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
Janine is 25 and has a good job at a biotechnology company. Janine estimates that she will need $875,000 in her total retirement nest egg by the time she is 65 in order to have retirement income of $20,000 a year. (She expects that Social Security will pay her an additional $15,000 a year.) She currently has $5,000 in an IRA, an important part of her retirement nest egg. She believes her IRA will grow at an annual rate of 8 percent, and she plans to leave it untouched until she retires at age 65. How much money will Janine have to accumulate in her company’s 401(k) plan over the next 40 years in order to reach her retirement income goal? Use Exhibit 1-A.
Answers
We have that for the Question, it can be said that
The amount of money Janine will have to accumulate in her company’s 401(k) plan over the next 40 years is
$ 766377.4
From the question we are told
Janine is 25 and has a good job at a biotechnology company.
Janine estimates that she will need $875,000 in her total retirement nest egg by the time she is 65 in order to have retirement income of $20,000 a year. (She expects that Social Security will pay her an additional $15,000 a year.) She currently has $5,000 in an IRA, an important part of her retirement nest egg. She believes her IRA will grow at an annual rate of 8 percent, and she plans to leave it untouched until she retires at age 65. How much money will Janine have to accumulate in her company’s 401(k) plan over the next 40 years in order to reach her retirement income goal? Use Exhibit 1-A.
IRA Sum = $ 5000 Growth ratio of IRA= 8%Growth period = 40 years
Generally the equation for the IRA Amount at Retirement is mathematically given as
IAR= 5000 * 1.08^(40)
= 108622.6
Since estimate is at
$875,000
Therefore
Accumulation in 401,000 = 875000 - 108622.6
Accumulation in 401,000 = 766377.4
Hence
The amount of money Janine will have to accumulate in her company’s 401(k) plan over the next 40 years is
$ 766377.4
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Use a system of equations to find the partial fraction decomposition of the rational expression. Solve the system using matrices.
[tex] \frac{3x ^{2} + 3x - 2 }{(x + 1)^{2} (x - 1)} = \frac{a}{x + 1} + \frac{b}{x - 1} + \frac{c}{(x + 1)^{2} } [/tex]
A=
B=
C=
Answers
Combine the fractions on the left with a common denominator:
[tex]\dfrac a{x+1} + \dfrac b{x-1} + \dfrac c{(x+1)^2} = \dfrac{a(x+1)(x-1) + b(x+1)^2 + c(x-1)}{(x-1)(x+1)^2}[/tex]
It follows that
[tex]3x^2+3x-2 = a(x+1)(x-1) + b(x+1)^2 + c(x-1)[/tex]
Expand the right side and collect like powers of x :
[tex]3x^2+3x-2 = a(x^2-1) + b(x^2+2x+1) + c(x-1)[/tex]
[tex]3x^2+3x-2 = (a+b)x^2 + (2b + c)x -a +b - c[/tex]
Then we have the system of equations
[tex]\begin{cases}a+b=3\\2b+c=3\\-a+b-c=-2\end{cases}[/tex]
or in matrix form,
[tex]\begin{bmatrix}1&1&0\\0&2&1\\-1&1&-1\end{bmatrix} \begin{bmatrix}a\\b\\c\end{bmatrix} = \begin{bmatrix}3\\3\\-2\end{bmatrix}[/tex]
Compute the determinant of the coefficient matrix:
[tex]\det\begin{bmatrix}1&1&0\\0&2&1\\-1&1&-1\end{bmatrix} = -4[/tex]
Then the inverse of the coefficient matrix is equal 1/det times the adjugate of the coefficient matrix (a.k.a the transpose of the cofactor matrix):
[tex]\begin{bmatrix}1&1&0\\0&2&1\\-1&1&-1\end{bmatrix}^{-1} = \dfrac1{-4} \begin{bmatrix}-3 & -1 & 2 \\ 1 & -1 & -2 \\ 1 & -1 & 2\end{bmatrix}^\top = -\dfrac14 \begin{bmatrix}3&-1&-1\\1&1&1\\-2&2&-2\end{bmatrix}[/tex]
Multiply both sides of the equation by the inverse :
[tex]\begin{bmatrix}a\\b\\c\end{bmatrix} = -\dfrac14 \begin{bmatrix}3&-1&-1\\1&1&1\\-2&2&-2\end{bmatrix} \begin{bmatrix}3\\3\\-2\end{bmatrix} = \begin{bmatrix}2\\1\\1\end{bmatrix}[/tex]
So, we have a = 2 and b = c = 1, and the partial fraction decomposition is
[tex]\dfrac{3x^2+3x-2}{(x+1)^2(x-1)} = \dfrac 2{x+1} + \dfrac 1{x-1} + \dfrac 1{(x+1)^2}[/tex]
Answer:
A = 2B = 1C = 1Step-by-step explanation:
One can solve for a, b, c a little more directly than using a system of 3 equations.
