Section 8.1 Introduction to the Laplace Transforms
Problem 6.
Prove that if
[tex]f(t)↔ F(s)[/tex]
then
[tex] {t}^{k} f(t)↔ {( - 1)}^{k} {F}^{k} (s).[/tex]
Hint: Assume that it's permissable to the differentiate the integral
[tex]F(s)={∫}^{ \infty } _{0} {e}^{ - st} f(t)dt[/tex]
with respect to s under the integral sign.
![Section 8.1 Introduction To The Laplace Transforms Problem 6.Prove That If [tex]f(t) F(s)[/tex]then [tex]](https://brainimage.s3.us-east-005.backblazeb2.com/contents/attachments/3xD1mKMZRRPzS5hLXPbgS6jLzL6h0sp4.jpeg)
Answers
Let k = 1, for a start. By definition of the Laplace transform,
[tex]\displaystyle F(s) = \int_0^\infty f(t) e^{-st} \, dt[/tex]
Differentiate both sides with respect to s :
[tex]\displaystyle F'(s) = \frac{d}{ds} \int_0^\infty f(t) e^{-st} \, dt[/tex]
[tex]\displaystyle F'(s) = \int_0^\infty \frac{\partial}{\partial s} \left[f(t) e^{-st}\right] \, dt[/tex]
[tex]\displaystyle F'(s) = \int_0^\infty -t f(t) e^{-st} \, dt[/tex]
so that [tex]t f(t) \leftrightarrow (-1)^1 F^{(1)}(s) = -F'(s)[/tex] is indeed true.
Suppose the claim is true for arbitrary integer k = n, which is to say that [tex]t^n f(t) \leftrightarrow (-1)^n F^{(n)}(s)[/tex]. Then if k = n + 1, we have
[tex]F^{(n+1)}(s) = \dfrac{d}{ds} F^{(n)}(s)[/tex]
Consider the two cases:
• If k = n + 1 is even, then n is odd, so
[tex](-1)^n F^{(n)}(s) = -F^{(n)}(s) \leftrightarrow t^n f(t)[/tex]
and it follows that
[tex]F^{(n+1)}(s) = \displaystyle \frac{d}{ds} \left[-\int_0^\infty t^n f(t) e^{-st} \, dt \right][/tex]
[tex]F^{(n+1)}(s) = \displaystyle -\int_0^\infty \frac{\partial}{\partial s}\left[ t^n f(t) e^{-st} \right] \, dt[/tex]
[tex]F^{(n+1)}(s) = \displaystyle \frac{d}{ds} \left[-\int_0^\infty t^n f(t) e^{-st} \, dt \right][/tex]
[tex]F^{(n+1)}(s) = \displaystyle \int_0^\infty t^{n+1} f(t) e^{-st} \, dt[/tex]
[tex]\implies F^{(n+1)}(s) = (-1)^{n+1} F^{(n+1)}(s) \leftrightarrow t^{n+1}f(t)[/tex]
• Otherwise, if k = n + 1 is odd, then n is even, so
[tex](-1)^n F^{(n)}(s) = F^{(n)}(s) \leftrightarrow t^n f(t)[/tex]
The rest of the proof is the same as the previous case.
So we've proved the claim by induction:
• [tex]t f(t) \leftrightarrow -F(s)[/tex], and
• [tex]\bigg(t^n f(t) \leftrightarrow (-1)^n F^{(n)}(s)\bigg) \implies \bigg(t^{n+1} f(t) \leftrightarrow (-1)^{n+1} F^{(n+1)}(s)\bigg)[/tex]
Related Questions
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
Solve each equation for y. Then find the value of y for each value of x.
Answers
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
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Answer:
se escribe como en la imagen
Step-by-step explanation:
I really need help it worth a lot of points
Answers
I think it's C, sorry if it isn't!
A room is 14 yards by 18 yards. Find the cost to carpet this room if carpet costs $2.30 per square yard
Answers
Answer:
$579.60
Step-by-step explanation:
First, we need to find the area of the (floor) of the room.
Area = Length * Width
We can plug in the given values into the equation.
Area = 14 yd * 18 yd
Area = 252 yd²
We can set up a conversion factor to see how much it costs to carpet the room...
[tex]\frac{\$2.30}{1 \ yard^2}*252 \ yards^2[/tex]
Cancel out the yd²...
$2.30*252=$579.60
I need help now pls
Answers
Answer:
22/10 or 11/5
Step-by-step explanation:
There are two (2) whole and two tenths (2/10).
You have to multiply the 2 whole by the denominator and add it to the numerator.
Step by step:
1) 2 and 2/10
2) (2*10) = 20
3) add 20 to 2, then put it on the /10
4) 22/10, then reduce the fraction
5) 11/5
Final answer: 11/5
Find the area of the shaded region.
Answers
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.
Someone tell me the answer
Answers
Answer: 6:1
Step-by-step explanation: you’re welcome
A net of a rectangular prism with dimensions of 18 inches, 20 inches, and 35 inches is given What is the surface area of this rectangular prism? Enter your answer in the box __
Answers
Answer:
The answer is 3380
Step-by-step explanation:
i took the test
WILL GIVE 100 POINTS!!!!
On the map of an amusement park on a coordinate grid, the skating rink is at (−1.5, −1).
The coordinate grid is shown from negative 3 to positive 3 on the x-axis and negative 3 to positive 3 on the y-axis. There are increments of 1 over 2 for each grid line on each of the two axes. Only the whole numbers are labeled on either side of the axis.
Which of these shows how to plot the point to mark the skating rink?
From the origin, move 1.5 units to the left along the x-axis and 1 unit vertically down, and place the point.
From the origin, move 1 unit to the left along the x-axis and 1.5 units vertically down, and place the point.
From the origin, move 1 unit to the right along the x-axis and 1.5 units vertically down, and place the point.
From the origin, move 1.5 units to the right along the x-axis and 1 unit vertically down, and place the point.
Answers
Answer:
The first option is correct.
Moving 1.5 units to the left (or -1.5 units to the right).
Moving 1 unit down (or -1 unit up).
Therefore you get the result of (-1.5, 1) as is the location of the rink.
Step-by-step explanation:
PLS GiVE ME BRAINLIEST
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
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
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
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 :
https://brainly.com/question/18802258
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The closest chicken shack to the city of portales is more than 125 miles away. Which two inequalities represent d the distance of chicken shacks from portales
Answers
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]
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
To learn more on Expressions click:
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Angela's text messaging plan charges 30¢ for each message over 600 in addition to a $9 base charge. If
she owes $36.60 for text messaging in the month of July, how many text messages did she send that
month?
Answers
answer: 692
Step-by-step explanation:
36.6 - 9 = 27.6
27.6/0.3 = 92
600 + 92 = 692 text messages
A tree is 7 feet 5 inches tall. How tall is it in inches
Answers
Answer:
The tree is 89 inches
Please mark as Brainliest
Use the figure to solve the inequality
Answers
Answer:
c hope this heps
Step-by-step explanation:
If you draw a single card once from a deck of ordinary playing cards, what is the probability that it will be (a) The ace of diamonds
Answers
Answer:
1/52
Step-by-step explanation:
hope this helps
Nicole has completed 36 deliveries so far this week. She needs to make 45 deliveries for the week. What percentage of her deliveries has Nicole completed?
Answers
Percent = 18 / 45
Reduced to 2 / 5, which is 40 percent.
Maria has 40 percent of her work completed.
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
Carlos is on the swim team. Each week he swim a total of 2000 meters. How many kilometers does he swim each week?
Answers
Answer:
2 kilometers
Step-by-step explanation:
1000 meters = 1 kilometer
2000 meters = 2 kilometers
If you need to setup a conversion
[tex]\frac{2000m}{1}[/tex] × [tex]\frac{1km}{1000m}[/tex]
The meters cancel
[tex]\frac{2000km}{1000} \\[/tex]
Reduce
2 km
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:
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
find the sum of 4/10 + 20/100 = show work
Answers
Step-by-step explanation:
Multiply (4/10) by 10 to find the sum using common denominators:
[tex] \frac{4 \times 10}{10 \times 10} = \frac{40}{100} [/tex]
Add:
[tex] \frac{40}{100} + \frac{20}{100} = \frac{60}{100} [/tex]
60/100 is your unsimplified answer.
Divide (60/100) by 20:
[tex] \frac{60 \div 20}{100 \div 20} = \frac{3}{5} [/tex]
3/5 is your simplified answer.
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!
2(x+5)-7=3(x-2)
Pls help ! find value of x
Answers
Answer:
dasd
Step-by-step explanation:
Given:
2(x + 5)-7= 3(x - 2)
To Find:
• Value of x
Solution:
⇒ 2(x + 5)-7= 3(x - 2)
⇒ 2x+10-7= 3x - 6
Rearranging the like terms,
⇒ 3x-2x = 10 + 6-7
⇒x=16-7
⇒x=9
Verification:
⇒ 2(x + 5)-7= 3(x - 2)
Putting x = 9,
⇒2(9+5)-7= 3(9 - 2)
2(14)-7= 3(7)
⇒28-7=21
⇒ 21 = 21
→ LHS = RHS
HENCE VERIFIED!!!