Take an arbitrary interval [a, b], where a < b.
Compute the arc length L of y = f(x) over [a, b] :
[tex]L=\displaystyle\int_a^b\sqrt{1+\left(f'(x)\right)^2}\,\mathrm dx[/tex]
Now comptue the area A under the curve y = f(x) over [a, b] :
[tex]A=\displaystyle\int_a^bf(x)\,\mathrm dx[/tex]
We have
f (x) = 1/4 e ⁻ˣ + e ˣ → f ' (x) = -1/4 e ⁻ˣ + e ˣ
Then
√(1 + (f ' (x))²) = √(1 + (-1/4 e ⁻ˣ + e ˣ)²)
… = √(1 + 1/16 e ⁻²ˣ - 1/2 + e ²ˣ)
… = √(1/16 e ⁻²ˣ + 1/2 + e ²ˣ)
… = 1/4 √(e ⁻²ˣ + 8 + 16e ²ˣ)
… = 1/4 √((e ⁻ˣ + 4 e ˣ)²)
… = 1/4 (e ⁻ˣ + 4 e ˣ)
… = 1/4 e ⁻ˣ + e ˣ
… = f (x)
so both A = L for any choice of interval [a, b].
It is true that the arc length on any interval has the same value as the area under the curve.
How to prove the statementThe function is given as:
[tex]f(x) = \frac 14e^{-x} + e^x[/tex]
Differentiate the function
[tex]f'(x) = -\frac 14e^{-x} + e^x[/tex]
On any interval, the following must be true
[tex]f(x) =f'(x)[/tex]
and
[tex]f(x) = \sqrt{1 + (f'(x))^2}[/tex]
So, we have:
[tex]f(x) = \sqrt{1 + (-\frac 14e^{-x} + e^x)^2}[/tex]
Expand the exponents
[tex]f(x) = \sqrt{1 + (\frac{1}{16}e^{-2x} - \frac 12 + e^{2x})}[/tex]
Remove the bracket
[tex]f(x) = \sqrt{1 + \frac{1}{16}e^{-2x} - \frac 12 + e^{2x}}[/tex]
Evaluate the like terms
[tex]f(x) = \sqrt{\frac{1}{16}e^{-2x} + \frac 12 + e^{2x}}[/tex]
Multiply by 16/16
[tex]f(x) = \sqrt{\frac{16}{16}(\frac{1}{16}e^{-2x} + \frac 12 + e^{2x})}[/tex]
So, we have:
[tex]f(x) = \sqrt{\frac{1}{16}(e^{-2x} + 8 + 16e^{2x})}[/tex]
Take the square root of 1/16
[tex]f(x) = \frac{1}{4}\sqrt{e^{-2x} + 8 + 16e^{2x}}[/tex]
Express the radical as a perfect square
[tex]f(x) = \frac{1}{4}\sqrt{(e^{-x} + 4e^{x})^2}[/tex]
Evaluate the exponents
[tex]f(x) = \frac{1}{4} * (e^{-x} + 4e^{x})[/tex]
Evaluate the products
[tex]f(x) = \frac{1}{4}e^{-x} + e^{x}[/tex]
Hence, it has been proved that the arc length on any interval has the same value as the area under the curve.
Read more about areas at:
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Can somebody please help me!!!!!
Answer:
Varies
Step-by-step explanation:
What this question is asking you to do is to measure the angle of the blue dot and find the 90 degree angle, so you need to get a protractor and measure the 90 degree angle in order to get the answer
Answer:
Step-by-step explanation:
Please help I’ll mark you as brainliest if correct
Answer:
Y= 250x+45
Step-by-step explanation:
answer = 250 per hour (X) + 45
write an ordered pair corresponding to the point
Answer:
(-3,-1)
Step-by-step explanation:
go back 3 for x
and down 1 for y
Which of the following equations will produce the graph shown below?
Answer:
C
Step-by-step explanation:
don't ask why, i used the goegebra app, and wrote each one in it
Ringo deposited some money into a savings account. He earned 4% interest. After 6 months he earned $21 in interest. How much money did he start with?
I NEED HELP PLEASE!!
Answer:
$1050
Step-by-step explanation:
I = PRT / 100
I = 21
P = ?
R = 4%
T = 6 months = 0.5 yr
P = 100I / RT
P = (100 * 21) / (4 * 0.5)
P = $1050
Solve and graph 4<-1/3n
Answer:
n<-12
Step-by-step explanation:
4<-1/3n
4<-n/3
4×3<-n
12<-n
-12<n
n<-12
Find the area of the composite figure.
