Answers
a. By the chain rule,
[tex]\dfrac{dy}{dx} = \dfrac{dy}{dt}\cdot\dfrac{dt}{dx} = \dfrac{dy}{dt}\cdot\dfrac{1}{\frac{dx}{dt}}[/tex]
Given that [tex]y=e^t+e^{-t}[/tex] and [tex]x=e^{-t}[/tex], we have the derivatives
[tex]\dfrac{dy}{dt} = e^t -e^{-t}[/tex]
[tex]\dfrac{dx}{dt} = -e^{-t}[/tex]
and so
[tex]\dfrac{dy}{dx} = \dfrac{e^t - e^{-t}}{-e^{-t}} = 1-e^{2t}[/tex]
Set this equal to zero and solve for t :
[tex]1-e^{2t} = 0[/tex]
[tex]1 = e^{2t}[/tex]
[tex]\ln(1) = \ln\left(e^{2t}\right)[/tex]
[tex]0 = 2t \ln(e)[/tex]
[tex]0 = 2t[/tex]
[tex]t=0[/tex]
This value of t corresponds to x = e⁰ = 1 and y = e⁰ - 1/e⁰ = 1 - 1 = 0. So the only point on the curve where the derivative dy/dx is zero is (1, 0).
b. Compute the second derivative. Since dy/dx is a function of t, we'll momentarily replace it with f(t). By the chain rule,
[tex]\dfrac{d^2}{dx^2} = \dfrac d{dx} \dfrac{dy}{dx} = \dfrac{df}{dx} = \dfrac{df}{dt}\cdot\dfrac{dt}{dx} = \dfrac{df}{dt}\cdot\dfrac{1}{\frac{dx}{dt}}[/tex]
We have
[tex]\dfrac{df}{dt} = -2e^{2t}[/tex]
and we've already committed dx/dt. So
[tex]\dfrac{d^2}{dx^2} = \dfrac{-2e^{2t}}{-e^{-t}} = 2e^{3t} [/tex]
Substitute the first and second derivative into the differential equation:
[tex]\left(\dfrac{d^2y}{dx^2}\right)^2 + \dfrac{dy}{dx} - 1 = 0[/tex]
[tex]\left(2e^{3t}\right)^2 + (1-e^{2t}) - 1 = 0[/tex]
[tex]4e^{6t} -e^{2t}= 0[/tex]
[tex]e^{2t} (4e^{4t} - 1) = 0[/tex]
[tex]4e^{4t} - 1 = 0[/tex]
[tex]4e^{4t} = 1[/tex]
[tex]e^{4t} = \dfrac14[/tex]
[tex]\ln\left(e^{4t}\right) = \ln\left( \dfrac14\right)[/tex]
[tex]4t \ln(e) = -\ln(4)[/tex]
[tex]4t = -\ln(4)[/tex]
[tex]t = -\dfrac{\ln(4)}4[/tex]
Related Questions
it is predicted that by 2050 there will be 10^10 people living on earth. approximatrly will be 10^12 trees. how many trees for each person?
Answers
Interesting question. Does that mean near the end of humanity?
Solution 1:
10^10 = 1,000,000,000
10^12 = 100,000,000,000
100,000,000,000/1,000,000,000 = 100 trees per person
Solution 2:
10^12 / 10^10 --- 10^10 / 10^10
10^2/1 = 10^2 = 100 trees per person
Please help! Grade 7 math
Answers
Answer:
6 apple pies
Step-by-step explanation:
A satellite orbits the Earth with an elliptical orbit modeled by x squared over 47 million three hundred thirty four thousand four hundred plus y squared over forty three million nine hundred fifty six thousand nine hundred equals 1 comma where the distances are measured in km. The Earth shares the same center as the orbit. If the radius of the Earth is 6,370 km, what is the maximum distance between the satellite and the Earth?
6,880 km
6,630 km
510 km
260 km
Answers
An elliptical of the satellite in the field of astrodynamics is known as a
Kepler orbit.
The distance between the satellite and the Earth is; 510 km.
Reasons:
The function that represents the elliptical orbit of the satellite is presented
as follows;
[tex]\dfrac{x^2}{47,334,400} + \dfrac{y^2}{43,956,900} = 1[/tex]
The radius of the Earth, R = 6,370 km.
The equation of an ellipse is presented as follows;
[tex]\mathbf{\dfrac{(x - h)^2}{a^2} + \dfrac{(y - k)^2}{b^2}} = 1[/tex]
Where, in a horizontal ellipse;
a = The major axis (the longer axis), which gives the maximum distance
between the satellite and Earth.
b = The minor axis (the shorter axis)
By comparison, we get;
a² = 47,334,400 km²
Therefore;
a = √(47,334,400 km²) = 6,880 km.
The distance between the satellite and the surface of the Earth, d, is given
as follows;
d = a - R
Which gives;
d = 6,880 km - 6,370 km = 510 km
Distance between the satellite and the surface of the Earth, d = 510 km.
Learn more here:
https://brainly.com/question/13999216
Answer:510 KM
Step-by-step explanation:
Crafts Unlimited sells ribbon by the foot. Serena needs ribbon
for an art project. The cost of the ribbon is 5 feet for $3.00.
Feet of Ribbon 1 5 10 15
Cost (in dollars) $3.00
a. Complete this ratio chart based on the information provided.
b. How many feet of ribbon can Serena buy for $12.00?
c. If Serena needs 17 feet of ribbon, how much will she pay?
How do you know?
Answers
The completed ratio chart is provided below:
Feet of Ribbon 1 5 10 15
Cost $0.60 $3 $6 $9
The amount Serena would get if she has $12 is 20 yards.
The amount Serena would pay if she wants 17 feet of the ribbon is $10.20.
The first step is to determine the unit cost of a ribbon.
Unit cost of a ribbon = $3 / 5 = $0.60
The cost of 10 feet of the ribbon = $0.60 x 10 = $6
The cost of 15 feet of the ribbon = $0.60 x 15 = $9
The feet of ribbon Serena can buy if she has $12 = $12 / $0.60 = 20 yards.
The cost of 17 feet of the ribbon = $0.60 x 17 = $10.20
To learn more, please check: https://brainly.com/question/12528427
B. DIRECTION: Read and analyze each problem. Give the best answer you can give in order to solve the problem
1. what is asked in the problem?____________________
2. What operation are you going to use to slove the problem?________
3. Show your solution___________________
Answers
Answer:
1: How much longer did he run than walk.
2: Subtraction
3: 1/8 hour
Step-by-step explanation:
1: The last sentence says what is being asked.
2: We want to see how much longer something is so it is subtraction.
3: 3/4 = 6/8 7/8 - 6/8 = 1/8
hello I need help please help me
Answers
Answer:
1) equation
2) equation
3) expression
4) expression
Step-by-step explanation:
An equation has a result, or something that equals to .
Numbers 1 and 2 have the word "is" which means equaled to, so they are equations. 3 and 4 don't, so they are only expressions.
Hope that helped :)
Step-by-step explanation:
Examples:
[tex]a+4;\ 3b;\ \dfrac{x}{4}-\text{these are the expressions}\\\\x=5;\ c+5=3c;\ p=q-\text{these are the equations}[/tex]
[tex]1.\ n+4=12-\bold{the\ equation}\\\\2.\ 5\cdot n=50-\bold{the\ equation}\\\\3.\ C\cdot 7-\bold{the\ expression}\\\\4.\ 6\cdot(12+t)-\bold{the\ expression}[/tex]
Determine the x-intercept for 3x + 2y = 14,
A)
(7,0)
B)
(0,7)
C)
2,0)
D)
(0,14
Answers
y-intercepts (0,7)
Answer:
X-intercept: [tex](\frac{14}{3}, 0)[/tex].
Y-intercept: (0, 7) ⇒ Option B.
Step-by-step explanation:
I'm not certain whether the question asks for either the x-intercept or the y-intercept, but I will provide the answers for both since none of the given options matches the coordinates of the x-intercept.
X-intercept:The x-intercept is the point on the graph where it crosses the x-axis, with coordinates of (a, 0). The x-intercept is the value of x when y = 0.
Given the linear equation in standard form, 3x + 2y = 14, set y = 0 to solve for the x-intercept:
3x + 2y = 14
3x + 2(0) = 14
3x + 0 = 14
3x = 14
Divide both sides by 3 to solve for x:
[tex]\frac{3x}{3} = \frac{14}{3}[/tex]
[tex]x = \frac{14}{3}[/tex]
Therefore, the x-intercept is [tex](\frac{14}{3}, 0)[/tex]. None of the given option matches the value of the x-intercept.
Y-intercept:The y-intercept is the point on the graph where it crosses the y-axis, and has coordinates of (0, b). It is the value of y when x = 0.
Set x = 0, and substitute into the given equation.
3x + 2y = 14
3(0) + 2y = 14
0 + 2y = 14
2y = 14
Divide both sides by 2 to solve for y:
[tex]\frac{2y}{2} = \frac{14}{2}[/tex]
y = 7
Therefore, the y-intercept is (0, 7), which matches Option B.
Andrew made a list of chores he needed to complete before his father comes home from work. On the chart, he showed the time it would take him to complete each chore.
