Use the definition of the laplace transform to show that if f(x) = 0 then
\(l[f(x)] = 0\)
show that f(x)= 1 then
\(l[f(x)] = \frac{1}{s} \)
show that f(x)= x then
\(l[f(x)] = \frac{1}{ {s}^{2} } \)
show that f(x)= e^ax then
\(l[f(x)] = \frac{1}{s - a} \)
provide the steps by using the definition and evaluating the integral.
Answers
Answer:
Step-by-step explanation:
the Laplace transform of the function f(x) = e^(ax) is 1/(a-s).
The definition of the Laplace transform of a function f(t) is given by:
L{f(t)} = F(s) = ∫_0^∞ e^(-st) f(t) dt
where s is a complex number.
If f(x) = 0, then we have:
L{f(x)} = L{0} = ∫_0^∞ e^(-st) 0 dt = 0
Therefore, the Laplace transform of the zero function is zero.
If f(x) = 1, then we have:
L{f(x)} = L{1} = ∫_0^∞ e^(-st) dt
Using integration by parts, we get:
L{1} = ∫_0^∞ e^(-st) dt = [-e^(-st)/s]_0^∞ = [0 - (-1/s)] = 1/s
Therefore, the Laplace transform of the constant function 1 is 1/s.
If f(x) = x, then we have:
L{f(x)} = L{x} = ∫_0^∞ e^(-st) x dt
Using integration by parts again, we get:
L{x} = ∫_0^∞ e^(-st) x dt = [(-e^(-st) x)/s]_0^∞ + (1/s) ∫_0^∞ e^(-st) dt
Since e^(-st) x approaches zero as t approaches infinity, the first term evaluates to zero. We can then simplify the second term using the result from part 2:
L{x} = (1/s) ∫_0^∞ e^(-st) dt = 1/s * (1/s) = 1/s^2
Therefore, the Laplace transform of the function f(x) = x is 1/s^2.
If f(x) = e^(ax), then we have:
L{f(x)} = L{e^(ax)} = ∫_0^∞ e^(-st) e^(ax) dt
Simplifying the integrand, we get:
L{e^(ax)} = ∫_0^∞ e^((a-s)t) dt
We can evaluate this integral using the formula:
∫_0^∞ e^(-bx) dx = 1/b
Setting b = a - s, we get:
L{e^(ax)} = ∫_0^∞ e^((a-s)t) dt = 1/(a-s)
Therefore, the Laplace transform of the function f(x) = e^(ax) is 1/(a-s).
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Related Questions
Simplify the expression. Write your answer using an improper fraction
with no spaces.
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Answer: 59/3k+14
Step-by-step explanation:
Simplify √-24 usingimaginary i
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Given:
\(\sqrt{-24}\)Apply the laws of exponents:
\(=\sqrt{-1}\sqrt{24}\)Apply the properties of imaginary numbers:
\(i\sqrt{24}\)Rewrite in standard complex form:
\(2\sqrt{6}i\)Answer:
\(2\sqrt{6}\text{ i}\)What is the distance on the
coordinate plane between the
points (-2, 16) and (-2,-5)?
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the distance on the
coordinate plane between the
points (-2, 16) and (-2,-5) is 21
how do i give brainliest? also please solve the problem :)
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Answer:
When there are 2 answers there will be a crown you can click and that will give brainlest
Step-by-step explanation:
=
a + b + c
2
=
8 + 10 + 5
2
= 11.5 inches
Area = √s(s-a)(s-b)(s-c)
= √11.5×(11.5 - 8)×(11.5 - 10)×(11.5 - 5)
= 19.810035335658 inches
Answer:
the answer is 19.81 inches
State whether the following statement is true or false. The Law of Sines can be used to solve triangles where three sides are known Choose the correct answer below. A. False, because to use the Law of Sines, all three angles must be known B. True, because to use the Law of Sines, all three sides must be known. C. True, because to use the Law of Sines, at least two sides must be known D. False, because to use the Law of Sines, two angles and one side or two sides and one angle must be known.
Answers
C. True, because to use the Law of Sines, at least two sides must be known.
The Law of Sines is a trigonometric rule that relates the sides and angles of any triangle. It states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle.
To use the Law of Sines, at least two sides and the angle opposite one of them (or two angles and one side) must be known.
Therefore, the statement "The Law of Sines can be used to solve triangles where three sides are known" is false, as it is not necessary to use the Law of Sines to solve a triangle where all three sides are known.
In summary, the correct answer is C. True, because to use the Law of Sines, at least two sides must be known.
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What is the value of the lower quartile?
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Answer:
not sure
Step-by-step explanation:
lolshjshebshuzhsbbsjxhgdpsizhgxe
Calculate all four second-order partial derivatives and check that . Assume the variables are restricted to a domain on which the function is defined.
Answers
The function is defined on the given domain, we need to make sure that all the partial derivatives are defined and continuous within the domain.
To calculate the four second-order partial derivatives, we need to differentiate the function twice with respect to each variable. Let's denote the function as f(x, y, z).
The four second-order partial derivatives are:
1. ∂²f/∂x²: Differentiate f with respect to x twice, while keeping y and z constant.
2. ∂²f/∂y²: Differentiate f with respect to y twice, while keeping x and z constant.
3. ∂²f/∂z²: Differentiate f with respect to z twice, while keeping x and y constant.
4. ∂²f/∂x∂y: Differentiate f with respect to x first, then differentiate the result with respect to y, while keeping z constant.
To check that the function is defined on the given domain, we need to make sure that all the partial derivatives are defined and continuous within the domain.
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Plz i wanna answer of this question i will mark a brainlist for how will solve it completely
Answers
Answer:
by construction —c = 7 unitsb = 13 unitsHop it's helpful!
Step-by-step explanation:
check my attachment for detailed explanation!
B= 12 units
please help me and show your working out
Answers
Answer:
i can give you tuturing 1 on 1 if you give me brainliest
Step-by-step explanation:
30*4 =120 +50
30*9=270+50
Write the equation in standard form for the circle with radius 4 centered at the origin.
Answers
Answer:
(x-0)²+(y-0)²=16
Step-by-step explanation:
The formula for the equation of a circle is (x-a)²+(y-b)²=r², with a,b=center coordinates, and r=radius.
Hope this helps!
We need to write the standard equation of circle with , center (0,0) and radius 4 unit . The Standard equation of the circle is:
\(\longrightarrow ( x - h)^2 + (y -k)^2 = r^2\)
\(\longrightarrow( x -0)^2 + (y-0)^2 = 4^2\)
\(\longrightarrow \bold{x^2 + y^2 = 16 }\)
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I need help please!!!
Answers
Answer:
D
Step-by-step explanation:
A translation 4 units to the right and 6 units down means that all of the x values increase by 4 and all of the y values decrease by 6.
Adding 4 to all of the x values and subtracting 6 from all of the y values gives us the coordinates:
A' (-3, 0), B' (8, 3), C' (2, -9)
Give the equation of the graph of y = 2 cos 3x + 1 after it has been stretched vertically by a factor of 2, flipped upside down, and stretched horizontally by a factor of 2.
Answers
Answer:
Step-by-step explanation:
So we have the following chain of transformations
\(f(x)\to 2f(x)\to -2\f(x)\to -2f(\frac{1}{2}x)\\\)
So now
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Tom ate 1/3 of the pizza and Jayden ate 3/10 who ate more
Answers
Answer:
1/3 is your answer
Step-by-step explanation:
Reason is bc 1/3 is a bigger slice
:)
Hope this helps
*
What is a degree two polynomial function called?
Answers
Answer:
a quadratic function
Step-by-step explanation:
A quadratic function is a second-degree polynomialfunction, and when graphed is called a parabola.
does anyone know this?
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Answers
Answer:
-3
Step-by-step explanation:
Since x=1, we can simplify the equation to 9+7y=-12.
To find the value of 7y, we first have to find what is missing.
-12-9=-21, so 7y has to be equal to -21. -21/7 equals -3, so y=-3.
Estimate the sum by first rounding each
mixed number to the nearest whole number
and then adding.
5 and 3/8 + 4 and 7/10
Answers
4 7/10 rounds to 5
Your estimated sum is 10
Give the most specific name for the quadrilateral based off of the picture. Please don’t post random things thanks
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Step-by-step explanation:
parellogram
two opposite sides are equal
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Solve the equation 4x-12=(11-3x)
Answers
Answer:
Step-by-step explanation:
4x - 12 = 11 - 3x
Bringing like terms on one side
4x + 3x = 11 + 12
7x = 23
x = 23/7
The vertices of the quadrilateral JKLM are J(-2,0), K(-1,2), L(1,3), and M(0,1). Can you use the slopes
and/or the distance formula to prove each statement?
Select Yes or No for A-C.
A. Quadrilateral JKLM is a parallelogram.
А
Yes
B
No
B. Quadrilateral JKLM is a rhombus.
А
Yes
B
No
A. Quadrilateral JKLM is a rectangle.
A
Yes
B
No
Answers
Answer:
A. Yes
B. Yes
C. No
Here's what it would look like:
What is the answer to 200 x 32.6
Answers
Answer:
6520
Step-by-step explanation:
Hope this helps :3
Find the slope of the line passing through each of the following pairs of points.
(−3, 9), (−3, −5)
Answers
Answer:
-14/0
Step-by-step explanation:
Well, we know that the formula for slope is rise over run. You might be thinking well what does that mean. It means we are going to take our y values over our x values. So, it will look something like this: -5-9/-3-(-3). Pretty simple right! Now heres a tip, when you see two "negatives" like -3-(-3), you can automatically change it to look like: -3+3! Remember this won't work unless you have those two negatives next to each other. So it doesn't apply to -5-9. Okay lets get to solving. We know that -5-9 is equal to -14 so our equation will now look like: -14/-3+3. We also know that -3+3 will equal to zero. So now our equation looks like -14/0. This should be your answer because you cannot simplify it further.
Please let me know if I was incorrect or if you have any further questions!!! I hope I helped!!
Answer:
the slope is undefined
Step-by-step explanation:
anything over 0 is undefined
NUMBER 3!!!!!!!!!!!!!!!!!
Answers
Answer:
if 1 equal to x
and 2 equal to x-4
that mean
(x)+(x-4)=180
2x-4=180
2x=184
x=92
lets start to find the angles that marked with numbers
1=92
2=88
3=88
4=92
5=92
6=88
7=88
8=92
that's it
Have a great day
In pensacola in june, high tide was at noon. the water level at high tide was 12 feet and 2 feet at low tide. assuming the next high tide is exactly 12 hours later and that the height of the water can be modeled by a cosine curve, find an equation for water level in june for pensacola as a function of time (t). f(t) = 12 cospi over 2t 5 f(t) = 5 cospi over 2t 12 f(t) = 5 cospi over 6t 7 f(t) = 7 cospi over 6t 12
Answers
An equation for water level in june for pensacola as a function of time (t) is f(t) = 5 cos pi/6 t + 7.
Which equation of cos show period amplitude ?
The equation given below show aplitude and period
\(y = A cos bx + c\)
where A = amplitude,
b = 2 pi/Period,
Period = 12 hrs,
c = midline,
x = t and y = f(t)
We have to find the amplitude
What is the formula for the amplitude?
\(A = 1/2 (Xmax - Xmin)\)
\(12 - 2 / 2 = 10/2 = 5\)
\(b = 2 pi / 12 = pi/6\)
\(c = 1/2 (Xmax + Xmin)\)
\(12+2/2 = 7\)
Therefore, the an equation for water level in june for pensacola as a function of time (t)
\(f(t) = 5 cos pi/6 t + 7\)
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How do you solve this?? It says solve for X.
Answers
Answer:
x = 7
Step-by-step explanation:
Ok, so these angle are equal, because they are opposite angle
So we could write as x+16 = 4x - 5
Now we solve the equation:
x+16 = 4x - 5
16+5 = 4x -x
21 = 3x
7 = x
if 8 men can build a house in 90 days, in how many days can 20 men working under the same rate build the house?
Answers
It will take 36 days for 20 men to complete the building
What is the range of the function f(x) = -2(6x) + 3?
0 (--,-2]
• (-0,3)
o [-2.00
O (3.00)
The answer is B
Answers
(because you said so lol)
The range of the function plotted on the graph is (- ∞, ∞).
What is function?A function is a relation between a dependent and independent variable. We can write the examples of function as -
y = f(x) = ax + b
y = f(x, y, z) = ax + by + cz
Given is the function -
y = - 2(6x) + 3
Range of a function is the set of {y} - values that exist for the values in the domain of the function. The given function is -
y = - 2(6x) + 3
y = - 12x + 3
Now, the slope of the function is {m} = - 12. The {y} - intercept of the function is {c} = 3. The function represents a linear function. Now, the linear function exist for all the possible values of {x} along the entire domain from (- ∞, ∞). The graph will be a straight line extending from (- ∞, ∞). So, we can write the range as (- ∞, ∞).
Therefore, the range of the function plotted on the graph is (- ∞, ∞).
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cual es el factor de 10?
Answers
________
________
Answer:
Step-by-step explanation:
1,2,5
Si ya tienen 2/3 y 13/7, cuanto falta para completar 4 enteros?
Answers
To complete 4 whole numbers using the fractions 2/3 and 13/7, a total of 31/21 is needed.
To determine how much is needed to complete 4 whole numbers using the fractions 2/3 and 13/7, we need to find the sum of these fractions and subtract it from 4.
First, we need to find a common denominator for the fractions. In this case, the least common multiple (LCM) of 3 and 7 is 21.
We convert the fractions to have the same denominator:
(2/3) x (7/7) = 14/21
(13/7) x (3/3) = 39/21
Now we add the fractions:
14/21 + 39/21 = 53/21
We subtract this sum from 4 whole numbers:
4 - 53/21
To perform this subtraction, we need to have the same denominator:
4 = (4/1) x (21/21) = 84/21
We subtract the fractions:
84/21 - 53/21 = 31/21
Therefore, to complete 4 whole numbers using the fractions 2/3 and 13/7, a total of 31/21 is needed.
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Please help me with the question please ASAP ASAP please ASAP
Answers
Answer:
V = 672; SA = 544; LA = 448
Step-by-step explanation:
Yo can someone help me out please?
Answers
Answer:
56.52in^3
Step-by-step explanation:
V=πr^2h
V=3.14×3^2×6in/3
V=3.14×9in^2×6in/3
V=3.14×54in^3/3
V=3.14×18in^3
V=56.52in^3