Round to the nearest hundredth.
(10 Points)
20.9m
14.3m
22.7 m
Answer:
474.43 m²
Step-by-step explanation:
The shape is composed of a rectangle and a parallelogram
Area of the composite shape = area of the rectangle + area of the parallelogram
= Length*Width + base*height
Where,
Length = 20.9 m
Width = 14.3 m
base = 20.9 m
height = 22.7 - 14.3 = 8.4 m
Area = 20.9*14.3 + 20.9*8.4
Area = 298.87 + 175.56
= 474.43 m²
Read the description of a proportional relationship.
Ezra is finding the perimeter of different-sized regular pentagons. There is a proportional relationship between the side length of the regular pentagon in inches, x, and the perimeter of the regular pentagon in inches, y.
A regular pentagon with 1-inch side lengths has a perimeter of 5 inches. Write the equation for the relationship between x and y.
y=
Answer:
Step-by-step explanation:
The answer is 15
the equation for the relationship between x and y is,
y = 5x
What is perimeter?A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical applications.
Perimeter refers to the total outside length of an object,
The equation given in the problem is
y = 5x
where
the side length of the regular pentagon in inches, x,
the perimeter of the regular pentagon in inches, y
Hence, the equation for the relationship between x and y is,
y = 5x
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#SPJ2
What is the m
Need help please!
9514 1404 393
Answer:
27
Step-by-step explanation:
The measure of an inscribed angle is half the measure of the arc it subtends.
∠P = (RT)/2
∠P = 54/2
∠P = 27
Estimate 85.77-62.396 by first rounding each number to the nearest whole number
Answer:
24
Step-by-step explanation:
85.77-62.396 estimated by rounding each number to the nearest whole number is 24. I got this by first rounding 85.77 to 86. I then rounded 62.396 to 62. Next, I did 86-62= 24.
So, the answer is 24.
Answer:
23.000
Step-by-step explanation:
8-6=2
5-2=3
So you know that 23.000 is the roaded answer, you can just ignore the digits behind the decimal point in order to round
Vending machines can be adjusted to reject coins above and below certain weights. The weights of legal U.S. quarters have a normal distribution with a mean of 5.67 grams and a standard deviation of 0.07 gram. If a vending machine is adjusted to reject quarters that weigh more than 5.81 grams and less than 5.53 grams, what percentage of legal quarters will be rejected by the vending machine
Answer: The percentage of legal quarters will be rejected by the vending machine = 2.275%
Step-by-step explanation:
Given: The weights of legal U.S. quarters have a normal distribution with a mean of 5.67 grams and a standard deviation of 0.07 gram.
Let x be the weights of legal U.S. quarters .
Required probability: [tex]P(x<5.53)+P(x>5.81)[/tex]
[tex]=P(\dfrac{x-\mu}{\sigma}<\dfrac{5.33-5.67}{0.07})+P(\dfrac{x-\mu}{\sigma}>\dfrac{5.81-5.67}{0.07})\\\\=P(z<-4.857)+P(z>2)\approx0.02275 \ \ \ [By\ P-value\ calculator][/tex]
The percentage of legal quarters will be rejected by the vending machine = 2.275%
Please help.
You will get brainliest.
Answer:
a
Step-by-step explanation:
1.what is the place value of 3 in 17.32
2.write as decimal 6+4/10
Answer:
1. Tenth.
2. 6+ 0.4=6.4.
Evaluate 2x3 - 3x2 + 4 for x = - 2.
Answer:
0 or 1.110838652
Step-by-step explanation:
The monthly cost for 46 minutes of calls is 12.75. What is the monthly cost for 52 minutes of calls?
Answer:
$14.41
Step-by-step explanation:
set up a proportion then cross-multiply:
12.75 / 46 = x / 52
46x = 663
x = 14.41
I'm ABC, the measure of
Answer: D,E,F,G
It's self explanitory
Answer:
I would say an angle.
Step-by-step explanation:
Three people take 2 1⁄2( two and a half ) hours to deliver leaflets to 270 houses
Assuming all people deliver leaflets at the same rate, how long will it take five people to deliver leaflets to 450 houses?
Give your answer in hours and minutes
9514 1404 393
Answer:
2 hours 30 minutes
Step-by-step explanation:
Effort is measured in people-hours per house. Here, the effort to deliver one leaflet is ...
(3 people)(2.5 hours)/(270 houses) = 1/36 people-hours/house
Then the number of hours (h) for 5 people and 450 houses is ...
(5 people)(h hours)/(450 houses) = 1/36 people-hours/house
h = (1/36)(450/5) = 2.5
It will take 2 hours 30 minutes to deliver to 450 houses.
_____
Alternate solution
The time is (assumed) inversely proportional to the number of people, so increasing the number of people from 3 to 5 multiplies the time by 3/5. The time is proportional to the number of houses, so increasing the number of houses from 270 to 450 multiplies the time by 450/270 = 5/3. The time for the second task is (3/5)(5/3) = 1 times the time for the first task.
2 1/2 hours is 2 hours 30 minutes.
Sophie drew a scale drawing of a house and its lot. The back patio is 3 centimeters wide in the drawing. The actual patio is 18 meters wide. What scale did Sophie use for the drawing?
Answer: 45?
Step-by-step explanation: Sorry If I'm wrong, I haven't learned this yet but I tried my best.
star time 8:00 a.m elapsed time 5 hours, 15 minutos end time is
Answer:
1:15 P.M
Step-by-step explanation:
After 5 hours if will be 1:00 P.M and 25 minutes after it turns 1:00 P.M will be 1:15 P.M
Answer:
1 : 15p.m
Step-by-step explanation:
PLS HELP I NEED THIS BY TOMORROW
Rearrange the parts of the parallelogram to form a rectangle. Find the area of the parallelogram and the area of the rectangle. How are the areas related?
Make a copy of the trapezoid to form a parallelogram. Find the area of the trapezoid and the area of the parallelogram. How are the two areas related?
When we do this, the base of the parallelogram has length b 1 + b 2, and the height is the same as the trapezoids, so the area of the parallelogram is (b 1 + b 2)*h. Since the two trapezoids of the same size created this parallelogram, the area of one of those trapezoids is one half the area of the parallelogram
the model below is shaded to represent an expression. which expression represents the model? PLS HELPPPPP
Answer:C
Step-by-step explanation:
The expression that most adequately expresses the model would be:
C). 3 × 2/5
What is a fraction?A fraction is described as an expression employed to denote a segment or section belonging to a larger whole.
In the given question, it can be observed that the model contains 10 parts in total with a division of 5 each into 2 parts.
Since the division of 2 is made thrice, it gives 3 × 2/5 which makes the third option the correct one.
Thus, option C is the correct answer.
Learn more about "Expression" here:
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10. Clifford Waxman purchased a snowmobile for $3,467. He made a 35 percent down payment and financed the
remainder. What amount did he finance?
$2,521.00
c. $2,345.44
b. $2,253.55
d. $2,433.11.
a.
Answer:
$3467 - $1213.45 (35%)
B
Which number line shows all the values of a that make the inequality - 3x +1 < 7 true?
Answer:
Third option
Step-by-step explanation:
Help due 30 mins....
What is the area of this figure?
Answer: C
Step-by-step explanation:
triangle: [(4 1/2) x 5] / 2
11.25
semicircle: (5/2)^2 x 3.14
(2.5^2 x 3.14) / 2
(6.25 x 3.14) / 2
19.625 / 2
9.8125
add them together:
9.8125 + 11.25 = 21.0625
rounded: 21.1
Two similar prisms have surface areas of 9
cm² and 16 cm ²?. Find the ratio of their
volumes.
Answer:
Simplified ratio is 9:16
Step-by-step explanation:
I WILL MARK YOU AS BRAINLIEST IF YOU ARE CORRECT! THIS QUESTION IS SUPER EASY! Look at the image below.
Answer:
the answer you are looking for is B
Answer:
A
Step-by-step explanation:
Its A.....unless im super dum lol
Given the following equations, determine the relationship of the lines they represent: y = 5 and x = 6 .
Answer:
The answer is perpendicular :]
Step-by-step explanation:
wont waste your time
I need the volume asap please homework due in 10mins question is a sphere has the inside of it a 9 so what's the volume
Answer:
Step-by-step explanation:
The volume should be 9 to the second power times 4×3.14
Solve the equation for all real solutions in simplest form.
-3w^2-10w+1=\,\,-4w^2−4w 2
Here is the answer. Let me know if you need any clarification!
