.
Find the percentage to the nearest tenth of 12-month-old baby boys who weigh between 19.7 and 24.4 pounds.
ANSWER -
Answer:
70.5%
Step-by-step explanation:
If a continuous random variable X is normally distributed with mean μ and variance σ², it is written as:
[tex]\large\boxed{X \sim\text{N}(\mu,\sigma^2)}[/tex]
Given:
Mean μ = 22.5Standard deviation σ = 2.2Therefore, if the weights of the 12-month-old baby boys are normally distributed:
[tex]\boxed{X \sim\text{N} \left(22.5,2.2^2 \right)}[/tex]
where X is the weight of the baby boy.
To find the percentage to the nearest tenth of 12-month-old baby boys who weigh between 19.7 and 24.4 pounds find P(19.7 ≤ X ≤ 24.4).
Calculator input for "normal cumulative distribution function (cdf)":
Upper bound: x = 19.7Lower bound: x = 24.4σ = 2.2μ = 22.5[tex]\implies \text{P}(19.7 \leq X \leq 24.4)=0.704548741[/tex]
[tex]\implies \text{P}(19.7 \leq X \leq 24.4)=70.5\%[/tex]
Therefore, the percentage of 12-month-old baby boys who weigh between 19.7 and 24.4 pounds is 70.5% (nearest tenth).