Section 8.1 Introduction to the Laplace Transforms
Problem 2.
Use the table of Laplace transforms to find the Laplace transforms of the following functions.
[tex](a)cosh \: t \: sin \: t[/tex]
[tex](b) {sin}^{2} t[/tex]
[tex](c) {cos}^{2} 2t[/tex]
[tex](d) {cosh}^{2} t[/tex]
[tex](e)t \: sinh \: 2t[/tex]
[tex](f)sin \: t \: cos \: t[/tex]
[tex](g)sin(t + \frac{\pi}{4} )[/tex]
[tex](h)cos2t - cos3t[/tex]
[tex](i)sin2t + cos4t[/tex]
Answers
I don't know what table you have as reference, but I suspect it includes the following transforms:
[tex]1 \leftrightarrow \dfrac1s[/tex]
[tex]e^{at} \leftrightarrow \dfrac{1}{s-a}[/tex]
[tex]\cos(at) \leftrightarrow \dfrac{s}{s^2+a^2}[/tex]
[tex]\sin(at) \leftrightarrow \dfrac{a}{s^2+a^2}[/tex]
[tex]\cosh(at) \leftrigharrow \dfrac{s}{s^2-a^2}[/tex]
[tex]\sinh(at) \leftrigharrow \dfrac{a}{s^2-a^2}[/tex]
It probably also includes some more general properties, like
[tex]t f(t) \leftrightarrow -F'(s)[/tex]
[tex]e^{at} f(t) \leftrightarrow F(s-a)[/tex]
where F(s) is the Laplace transform of f(t).
Beyond these, you should also know the following identities:
[tex]\cosh(t) = \dfrac{e^t + e^{-t}}2[/tex]
[tex]\cosh^2(t) = \dfrac{1 + \cosh(2t)}2[/tex]
[tex]\cos^2(t) = \dfrac{1 + \cos(2t)}2[/tex]
[tex]\sin^2(t) = \dfrac{1 - \cos(2t)}2[/tex]
[tex]\sin(2t) = 2 \sin(t) \cos(t)[/tex]
[tex]\sin(t \pm T) = \sin(t) \cos(T) \pm \cos(t) \sin(T)[/tex]
Putting everything together, we have
• (a)
[tex]\cosh(t) \sin(t) = \dfrac{e^t + e^{-t}}2 \times \sin(t) = \dfrac12 e^t \sin(t) + \dfrac12 e^{-t} \sin(t)[/tex]
and the Laplace transform is
[tex]\dfrac12 F(s - 1) + \dfrac12 F(s + 1)[/tex]
where F(s) is the transform of sin(t),
[tex]F(s) = \dfrac{1}{s^2 + 1}[/tex]
Then
[tex]\cosh(t) \sin(t) \leftrightarrow \dfrac{\frac1{(s-1)^2+1} + \frac1{(s+1)^2+1}}2 = \boxed{\dfrac{s^2+2}{s^4+4}}[/tex]
• (b)
[tex]\sin^2(t) = \dfrac12 \left(1 - \cos(2t)\right)[/tex]
and the transform is
[tex]F(s) = \dfrac12 \left(\dfrac1s - \dfrac{s}{s^2+4}\right) = \boxed{\dfrac{2}{s^3+4s}}[/tex]
• (c)
[tex]\cos^2(2t) = \dfrac12 \left(1 + \cos(4t)\right)[/tex]
with transform
[tex]F(s) = \dfrac12 \left(\dfrac1s + \dfrac{s}{s^2+16}\right) = \boxed{\dfrac{s^2+8}{s^3+16s}}[/tex]
• (d)
[tex]\cosh^2(t) = \dfrac12 \left(1 + \cosh(2t)\right)[/tex]
with transform
[tex]F(s) = \dfrac12 \left(\dfrac1s + \dfrac{s}{s^2-4}\right) = \boxed{-\dfrac2{s^3-4s}}[/tex]
• (e)
[tex]t\sinh(2t) \leftrightarrow -F'(s)[/tex]
where F(s) is the Laplace transform of sinh(2t),
[tex]F(s) = \dfrac{2}{s^2 - 4} \implies -F'(s) = \boxed{\dfrac{4s}{(s^2-4)^2}}[/tex]
• (f)
[tex]\sin(t) \cos(t) = \dfrac12 \left(2\sin(t) \cos(t)\right) = \dfrac12 \sin(2t)[/tex]
with transform
[tex]F(s) = \dfrac12 \times \dfrac{2}{s^2+4} = \boxed{\dfrac1{s^2+4}}[/tex]
• (g)
[tex]\sin\left(t+\dfrac\pi4\right) = \sin(t) \cos\left(\dfrac\pi4\right) + \cos(t) \sin\left(\dfrac\pi4\right) = \dfrac1{\sqrt2} \left(\sin(t) + \cos(t)\right)[/tex]
with transform
[tex]F(s) = \dfrac1{\sqrt2} \left(\dfrac1{s^2+1} + \dfrac{s}{s^2+1}\right) = \boxed{\dfrac{s+1}{\sqrt2 (s^2+1)}}[/tex]
The last two are trivial and follow directly from the properties listed above.
• (h)
[tex]\cos(2t) - \cos(3t) \leftrightarrow \dfrac{s}{s^2+4} - \dfrac{s}{s^2+9} = \boxed{\dfrac{5s}{s^4 + 13s^2 + 36}}[/tex]
• (i)
[tex]\sin(2t) + \cos(4t) \leftrightarrow \dfrac2{s^2+4} + \dfrac{s}{s^2+16} = \boxed{\dfrac{s^3+2s^2+4s+32}{s^4+20s^2+64}}[/tex]
Related Questions
i dont know how to do this I will give brainleist
Answers
Answer:
y = x+14
Step-by-step explanation:
At x =0 y is 14. The pattern of y is 14 greater than x continues.
Answer:
y=x+14
Step-by-step explanation:
Each of the x numbers are being added to 14 to get the y number.
Let us check if this is the answer:
the first numbers in the table are 0 and 14
0 plus 14 equals 14.
Now the next two numbers on the table:
1 plus 14 equals 15
And one more: the third numbers on the table:
2 plus 14 equals 16
video explaining this-watch it, it is short and easy to understand:
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-functions/8th-x-and-y-intercepts/v/finding-intercepts-for-a-linear-function-from-a-table
PLEASE HELP 4 3/5+7 8/15
Answers
Answer:
Step-by-step explanation:
4 3/5 ---> 23/5
7 8/15---> 113/15
23*3/5*3=69/15
113/15+69/15=182/15---->12 2/15
xnxx
What is the first time in the volume of a sphere?
Answers
Answer:
The formula for the volume of a sphere is V = 4/3 πr³. See the formula used in an example where we are given the diameter of the sphere.
Step-by-step explanation:
I hope it will help you....
[tex]kai6418[/tex]
#carryonleraning
HELP!!! The graph shows the relationship between the distance that a snail crawls and the time that it crawls. What numbers will correctly complete the table below?
Answers
Answer:
its c because i took the quiz
Step-by-step explanation:
I took the quiz
I need to know! What is 2+5?
Answers
Answer:
seven
Step-by-step explanation:
1+1=2 and 3+2=5 so 2+5=7
Answer:
2+5=the answer is 7 you welcome
What inequality is represented by this graph?
1
2
3
4
5 6 7 8
9
O x>6
O x56
O x <6.
O x26
Answers
x < 6
This is because there is an open hole at 6, meaning we exclude it from the solution set. Stuff to the left of 6 (ie smaller than 6) is shaded, which is why we go for x < 6
The inequality represented in the given graph is x < 6.
Given is a number line where an inequality is marked.
We have to find the inequality which represents in the number line given.
Here, a graph or a number line where the numbers are marked from 1 to 9 is drawn.
A blue line is drawn prior to the number 6. So the inequality sign is "<".
An open circle is over the number 6. So 6 is not included in the inequality.
So, the required inequality is x < 6.
The correct option is C. x < 6.
Learn more about Inequalities here :
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Evaluate $\dfrac{2\sqrt{72}}{\sqrt{8}+\sqrt{2}}$.
Answers
========================================================
Work Shown:
[tex]\frac{2\sqrt{72}}{\sqrt{8}+\sqrt{2}}\\\\\frac{2\sqrt{36*2}}{\sqrt{4*2}+\sqrt{2}}\\\\\frac{2\sqrt{36}*\sqrt{2}}{\sqrt{4}*\sqrt{2}+\sqrt{2}}\\\\\frac{2*6*\sqrt{2}}{2*\sqrt{2}+\sqrt{2}}\\\\\frac{12\sqrt{2}}{2\sqrt{2}+\sqrt{2}}\\\\\frac{12\sqrt{2}}{3\sqrt{2}}\\\\\frac{12}{3}\\\\4[/tex]
Note in step 2, I factored each number in the square root to pull out the largest perfect square factor. From there, I used the rule that [tex]\sqrt{A*B} = \sqrt{A}*\sqrt{B}[/tex] to break up the roots.
Answer:
4
Step-by-step explanation:
We first simplify $\sqrt{72}$ and $\sqrt{8}$:\begin{align*}
\sqrt{72} &=\sqrt{36\cdot 2} = \sqrt{36}\cdot \sqrt{2} = 6\sqrt{2},\\
\sqrt{8} &= \sqrt{4\cdot 2} = \sqrt{4}\cdot \sqrt{2} = 2\sqrt{2}.
\end{align*}Then, we have
\[\dfrac{2\sqrt{72}}{\sqrt{8}+\sqrt{2}} = \dfrac{2\left(6\sqrt{2}\right)}{2\sqrt{2} + \sqrt{2}}
= \dfrac{12\sqrt{2}}{3\sqrt{2}} = \dfrac{12}{3}\cdot \dfrac{\sqrt{2}}{\sqrt{2}} = \boxed{4}.\]
What is the range of {(1,3) (4,3) (6,2))?
Answers
Answer:
5
Step-by-step explanation:
Minus the biggest number with the smallest number
Solve each equation for y. Then find the value of y for each value of x.
Answers
Cisca, 12 years old, is three times as old as Catherine. How old will she be when she's twice as old as Catherine?
Answers
Step-by-step explanation:
Divide 12 by 3 to determine Catherine's age.
[tex]12 \div 3 = 4[/tex]
Catherine is 4 years old when Cisca is 12 years old.
Let's identify when Cisca will be twice as old as Catherine by representing the ages per future years.
Year 1: 13, 5
Year 2: 14, 6
Year 3: 15, 7
Year 4: 16, 8
Year 5: 17, 9
Year 6: 18, 10
In 4 years, Cisca will be twice as old as Catherine.
Her age is 16.
She would be 4 years
Calculation of ageAge of Catherine = X
Age of Cisca = 12yrs = 3x
Therefore X = 12/3= 4 years.
Actual age of Catherine = 4 years.
Twice ( 2x) age of Catherine = 2×4= 8
Therefore, she will be 4 years to be twice as old as Catherine
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b. Cai thinks of the area of the poster as the sum of the areas of the three separate sections. Write an expression that represents Cai's way of thinking. What does Cai's expression let you see about the layout that Brett's expression does not?
Answers
On solving the provided question, we can say that Cai's thinking expression will be area of the poster k = Ax Bt +Cz => K = x² + lb + 1/2wm
What is Expression?An expression is combination of variables, numbers and operators.
Cai's thinking expression will be.
The three sections are of different shapes which includes a square, rectangle and a triangle.
The area of square is side times of side= x²
The area of rectangle is length times of breadth (lb)
The area of triangle=1/2length times of breadth(1/2lb)
Area of the poster k = Ax+By+Cz
Ax = x²
Bx = lb
Cz = 1/2wm
K = x² + lb + 1/2wm
Hence, Cai's thinking expression will be area of the poster K = x² + lb +1/2wm
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Simplify.
(fg2)4
Can you please help me? I'm stuck on this question
Answers
Answer:
[tex]4fg^{2}[/tex]
Step-by-step explanation:
1) Remove parentheses.
[tex]fg^{2} *4[/tex]
2) Regroup terms.
[tex]4fg^{2}[/tex]
Answer:
The correct answer is...
Step-by-step explanation:
"F^4g^8" Hope this helps!
If P(E) = 0.32, then what are the odds against E?
Answers
Answer:
0.68 : 0.32
Step-by-step explanation:
the probability of all the outcomes of an event is always 1 .
if P(E) = 0.32,
P(E') = 1 - 0.32
= 0.68
N.B that P(E') is the same as P( not E ).
since your answer should be given in the ratio form, as the odds against E ( probability not E : probability E),
P(E') : P(E)
0.68 : 0.32
hope this helps you!
-s.
Convert 30 C to F using the formula (9C/5)+32=F
Answers
Answer:
answer below
Step-by-step explanation:
C=5/9(F--32)
9C/5=F--32
F=9C/5 +32
When C=30
F=(9*30/5)+32=9*6+32=54+32=86
F=86
so the answer is 86
hope this helps ^^
Answer:
What is the real increase if VAT increases from 10% to 14%?
Step-by-step explanation:
What is the real increase if VAT increases from 10% to 14%?
Write 2,748 in Mayan numerals. Show work. Take a screen shot or picture and upload using the Insert Image link (the tree icon). I don’t know what to do
Answers
Answer:
se escribe como en la imagen
Step-by-step explanation:
What’s the correct answer for this?
Answers
Answer:
6 pounds and 15 ounces
Step-by-step explanation
1 pound is equal to 16 ounces. Divide 111 by 16 to get 6 remainder 15
PLEASE HELP
ILL MARK AS BRAIN LEIST!
Answers
Answer:
C. 707 cm^2
Step-by-step explanation:
Three friends went apple picking and then counted the apples they picked.
Who has the largest red to green apple ratio? Use original complete sentences.
!! WILL GIVE BRAINLIEST IF CORRECT !!
Answers
Answer:
Carmen has the biggest red to green apple ratio
Step-by-step explanation:
Red to green apple ratios are, in order, 20:12, (5:3), 12:10 (6:5), and 16:20 (4:5). 5:3 is the biggest red to green apple ratio because for every 5 red apples Carmen has 3 green apples.
100 pts and mark brainliest
Use the quadratic formula to solve
x^2-4x+3
Show your work. Then describe the solution. if you dont describe the solution it will get reported.
Answers
Answer:
x = 3, x = 1
The points at which the curve crosses the x-axis are (3, 0) and (1, 0)
Step-by-step explanation:
Quadratic formula:
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac} }{2a}[/tex]
Quadratic equation: [tex]ax^2+bx+c[/tex]
Therefore, for [tex]x^2-4x+3[/tex]:
a = 1 b = -4 c = 3
Inputting values into quadratic formula:
[tex]x=\dfrac{4\pm\sqrt{(-4)^2-(4\times1\times3)} }{2\times1}[/tex]
[tex]\implies x=\dfrac{4\pm\sqrt{16-12} }{2}[/tex]
[tex]\implies x=\dfrac{4\pm\sqrt{4} }{2}[/tex]
[tex]\implies x=\dfrac{4\pm2 }{2}[/tex]
[tex]\implies x=2\pm1[/tex]
[tex]\implies x=3, x=1[/tex]
Therefore, the points at which the curve crosses the x-axis are (3, 0) and (1, 0)
[tex]\\ \tt\hookrightarrow x^2-4x+3=0[/tex]
a=1b=-4c=3[tex]\\ \tt\hookrightarrow x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
[tex]\\ \tt\hookrightarrow x=\dfrac{4\pm \sqrt{16-12}}{2}[/tex]
[tex]\\ \tt\hookrightarrow x=\dfrac{4\pm 2}{2}[/tex]
x=6/2 or 2/2x=3 or 1Leni opened a savings account that pays 0.8% interest, compounded quarterly. Her initial deposit was $50. How long would she have to leave the money in the account for it to double?
Answers
The number of years it would take the money to double is 86.6 years.
What is the time it would take the deposit to double in value?Number of years = (In fv / pv) / r
Where:
FV = future valuePV = present valuer = interest rateFV / PV = 2IN 2 / 0.008 = 86.6 years
To learn more about how to determine the number of years, please check: https://brainly.com/question/21841217
write and equation of the line that passes through the given points point (-2, 4) and is parallel to the line y=4x-4
Answers
Answer:
y=4x+12
Step-by-step explanation:
m=4
y-(4)=4(x-(-2))
y=4x+12
A motel rents double rooms at $34 per day and single rooms at $26 per day. If 27 rooms were rented one day for a total of $846, how many rooms of each kind were rented
single rooms
double rooms
Answers
Answer:
double roooms
Step-by-step explanation:
it is double rooms thank me later
Explanation:
Double+Single rooms= 27 rented rooms
x+y=27
$34 per day X amount of double rooms, $26 per day Y amount of single rooms= $846
34x+26y=846
Express Y in X terms
34x+26(27-x)=846
34x+702-26x=846
8x=144
x=18
Plug the value of x=18 in the original equation
18+y=27
y=27-18
y=9
This was an exceptionally dry year for portions of the southwestern United States. Monthly precipitation in Phoenix, Arizona, was recorded in the table and is modeled by y = –0.04088x2 + 0.4485x + 1.862.
In what month did Phoenix receive the highest amount of precipitation?
Month (x)
Precipitation
January
2.27 inches
February
?
March
?
April
?
May
?
June
?
July
?
August
?
September
2.59 inches
October
?
November
?
December
?
Sketch a graph or fill in the table to answer the question.
April
May
June
July
Answers
Using the vertex of a quadratic function, it is found that Phoenix received the highest amount of precipitation on the month of May.
What is the vertex of a quadratic equation?A quadratic equation is modeled by:
[tex]y = ax^2 + bx + c[/tex]
The vertex is given by:
[tex](x_v, y_v)[/tex]
In which:
[tex]x_v = -\frac{b}{2a}[/tex]
[tex]y_v = -\frac{b^2 - 4ac}{4a}[/tex]
Considering the coefficient a, we have that:
If a < 0, the vertex is a maximum point.If a > 0, the vertex is a minimum point.In this problem, the precipitation each month x in Phoenix is given by:
[tex]y = -0.04088x^2 + 0.4485x + 1.862[/tex]
Hence the coefficients are a = -0.04088, b = 0.4485, c = 1.862, and the month for which the precipitation will be highest is given by:
[tex]x_v = -\frac{0.4485}{2(-0.04088)} = 5.48[/tex]
Hence during the month of May.
More can be learned about the vertex of a quadratic function at https://brainly.com/question/24737967
RSTU and VXYZ are quadrilaterals. Given RSTU ~ VXYZ , describe a sequence of rigid motions followed by a dilation with center (0,0) that maps RSTU to VXYZ
Answers
Quadrilateral RSTU was rotated, translated and dilated to form quadrilateral VXYZ.
TransformationTransformation is the movement of a point from its initial location to new location. Types of transformation are rotation, reflection, translation and dilation.
Dilation is the increase or decrease in the size of a figure to create an image.
Quadrilateral RSTU was rotated, translated and dilated to form quadrilateral VXYZ.
Find out more on Transformation at: https://brainly.com/question/1462871
Need help asap giving brainliest PLSSS
Answers
Answer:
2144 in^3
Step-by-step explanation:
[tex]\frac{4}{3}\pi r^3 > \frac{4}{3}\* *\pi * 8^3 > 2144.660[/tex]
Use the associative property to rewrite the following expression.
3+(a+b)
Answers
Answer:
Associative Property of Addition a + (b + c) = (a + b) + c
Distributive property
[tex]a (b + c) = ab + ac[/tex]
Determine the average yards per carry for the following running back. Round to the nearest tenth.
Vic: 93 yd., 3 yd., 6 yd.
34 yd.
102 yd.
32 yd.
Answers
Answer:
45
Step-by-step explanation:
93+3+6+34+102+32=270270/6
How to find average/mean:
Add all the numbers, then divide by the amount of numbers.
Brainliest if correct
Answers
Answer:
-3
(negative 3. ) in word form
Since they give you a value, n = 4, we will plug in that value into the overall equation.
(4)^2 - 11
16 - 11 = 5
So, our final answer is 5!
The radius of a circle is 1 mile. What is the circle's area?
r=1 mi
Use 3.14 for . PLS HURRY!!!!!!
Answers
Answer:
$$6.28$$ miles Explanation: Circumference of circle with a radius $$r=1$$ mile then the circumference of circle $$=2\pi r$$ $$=2\cdot 3.14\cdot 1$$ $$=6.28\
Step-by-step explanation:
the graph of the function f(x)=mx+b contains the points (-1,3) and (-2,4) find m and b
Answers
Answer:
m = -1; b = 2
Step-by-step explanation:
the first step in solving this equation is to find the slope
so we use the equation (y2-y1)/(x2-x1)
so with this the equation is
(4 - 3)/(-2 + 1)
then it would be 1/-1
that is -1
so the function so far is f(x) = -x + b
to find the b you would plug in the coordinates
we can plug in the first set
3 = 1 + b
b = 2
so the m is -1 and the b is 2