If we multiply the rational expression by (x+1)², we get ...
(3x² +3x -2)/(x -1) = (x+1)²(a/(x+1) +b/(x-1)) +c
Evaluating this for x = -1 gives ...
(3(-1)² +3(-1) -2)/(-1 -1) = c
-2/-2 = 1 = c
Similarly, multiplying by (x -1) gives ...
(3x² +3x -2)/(x +1)² = (x -1)(a/(x +1) +c/(x +1)²) + b
Evaluating this for x = 1 gives ...
(3·1² +3·1 -2)/(1 +1)² = b
4/4 = 1 = b
Now, we need to find the value of 'a'. The identity will hold true for any value of x, so we can see what happens when we substitute x=0. We can use the values of 'b' and 'c' that we found above.
(3·0² +3·0 -2)/((0 +1)²(0 -1)) = a/(0 +1) +1/(0 -1) +1/(0 +1)²
-2/-1 = a -1 +1 ⇒ a = 2
_____
System of equations solution
When the terms of the right-side expansion are combined, the numerator of the result is ...
a(x +1)(x -1) +b(x +1)^2 +c(x -1) = (a+b)x² +(2b+c)x +(-a+b-c) ≡ 3x² +3x -2
Equating the coefficients gives the system of equations whose augmented matrix is:
[tex]\left[\begin{array}{ccc|c}1&1&0&3\\0&2&1&3\\-1&1&-1&-2\end{array}\right][/tex]
Transforming this to reduced row-echelon form using any of a variety of available tools gives ...
[tex]\left[\begin{array}{ccc|c}1&0&0&2\\0&1&0&1\\0&0&1&1\end{array}\right][/tex]
which tells you the solution is (A, B, C) = (2, 1, 1).
ple help me quick thank you:)
Answers
Answer:
z> 5/4
To graph, use the open dotted line and put it at point 5/4 aka 1.25 going towards the right because it is greater than.
Step-by-step explanation:
Isolate your variable (z) by multiplying both sides of the equation by 5/3.
z> 5/4
A person working with his cousin can refinish a table in 3 hours. Working alone, his cousin can complete the job in 12 hoursHow long would it take the person to refinish the table working alone?
Answers
Answer:
It is either 4 or 9.
Step-by-step explanation:
because you have to subtract or divide
Two points are shown on the number line. Which value is best represented by the point F? 
Answers
Answer:
I believe its d
I 13.12.21 Mather asianmen I choose the best answert the next two numbers in the pattern 1, 4, 9, 16 are. (A) 26,36 (B) 25, 36 (C) 24,35 (D) 36,48
Answers
Answer:
(B) 25, 36
Step-by-step explanation:
1 = 1²
4 = 2²
9 = 3²
16 = 4²
25 = 5²
35 = 6²
Precalc!! Please help
Answers
Answer:
Step-by-step explanation:
1
an investment had return of 13% if inflation was 4.1% what was your real return ?
Answers
Answer: 8.9%
Step-by-step explanation: 13 - 4.1
19. Round the number 347 500 to the nearest 1000; 10 000 and 100 000.
To the nearest 1000
To the nearest 10 000
To the nearest 100 000
Answers
Answer:
34800, 350000, 300000
Step-by-step explanation:
5>=5
34800
7>=5
350000
4<=5
300000
Which ones are true?
Answers
Answer:
The first, third and last one are true
which is the graph of y -3 equals -2/3 (x + 6)
NO LINKS
Answers
Step-by-step explanation:
This is point-slope form:
[tex]y-y1=m(x-x1)[/tex]
in this case:
[tex]y-3=-\frac{2}{3} (x+6)[/tex]
Looking at the structure of point slope form up top, you can find the slope and one point.
[tex]m=-\frac{2}{3} \\x1=-6\\y1=3\\(-6,3)[/tex]
Now, just find the graph with a slope of [tex]-\frac{2}{3}[/tex] that passes through (-6, 3). It's hard to see on the image, but that looks like the top left.
By the way, what's the "no links" thing about? I just came back to this site after a year away
1. Tim's phone service charges $27.42 plus an additional $0.18 for each text message sent per
month. If Tim's phone bill was $31.92, how many text messages did he send that months
Answers
Answer:
25 text messages
Step-by-step explanation:
First of all you need to find how much money was added to his phone bill. 31.92-27.42 would equal 4.5. Then you would divide that number by 0.18 to find out the total text messages, which is 25
Use the distributive property to write an expression equivalent to 7(48).
Drag and drop the appropriate number into each box.
Answers
Answer:
336
Step-by-step explanation:
7x48=
7(8)= 56
7(4)+5= 33
7(48) = 336
Answer:
336
Step-by-step explanation:
In the equation y ≈7x-9,if x≥2, then what is the value of y?
Answers
Answer:
y[tex]\geq[/tex]5
Step-by-step explanation:
7(2)-9
14-9
5
y[tex]\geq[/tex]5
Sally Jones had a policy with a $250 deductible which paid 80% of her covered charges less deductible. She had medical expenses of $18,240.00. What is S
Answers
Answer
Find out the insurance company's payment .
To proof
Deductible
The deductible is the amount of money you will pay in an insurance claim before the company starts paying you insurance coverage .
As given
Sally Jones had a policy with a $250 deductible which paid 80% of her .covered charges less deductible
She had medical expenses of $18,240.00.
Amount becomes after deducting deductible = $18,240 - $250
= $17990
80 % is written in the decimal form
now 80% of the Amount after deducting deductible
= $ 14392
Therefore $ 14392 is the insurance company's payment .
Remember To Stay In School
Wink Wink
Answer:3,848
Step-by-step explanation:I took the test.
Which value for x satisfies the equation 3x−6=15?
Answers
Answer:
x= 7
Step-by-step explanation:
1. Add 6 to both sides
2. simplify
3. Divide by the same term
4. simplify
Answer:
7
Step-by-step explanation:
3x7=21
= 21 - 6 = 15
I hope you find it helpful ..
Good luck
100.00 decreased by 20%
Please help
Answers
Answer: If Percentage decrease % = 20%, (1 - Percentage decrease %) = 1 - 20% = 100/100 - 20/100 = 80/100 = 0.8 => Percentage decreased number = 0.8 × Initial value
Step-by-step explanation: Number 100 decreased by 20% (20 percent) of its value (percentage decrease) and calculated absolute change (actual difference)
Find the equation of the line.
Answers
Answer:
y= 4/1x +1
Step-by-step explanation:
the number on the y axis is the b and you do rise over run to find m you rise (so you go up or down) 4 times and run( go to the sides) once.
rise 4
-------- = -----
run 1
Answer:
y = 3x + 1
Step-by-step explanation:
Slope: As it rises 3, it runs 1
Slope = Change in y/change in x
Slope = 3/1
y - intercept = 1 ,the point of intersection
Equation : y = mx + b
m = slope
b = y-intercept
Equation : y = 3x + 1
-Chetan K
which matrix multiplication is defined?
Answers
Find the simplified product. 5/n+1 times n+1/n+3
Answers
[tex]\left(\dfrac{5}{n+1}\right) \left( \dfrac{n+1}{n+3} \right)\\\\\\=\dfrac{5(n+1)}{(n+1)(n+3)} \\\\\\=\dfrac{5}{n+3}[/tex]
Plz solve. The question is in the picture. 100 points and brainliest.
Answers
Answer:
A
Step-by-step explanation:
a scale diagram, 0.125 inch represe
5 feet. How many inches represent
pot?
0.001
0.01
0.1
0.12
Math
Answers
Answer:
the scale diagram is 9.12
Given: DE is a midsegment of ∆C. Show all work. Find the value of x and y. What is the length of C?
Answers
Answer:
Find out the value of x.
To prove
Definition of mid- segment
It is a segment connecting the midpoints of two sides of a triangle.
It is always parallel to the third side, and the length of the midsegment is half the length of the third side.
As shown in the triangle given below.
As given
DE = 5x and BC = 11x – 15
Thus
15 = 11x - 10x
15 = x
Therefore the value of x is 15 .
Step-by-step explanation:
through: (1, 3) and (-3,-5)
Answers
m = (Y2 - Y1)/(X2 - X1)
m = (-5 - 3)/(-3 - 1)
m = -8/-4
m = 2
Identify the type of function graphed to the right.
o linear
exponential
other
As x increases by 1 unit, what is the exponential
growth factor?
Answers
9514 1404 393
Answer:
exponentialgrowth factor 3Step-by-step explanation:
If the function showed linear growth, it would be a straight line. Since it curves upward, it will be either "exponential" or "other." The growth factor seems more or less constant, so we can claim the curve to be exponential.
When x=1, y=1; when x=2, y=3. The increase in x by 1 unit causes the value to grow by a factor of 3/1 = 3. When x=3, y=9, which is growth by another factor of 3. The growth factor is 3.
Answer:
Am working on that too this is for everyone! here you go
Step-by-step explanation:
Edge 2021!
The perimeter of the triangle is
equal to the area of the rectangle.
What is the value of x?
Answers
Answer:
x=-21
Step-by-step explanation:
make the equation
2x+9x-1+40=6(x-11)
simplify
11x+39=6x-66
solve
5x=-105
x=-21
I need to answer the bottom question please help me ASAP
Answers
Answer:
60 degrees
Step-by-step explanation:
Because line ED exists, the sum of 100, 20, and x must be 180. So, just solve for x in the equation 100 + 20 + x = 180
20 + x = 80
x = 60
If 12 cats can kill 12 mice in 12 hours, how many hours will it take for 2021 cats to kill 2021 mice
Answers
Step-by-step explanation:
168 hours I guess
have a food day
Four times the product of a number x and a number y is equal to 20.
Answers
Step-by-step explanation:
4x×y=20
.....
..................................
The expression for four times the product of a number x and a number y is equal to 20 is 4xy = 20.
Given,
Four times the product of a number x and a number y is equal to 20.
We need to write the expression.
What is the meaning of the mathematical terms?- 2 times M = 2 x M
- Product of M and N = M x N
- 3 more than M = M + 3
- 3 less than M = M - 3
Find the expression.
We have,
Four times the product of a number x and a number y is equal to 20.
Here,
The product of a number x and a number y can be written as:
= (x) x y
Four times the product of a number x and a number y can be written as:
= 4 x [ (x) x y ]
- is equal to 20 can be written as:
= 20
Now,
We get,
4xy = 20
Thus the expression for Four times the product of a number x and a number y is equal to 20 is 4xy = 20.
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A long-sleeve shirt has 8 front buttons and 2 cuff buttons; a short-sleeve shirt has 6 front buttons and no cuff buttons. The factory which makes ‘Slimboy Shirts’ uses 10 times as many front buttons as cuff buttons. What is the ratio of long-sleeve shirts to short-sleeve shirts produced by the factory?
Leave your ratio in its simplest form.
Answers
Answer:
ratio of long-sleeved shirts to short-sleeved shirts is 1:2 this is my answer that I thought of if you don't feel comfortable believing me just wait for another person to answer and see what they say.
Step-by-step explanation: