First, [tex]f(x)[/tex] is continuous on its domain, so
[tex]\displaystyle \lim_{x\to\alpha^+} f(g(x)) = f\left(\lim_{x\to\alpha^+} g(x)\right) \\\\ ~~~~~~~~~~~~~~~~~~ = \sin\left(\frac\pi6 \lim_{x\to\alpha^+} \frac{\ln\left(\sqrt x - \sqrt\alpha\right)}{\ln\left(e^{\sqrt x} - e^{\sqrt\alpha}\right)}\right)[/tex]
As [tex]x\to\alpha^+[/tex], [tex]\sqrt x-\sqrt\alpha\to0[/tex] and [tex]e^{\sqrt x}-e^{\sqrt\alpha}\to0[/tex], so overall we have an indeterminate form ∞/∞. Apply l'Hôpital's rule and simplify.
[tex]\displaystyle \lim_{x\to\alpha^+} f(g(x)) = \sin\left(\frac\pi6 \lim_{x\to\alpha^+} \frac{\frac1{2\sqrt x\left(\sqrt x - \sqrt\alpha\right)}}{\frac{e^{\sqrt x}}{2\sqrt x\left(e^{\sqrt x} - e^{\sqrt\alpha}\right)}}\right) \\\\ ~~~~~~~~~~~~~~~~~~ = \sin\left(-\frac\pi6 \lim_{x\to\alpha^+} \frac{e^{-(\sqrt x-\sqrt\alpha)} - 1}{\sqrt x - \sqrt\alpha}\right)[/tex]
Substitute [tex]y=\sqrt x-\sqrt\alpha[/tex], so that [tex]x\to\alpha^+\implies y\to0^+[/tex].
[tex]\displaystyle \lim_{x\to\alpha^+} f(g(x)) = \sin\left(-\frac\pi6 \lim_{y\to0^+} \frac{e^{-y} - 1}y\right)[/tex]
The remaining limit is the right-derivative of [tex]e^{-y}[/tex] at [tex]y=0[/tex], so
[tex]\displaystyle \lim_{x\to\alpha^+} f(g(x)) = \sin\left(-\frac\pi6\frac{de^{-y}}{dy}\bigg|_{y\to0^+}\right) \\\\ ~~~~~~~~~~~~~~~~~~ = \sin\left(\frac\pi6\right) = \boxed{\frac12}[/tex]
Find the distance between the points A (13, 2) and B (7, 10).
Answer:
[tex]\sqrt{89}[/tex]units
Step-by-step explanation:
[tex]\sqrt{(7-13)^2+(10-2)^2}[/tex]
(-5)^2+(8)^2=25+64
25+64=89
[tex]\sqrt{89}[/tex]
the answer would be [tex]\sqrt{89}[/tex]
edie earned a gross pay of $927.88 last week. determine the amount that edie contributed to both social security and medicare
Based on the gross pay to Edie of $927.88, the amount contributed by Edie Social Security was $57.53 and the amount to Medicare was $13.45
How to find contribution to social security?The current Social Security tax rate is 6.2% which means that to find the Social Security tax that Edie will contribute will be:
= Social security tax rate x Gross pay of Edie
= 927.88 x 6.2%
= $57.52856
= $57.53
The Medicare tax rate is 1.45%. The Medicare tax can therefore be found by multiplying the gross pay by the Medicare tax rate:
= 927.88 x 1.45%
= $13.45
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Follow the guided instructions below to create the line of reflection that would map the pink figure onto the blue figure.
Answer:
Slope of one of the dotted lines:
[tex]y=x[/tex] or [tex]y=x+5[/tex]
Slope of the line of reflection:
[tex]y=-x[/tex]
Step-by-step explanation:
When seeing the dotted lines, both of them have a rise & run of 1, indicating that y = x. However, one of the lines has a y-intercept of 5, making it y = x + 5. So, the slopes for the dotted lines are [tex]y=x[/tex] or [tex]y=x+5[/tex].
The line of reflection has a rise & run of -1, and goes through the origin, indicating that the equation is y = -x.
Part A
The cost of having one pizza delivered is $14. Each additional pizza costs $9 more. Choose the explicit formula to represent the situation.
A. an = 14 + 9(n + 1)
B. an = 5 + 9n
C. an = 14 + 9n
D. an = 5 + 9(n + 1)
Part B
The cost of having one pizza delivered is $14. Each additional pizza costs $9 more. Choose the recursive formula to represent the situation.
A. an = an – 1 + 5; a1 = 14
B. an = an – 1 + 23; a1 = 9
C. an = an – 1 + 14; a1 = 9
D. an = an – 1 + 9; a1 = 14
Answer:
C. an = 14 + 9n
Step-by-step explanation:
The cost of having one pizza delivered is $14. Each additional pizza costs $9 more.
Let n= additional pizza
an = 14 + (9 × n)
an = 14 + 9n
Question 2 (1 point)
To find which number is one thousand times larger than 2.3 x 10^2, you would do the following steps.
O a
O b
O c
Od
Change 1,000 to Scientific notation and then add to (2.3 x 10^2)
Change 1,000 to Scientific notation and then multiply by (2.3 x 10^6)
Change 1,000 to Scientific notation and then divide by (2.3 x 10^6)
Change 1,000 to Scientific notation and then subtract from (2.3 x 10^6)
The correct answer choice as required in the task content is; Choice B; Change 1,000 to Scientific notation and then multiply by (2.3 x 10^6).
To find which number is one thousand times larger than 2.3 x 10², Change 1,000 to Scientific notation and then multiply by (2.3 x 10²).
Scientific notationThis is a method of writing or of displaying real numbers as a decimal number between 1 and 10 followed by an integer power of 10.
which number is one thousand times larger than 2.3 x 10²
Change 1,000 to Scientific notation and then multiply by (2.3 x 10²)1000 to scientific notation
= 10³
2.3 x 10² × 1000
= 2.3 x 10² × 10³
= 2.3 × 10^5
Furthermore, scientific notation helps us to express numbers in decimal number between 1 and 10 followed by an integer power of 10. It also help us to represent numbers in this pattern.
Therefore, to find which number is one thousand times larger than 2.3 x 10², Change 1,000 to Scientific notation and then multiply by (2.3 x 10²).
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PLEASE HELP FAST I ONLY HAVE ONE HOUR!!!!!!
FG is perpendicular to DM, if DF = 4R, FM = r + 21, and DF = FM, what is the value of R?
Answer:
(r+21 ) / 4
Step-by-step explanation:
as we can see DF = FM
and DF = 4R , FM =r+21
so,
4R = r+21
=> R= (r+21)/4
The cost to rent a storage unit is different for two storage companies. Stack n’ Store charges $150 per month, plus a non refundable deposit of $175. Smart Storage charges $185 each month with no deposit. Complete the table showing the total cost for each company during the first six months and use it to answer the questions.
Lee plans to store clothes for four months. Which company offers the better deal?
After how many months is the total cost the same at both storage companies?
Month Cost at Stack n’ Store Cost at Smart Storage
1 $325 $185
2 $475 $370
3 $625 $555
4 $775 $740
5 $925 $925
6 $1075 $1100
The company that offers the better deal is Smart Storage.
After 5 months the cost would be the same at both storage companies.
What are the total costs?The equation that represent the total cost of renting at Stack n’ Store is:
Total cost = non refundable deposit + charge per month
Total cost in the first month : $175 x ($150 x 1 ) = $325
Total cost in the second month : $175 x ($150 x 2 ) = $475
Total cost in the third month : $175 x ($150 x 3 ) = $625
Total cost in the fourth month : $175 x ($150 x 4) = $775
Total cost in the fifth month : $175 x ($150 x 5) = $925
Total cost in the sixth month : $175 x ($150 x 6) = $1075
The equation that represent the total cost of renting at Smart Storage is:
Total cost = cost per month x number of months
Total cost in the first month : $185 x 1 = $185
Total cost in the second month : $185 x 2 = $370
Total cost in the third month : $185 x 3 = $555
Total cost in the fourth month : $185 x 4 = $740
Total cost in the fifth month : $185 x 5 = $925
Total cost in the sixth month : $185 x 6 = $1,110
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please help with this! i am stuck
The required simplification of the given expression [tex]=\lim_{x \to \infty} \frac{x^2-4x +4}{x^2-4}[/tex] is 1.
Given that,
To determine the simplified value of the expression [tex]=\lim_{x \to \infty} \frac{x^2-4x +4}{x^2-4}[/tex].
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,
Given expression,
[tex]=\lim_{x \to \infty} \frac{x^2-4x +4}{x^2-4}[/tex]
[tex]=\lim_{x \to \infty} \frac{x^2(1-4/x +4/x^2)}{x^2(1-4/x^2)}[/tex]
Now put limits, so 1 / x², 1 / x becomes zero so,
= 1
Thus, the required simplification of the given expression is 1.
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HELP Trapezoid A'B'C'D' is the image of trapezoid ABCD under a rigid transformation.
The Image of the trapezoid under rigid transformation is given by A'B'C'D' attached below.
The unique process of changing an existing graph or graphed equation to create a completely different version of the given graph is known as graph transformation.
Without changing the figure, a transformation shifts it up, down, to the right, to the left, or diagonally.Rotation is the term used to describe the transformation of a figure around a specific point. A fixed point is used to rotate the figure either clockwise or counterclockwise.In the given figure we can see that the Trapezoid ABCD is rotated around a fixed point to get the image of the trapezoid A'B'C'D'.
So the figure gets a rotational transformation.
From the figure we can assume that the figure is rotated about 135° clockwise about a fixed point.
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How many times larger is 5.025 x 1010 than 1.5 x 108?
335
299
29.9
3.35
[tex]\frac{5.025 \times 10^{10}}{1.5 \times 10^{8}} \\ \\ =\frac{5.025}{1.5}(100) \\ \\ =\boxed{335}[/tex]
Option A is correct, 5.025 x 10¹⁰ is 335 times larger than 1.5 x 10⁸.
To determine how many times larger one number is than another, we divide the larger number by the smaller number.
In this case, we want to find how many times larger 5.025 x 10¹⁰ is than 1.5 x 10⁸.
Dividing the larger number by the smaller number:
(5.025 x 10¹⁰) / (1.5 x 10⁸)
To simplify, we can divide the coefficients (5.025 / 1.5) and subtract the exponents (10¹⁰ - 10⁸):
(5.025 / 1.5) x 10¹⁰⁻⁸
Evaluating the division of the coefficients:
(3.35) x 10²
Therefore, 5.025 x 10¹⁰ is 3.35 x 10² times larger than 1.5 x 10⁸.
In decimal form, this can be expressed as:
3.35 × 10² = 335
Therefore, 5.025 x 10¹⁰ is 335 times larger than 1.5 x 10⁸.
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a contractor purchases 3 dozen pairs of padded work gloves for $44.28. she incorrectly calculates the unit price as $14.76 per pair of the expense report. what is the correct unit price? what is the error?
The unit price of the pairs of padded will be $1.23 per pair. The error is the price of the pairs of pads should have been divided by 12.
What is the value of the expression?When the relevant factors and natural laws of a mathematical model are given values, the outcome of the calculation it describes is the expression's outcome.
For $44.28 a contractor buys 3 dozen pairs of cushioned work gloves.
The unit price of the pairs of padded will be given as,
⇒ $44.28 / (12 x 3)
⇒ $1.23 per pair
The unit price of the pairs of padded will be $1.23 per pair.
The error is the price of the pairs of pads should have been divided by 12.
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Solve these questions
1) x +2x + x = 68
2) x+y+3y= 45
3) x+y-4z= -32
4) x+y+z= ?
5) -x + 8x= 63
6) 4x+3x-x=42
Same questions as the pic
Answer:
here is the answers
Step-by-step explanation:
sample:
1. 4x = 68
x = 68/4
x = 17
Solution :
1) x + 2x + x = 68
>> 4x = 68
>> x = 68 / 4
>> x = 17
2) x + y + 3y = 45
>> x + 4y = 45
>> 17 + 4y = 45
>> 4y = 45 - 17
>> 4y = 28
>> y = 28 / 4
>> y = 7
3) x + y - 4z = -32
>> 17 + 7 - 4z = -32
>> 24 - 4z = -32
>> -4z = -32 - 24
>> -4z = -56
>> z = 56 / 4
>> z = 14
4) x + y + z = ?
>> 17 + 7 + 14
>> 24 + 14
>> 38
5) -x + 8x = 63
>> 7x = 63
>> x = 63 / 7
>> x = 9
6) 4x + 3x - x = 42
>> 7x - x = 42
>> 6x = 42
>> x = 42 / 6
>> x = 7
ILL GIVE BRANNIEST TO THE CORRECT ANSWER Find the coordinates of the point
7/10 of the way from A to B.
Answer: X = 8.4 Y= 3.5
Step-by-step explanation:
How much is 2 tablets 3 x per day for 14 days
Answer:
84 tablets with 6 per day
If f(1)=6f(1)=6 and f(n)=f(n-1)-2f(n)=f(n−1)−2 then find the value of f(6)
The value of function f(n) = f(n - 1) - 2, at n = 6 is f(6) = -4
In this question,
we have been given a function f(n) = f(n - 1) - 2
Also, we have been given the value of function f(n) for n = 1
i.e., f(1) = 6
We need to find the value of f(6).
For n = 2,
f(2) = f(2 - 1) - 2
f(2) = f(1) - 2
f(2) = 6 - 2
f(2) = 4
For n = 3,
f(3) = f(3 - 1) - 2
f(3) = f(2) - 2
f(3) = 4 - 2
f(3) = 2
For n = 4,
f(4) = f(4 - 1) - 2
f(4) = f(3) - 2
f(4) = 2 - 2
f(4) = 0
For n = 5,
f(5) = f(5 - 1) - 2
f(5) = f(4) - 2
f(5) = 0 - 2
f(5) = -2
For n = 6,
f(6) = f(6 - 1) - 2
f(6) = f(5) - 2
f(6) = -2 - 2
f(6) = -4
Therefore, the value of function f(n) = f(n - 1) - 2, at n = 6 is f(6) = -4
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Express the given function
Answer:
f(x) = [tex]{\displaystyle {\sqrt[8]{x}}}[/tex]
Step-by-step explanation:
If we let
[tex]g(x) = x^2 - 3\\\\\rm Then \;f(g(x) \;also \;written\; as\; (f\;\circ \rm \;g) \;is\; f(x^2 - 3)\\\\For\; this \;to \;be\; \sqrt[8]{x^2-3} , \\\\\rm f(x) \;has\;to\;be\;\sqrt[8]{x}[/tex]
4. Given 6 (1-x) what is the value of the
expression when x = -8?
Answer:
let x be -8
6(1-8)
6(7)
6x7=42
answer=42
I need help with this question
Answer:
Step One: Evaluate the Function for A and B
f(1) = -6
f(2) = -1.5
Step Two: Use the Average Rate of Change Formula
f(b) - f(a) / b - a
-1.5 - -6 / 2 - 1
4.5 / 1
4.5
The average rate of change is equal to 4.5
In short, use
[tex] \frac{f(2) -f(1) }{2 - 1} [/tex]
Step-by-step explanation:
To explain, substitute 1 and 2 for f(x) then divide the difference of the two answers by the difference of the two x coordinates above to find the average rate of change
i need help for a good grade thank youuu
Answer:
Here us the answer in the attachment
Answer:
the person above is correct
Step- by-step explanation:
K
Convert the angle to a decimal in degrees.
16°43'27"
The value of the angle 16°43'27"is approximately 16.724167 degrees.
A plane angle is measured in degrees, with the letter ° typically used to indicate that one full rotation is equal to 360 degrees.
The radian is the SI unit for measuring angles.1 degree = [tex]\frac{\pi}{180}[/tex] radiansA complete rotation around a circle is 180°.A degree is a narrow enough angle for many practical tasks where full degrees are enough for precision. In those instances when this is not the case, such as in astronomy or for geographic coordinates (latitude and longitude), degree measurements may be expressed in decimal degrees.To convert a degree from minutes and seconds to decimal we must know the relation between degree and minutes and seconds.
1 degree=60 minutes.
1 minute = 60 seconds.
The given angle is 16°43'27"
27 seconds = 27 ÷ 60 minutes = 0.45 minutes = 0.45 ÷ 60 degree =0.0075 degree
43 minutes= 43 ÷ 60 degree = 0.71666666..
Total angle in decimal = 16+0.71666666+0.0075 =16.7241666...
Therefore required angle ≈ 16.7241667 degrees.
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These questions are about calculus, I would greatly appreciate your help on both questions!
asap, please!
For the function f(x) =(4 sin x)/(1+cosx) , then f'(x) = 4/(1+cosx) using quotient rule of differentiation and f'(2) = 2.00
What is quotient rule for differentiation?Quotient rule for differentiation states that the differentiation the division of two functions u, v is given as
(u/v)' = (u' v - u v')/v² , where ' represents first derivative with respect to independent variable
Given f(x) = (4sinx)/(1+cosx)
let u =4 sin x , v = 1+cos x , u'=4cos x , v' = -sin x
Then f(x) =u / v
Using quotient rule of differentiation:
f'(x) = ( u' v -u v' )/v² = (4cosx(1+cosx) +4sin²x )/(1+cos x)²
f'(x) = = 4(cos x +cos² x +sin² x)/(1+ cos x)² = 4(cos x +1)/(1+ cos x)²
f'(x) = 4/(1+ cos x)
To find value of f'(x) at x= 2
f'(5) = 4/(1+cos2) = 2.000609= 2.00(rounding off )
Therefore if f(x) = (4x² tan x)/sec x then f'(x) = 4/(1+cos x) and f'(2) =2.00
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Solve for n.
n+m= 16
Answer:n=16-m
Step-by-step explanation:
n+m=16
Subtract m from both sides.
n=16−m
3. Jerry purchased two cans of paint. He used 6/4 of it. How much did Jerry have
used in decimal?
A. 6.40
B. 4.60
C. 1.50
D. 0.67
The amount used in decimal is 1.50
How to determine the amount used?The given parameters are
Cans = 2
Amount used = 6/4
The amount used in decimal is calculated as:
Amount used = 6/4
Divide 6 by 4
So, we have
Amount used = 1.50
Hence, the amount used in decimal is 1.50
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Which of the following equations has no solutions?
4x - 4 = -4(-x + 1)
-9x + 2 = -3(3x + 2)
-6x + 1 = -3(2x + 1) + x
-5 + 14x = 7(2x) - 5
Answer:
-9x + 2 = -3(3x + 2).
Step-by-step explanation:
-9x + 2 = -3(3x + 2)
-9x + 2 = -9x - 6
0 = -8, which is absurd, so :-
No solutions.
Write equation in slope-intercept form (9,-5) slope -4/3
The equation of the line in slope intercept form is y = (-4/3)x + 7.
According to the slope intercept form, the equation of a line is given as-
y = mx + c
where m is the slope and c is the y intercept.
Here we are given that an equation passes through the point (9,-5) and has a slope of -4/3.
According to the point slope form, the equation of a line passing through (x1, y1) and slope m is given as-
(y - y1) = m(x - x1)
Here,
x1 = 9, y1 = -5 and m = -4/3
Thus,
(y + 5) = -4/3(x - 9)
3(y + 5) = -4(x - 9)
3y + 15 = -4x + 36
3y = -4x + 21
y = (-4/3)x + 21/3
y = (-4/3)x + 7
This is our equation in slope intercept format as m = -4/3 and c = 7
Thus, the equation of the line in slope intercept form is y = (-4/3)x + 7.
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Help asap please!! I don’t understand
Answer: 2+[tex]\frac{6x-2}{x(x-3)}[/tex] ==> x[tex]\neq[/tex]0, x[tex]\neq[/tex]3
Step-by-step explanation:
f(x)=2x^2-2
g(x)=x^2-3x
(f/g)(x)=
[tex]\frac{2x^2-2}{x^2-3x}[/tex]=
[tex]\frac{2x^2-6x+6x-2}{x^2-3x}[/tex]=
[tex]\frac{2x^2-6x+6x-2}{x^2-3x}[/tex]=
2+[tex]\frac{6x-2}{x^2-3x}[/tex]=2+[tex]\frac{6x-2}{x(x-3)}[/tex] ==> x[tex]\neq[/tex]0, x[tex]\neq[/tex]3
Find all the solutions to the equation 4x3 + 16x2 + 28x = 0. 0, negative 4 + 2 i StartRoot 3 EndRoot, negative 4 negative 2 i StartRoot 3 EndRoot 0, 4 + 2 i StartRoot 3 EndRoot, 4 minus 2 i StartRoot 3 EndRoot 0, negative 2 + i StartRoot 3 EndRoot, negative 2 minus i StartRoot 3 EndRoot 0, 2 + StartRoot 3 EndRoot, 2 minus i StartRoot 3 EndRoot
The required solution to the equation 4x^3 + 16x^2 + 28x = 0 is 0, -2 + i√3, -2 - i√3
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions by connecting them with the equal sign = .
Now the given equation is,
4x^3 + 16x^2 + 28x = 0
Taking x common we get,
x(4x^2 + 16x + 28) = 0
So, we get
x = 0
And 4x^2 + 16x + 28 = 0
or, x^2 + 4x + 7 = 0
Thus from the above equation is of the form
ax^2 + bx + c = 0
where, a = 1, b = 4 and c = 7
Therefore solution of the above equation is given as,
x = [-b ± √(b^2 - 4ac)] / 2a
Putting the values we get,
x = [-4 ± √(4^2 - 4*1*7)] / 2*1
so, x = [-4 ± √(16 - 28)] / 2
x = [-4 ± √(- 12)] / 2
x = [-4 ± 2√(- 3)] / 2
or, x = [-2 ± √(- 3)]
Since √-1 = i
∴ √(- 3) = i√3
Thus,
x = -2 ± i√3
So, the required values of x are 0, -2 + i√3, -2 - i√3.
Thus, the required solution to the equation 4x^3 + 16x^2 + 28x = 0 is 0, -2 + i√3, -2 - i√3.
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y = x – 2
y = –3x – 2
Answer:
y=3x-2
Step-by-step explanation:
if for y=-3x-2 is being solved for x your answer would be:
y=3x-2
3x-2=y
3x=y+2
3x/3=y+2/3
so your final answer for this question is: x=y+2/3
and for the first one y=x-2 there is no answer for it
The total attendance at a science museum is modeled by the function f(x)=17,371(1.041)x. If the initial total attendance (that is, the total attendance when x=0) was measured January 1, 2013, what was the initial total attendance?
The initial total attendance at a science museum is modeled by the function f(x)=17,371(1.041)x will be 17371.
How to calculate the value?It should be noted that from the information, the total attendance at a science museum is modeled by the function
In this case, the total attendance when x=0 will be:
f(x) = 17,371(1.041)x.
f(x) = 17,371(1.041^0)
= 17371 × 1
= 17371
Therefore, the total attendance when x = 0 will be 17371.
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