If Andrew’s father is expected home at 6:00 p.m., what is the latest time Andrew could begin his chores?
Use the Hint to view the conversion chart.
Answers
Answer: the answer is 3:20pm!
Step-by-step explanation:
i got the question correct just now!
Answer:
3;20
Step-by-step explanation:
Please help if you can!
Answers
Answer:
4-1
Step-by-step explanation:
i think thats right
Lillian misses 10% of the free throws she attempts in a season. How many total free
throws did she attempt if she missed 51
help fast
Answers
Answer:
510
Step-by-step explanation:
10/100 = 51
10*10/100 = 51*10
100/100 = 510
Jay has written 24 songs to date. He writes an average
of 6 songs per vear. Jenna started writing songs this
year and expects to write about 12 songs per year.
How many years from now will Jenna have written
as many songs as Jay Write and graph a system of
equations to find your answer.
Answers
Step-by-step explanation:
I'm sorry could you maybe send more information so I could help you
at does the point (2, 1.5) represent in this context?
Answers
In what context? Do you mean calculus If you are talking about graphing on a graph plane it would be the two accelerates on the x axis so (left or right) and then the 1.5 accelerates on the y axis so 1.5 on the y axis would be one and halfway to the 2 mark on the y axis so in the middle of 1 to 2 on the y axis the y axis goes (up and down)
. Bobignment In what year was the LAWMA Established-
Answers
Calculate the average of these values: 38, 48, 58, 68?
Answers
Answer:
53 is the average value.
Step-by-step explanation:
To find the average, or mean of multiple values you first add them all up:
38+48+58+68=212
Then you count how many different numbers there are, (there are 4 numbers here, if there are 2 of the same numbers you still count them both, just so you understand).
Then you need to divide the total you got, 212, by the total amount of numbers, 4, to get the average amount, 53.
I hope that my answer has helped you, have a great day, ask any questions you have below.
Determine whether the ratios 8:7 and 15:14 are equivalent
Answers
Answer:
They are not
Step-by-step explanation:
We can see that 14 / 2 = 7, so we can check for the other side by dividing both sides of the ratio by 2:
15:14
\/
7.5:7 ≠ 8:7
Have a nice day!
I hope this is what you are looking for, but if not - comment! I will edit and update my answer accordingly. (ノ^∇^)
- Heather
trevin puts 8 gallon of gas
Answers
Answer:
good for trevin?
Step-by-step explanation:
we'll need more to go on haha
3) ) Each of the functions below model the monetary value of an item. Which is the only that has a growth factor showing the item is increasing in value?
Answers
Answer:
Imagine that scientists have just discovered a non-bird dinosaur skeleton. They want to know whether the dinosaur was closely related to birds. What features in ...
Step-by-step explanation:
(A) use the Pythagorean theorean to determine the length of the unknown side of the triangle
Answers
Answer:
a. = 36km
b. Perimeter = 108km
c. Area = 486km^2
Step-by-step explanation:
Pythagoras theorem
a^2 + b^2 = c^2
27^2 + b^2 = 45^2
b^2 = 45^2 - 27^2
b^2 = 1296
b = [tex]\sqrt{1296}[/tex]
b = 36km
Perimeter
a + b + c
27 + 36 + 45 = 108
Perimeter = 108km
Area = [tex]\frac{ab}{2}[/tex]
Area = [tex]\frac{27 * 36}{2}[/tex]
Area = [tex]\frac{972}{2}[/tex]
Area = 486km^2
2. which of the following shows the cube of a binomial?
Answers
Answer:
Socratic app
Step-by-step explanation:
it will help you
Can anyone help me if you get it right i'll give 30!
Answers
[tex]LHS \\ = 8 + 7 - {2}^{2} \\ = 8 + 7 - 4 \\ = 15 - 4 \\ = 11[/tex]
[tex]RHS \\ = {3}^{2} + 2 \\ = 9 + 2 \\ 11[/tex]
So, LHS = RHS
Answer:=
Hope it helps.
Do comment if you have any query.
Answer:
Below!
Step-by-step explanation:
Let's try the 'equal' sign for this equation.
8 + 7 - 2² = 3² + 2Remember, when it comes to simplifying, exponents come first. This means that we need to simplify the exponents first.
=> 8 + 7 - (2 x 2) = 9 + 2=> 15 - 4 = 9 + 2=> 11 = 11Since the LHS and the RHS are equal, the 'equal' sign will fit into the equation. Hence, this is proven as true. Let's try the 'not equal' sign.
=> 8 + 7 - 2² ≠ 3² + 2=> 15 - 4 ≠ 9 + 2=> 11 ≠ 11Now, this is a definite false. The 'not equal' sign is used when two values are not equal to each other, however, in this situation, 11 does equal 11. In between them is a 'not equal' sign. Hence, this is proven false.
We can conclude that the sign you are looking for is an 'equal' sign.
Hoped this helped.
[tex]BrainiacUser1357[/tex]
2x+3-3=-6
x=3 ? Solve
Answers
[tex]2(x)+3-3=-6\\2(3)+3-3=-6\\6+3-3=-6\\6=-6\\[/tex]
[tex]x\neq 3[/tex]
As we can see, x cannot be equal to 3 as 6 cannot equal -6.
I hope I've helped! :)
Is the following relation a function?
X. Y
-1. -2.
2. 3
3. 1
6. -2
Answers
Answer:
Step-by-step explanation:
x y
−1 −2
2 3
3 1
6 −2
Answer:
no
Step-by-step explanation:
6 and -2 is not a function
12x-4=10x-10+30 find value of x
Answers
Answer:
12
Step-by-step explanation:
12x - 4 = 10x + 20
2x - 4 = 20
2x = 24
x = 12
The Cardinality of this set {7,8,9) includes
a. n(P) = 3
b. n(P) = 6
c. n(P) = 24
d. n(P) = 2
Answers
Answer:
The Cardinality of this set {7,8,9} is n(P) = 8.
Step-by-step explanation:
Because:
n(P) = {{}, {7}, {8}, {9}, {7, 8}, {8, 9}, {7, 9}, {7, 8, 9}}
Hence by the formula 2^k.
where, k = number of elements in the set.
The cardinality is 8.
a box can take 12 pencils. If 156 pencils are packed into such boxes, how many boxes will be fully packed
Answers
A box=12 pencils
156 boxes =?
156÷12=13 boxes
How do I do i’m not sure
Answers
[tex](a)\\\\m=6.1\times 10^{-5},~~ t=1.7 \times 10^{-6} \\G = \sqrt{\dfrac{m}{t^2}} = \sqrt{\dfrac{6.1\times 10^{-5}}{\left(1.7 \times 10^{-6}\right)^2} } ~=4590\\\\\\(b)\\\\m=6.1\times 10^{-5}(1+ 8\%) = 6.1\times 10^{-5}(1+0.08) = 0.0000648\\\\t=1.7\times 10^{-6}(1+ 5\%)= 1.7\times 10^{-6}(1+0.05) =0.000001785\\\\\\G=\sqrt{\dfrac{m}{t^2}} = \sqrt{\dfrac{0.0000648}{(0.000001785)^2}}=4510[/tex]
10) For the holidays, Judy wants to spray fake snow around
the perimeter of her front store window and along both
diagonals. How many feet of snow will she be spraying?
8 feet
15 feet
Answers
Answer:i dont know sorry
Step-by-step explanation:
An electrician charges a flat fee plus an hourly rate for a service call. She charges a total of $94 for a 3-hour service call on Monday and $130 for a 5-hour service call on Tuesday. How much should she charge for a 2-hour service call?
$36
$52
$76
$98
Answers
Answer:
$76.
Step-by-step explanation:
You're looking over your child's math homework. The directions say to
simplify the expression below. What feedback would you give your child?
Make sure to include if their answer is correct or not (1 pt) and a
description of why (1 pt). *
6x + 5x + 2y
= 11x + 2y
= 13xy
=
Your answer
Answers
Hello, hope you are having a nice day.
This answer is not correct, because we can't add different variables. For instance, if we have 11x+2y, we can't add x's and y's together.
Therefore, the correct answer would be
11x+2y
Hope it helps. Ask me if you have any queries.
~An emotional teen who helps others with joy
[tex]MagicalNature[/tex]
Good luck.
Answer:
Step-by-step explanation:
The first step is the answer, so you would have gotten 100% if you just left the question there. The next step, the second step, is quite right.
If it was, then why didn't you do that in the first step?
Think of all the possibilities
60xy
10xy
12xy
10xy + 5y
12xy + 5x
There must be many more.
And yet dear one, you have only made 1 mistake. You combined xs and ys in step 2. Here's the rule.
Don't. If they were added, let them stay added. xs go with other xs. ys go with other ys. They never go together when they are added.
Which of the following statements about resale price maintenance is correct? Resale price maintenance _______.
A. is considered by some economists to be efficient and by other economists to be inefficient
B. is also called horizontal price fixing
C. is legal under the Sherman Act
D. occurs when a manufacturer prints the price of its good directly on the packaging
Answers
Answer:
A. is considered by some economists to be efficient and by other economists to be inefficient
Step-by-step explanation: