5

Ezt 0 <t<2 (o t <0 2 <t<4 let f(t) = t 0 <t<t; 9) = ~2t 4 <t<5 sin(t) T < t 5 < t(a) Write each function in terms of the unit step ...

Question

Ezt 0 <t<2 (o t <0 2 <t<4 let f(t) = t 0 <t<t; 9) = ~2t 4 <t<5 sin(t) T < t 5 < t(a) Write each function in terms of the unit step function (b) Plot each function (c) Find the Laplace transform of f(t) and g(t) Find the inverse Laplace transform of the following functions:(a) (b)(c) (+1)+1

ezt 0 <t<2 (o t <0 2 <t<4 let f(t) = t 0 <t<t; 9) = ~2t 4 <t<5 sin(t) T < t 5 < t (a) Write each function in terms of the unit step function (b) Plot each function (c) Find the Laplace transform of f(t) and g(t) Find the inverse Laplace transform of the following functions: (a) (b) (c) (+1)+1



Answers

Determine the Laplace transform of each of the following functions:
(a) $f_{1}(t)=\left\{\begin{array}{ll}{t,} & {t=1,2,3, \ldots,} \\ {e^{t},} & {t \neq 1,2,3, \ldots.}\end{array}\right.$
(b) $f_{2}(t)=\left\{\begin{array}{ll}{e^{t},} & {t \neq 5,8,} \\ {6,} & {t=5}, \\ {0,} & {t=8.}\end{array}\right.$

In the booth. A if one D waas 01 team. Do you want to know that is not so First of all, um, if one day you may I wonder what is this? We have far me. Yeah. F duty is given us fire to the finest at was you have to take six empty they know because one and six So they left last in worse. And the is yes, my own is Do it is on one about We know we have the other part which is boxee Well, FTD they won t So the lesson worse are everybody is given that I was gonna bone his squad. So FDA is continuous. You don't know? Hey, this is NASA.

Hi. Today we're looking at the last blast transom of three different functions. If we look at the question, though, what's common to all of them is that they are equal to each of the tea. Except for finally many values. T since the last last transformer only depends on the values of the function except of finally many points. We know that the last trip it transforms any of the three functions is gonna be equal to one over X minus one, which is the same as the last blast. Transform off each of the tea. However, the inverse lap last transform of one over X minus one. It's only equal to eat the tea because the end of a slap US transform always takes the continuous function. Who's like last chance? One is the function given and this function e to the T, which is the third function, is the only function which is continuous at the three

Okay, so we have this function that's defined piece wise from zero to pi over two and then from pi over to two pi And it's also periodic in pie. So we've got that Big T is equal to pi so we want to figure out the LaPlace transform. So we use theorem 10.3 point three in the book. This gives us that it is 1/1 minus e to the minus see into from zero. So, um okay, so into So this is big t so big tease pie now. So it's that zero to pi of e to the minus s t times f of t d t. Of course, we'll have to split this integral up from going from zero to pi over two in them from pirate to two pi. So let's call this I So I is equal to zero to pi over two e to the minus esti times fft fft is to t over pie and then into grown from pi over to two pi of e to the minus s t times fft But fft now it's sine of t So this is okay now we've got to integrate this. So let me call this guy. I won. And then we call this I tune. Let's figure out what I want and I two r so as an aside, I won. We're going to do this by integration by parts. So we're gonna differentiate the tea and integrate the exponential. So that's gonna leave us with this for the boundary term, right? I So it's you Devi. So now this is U V. The minus integral. So v. So the integral of this is gonna be is going to give us plus one over us E to the minus esti. And then the derivative. This would just be one. And so what do we get? So let me swap the limits around to get rid of this minus sign. So we're going to get minus one over us times by pi over two times by e to the minus pi over two times s and thats plus one over us. Okay? And so that's minus pi over two. Yes, e to the minus pious, over to And then stop the limits here again just to get rid of the minus sign. That's plus one over r squared into one minus E to the minus pi t over. Pi s over to okay. And then we can combine this all into home and denominators two s squared that IHS So it's minus pi e to the minus pious over to okay. Plus, um oh, after multiplied by ass course minus pi s. And then this is plus to the minus two eats the minus pi over two. Now for integral to that's this. So I'm gonna different cheat this and integrate this so that's going to be minus caused t and then leave this alone for the boundary term. And then that's minus. So now I have to have to integrate this indifferent sheet. This so that's plus S e to the minus s t then caused t so that picks up another minus. Sign DT this boundary term right here. Zero because I know it isn't sorry. Careful. So this boundary term is eat the minus pious and the minus will cause a pirate too is zero. So this is just e to the minus pious, okay, and then this is minus asked times. Now we do integration by parts again on this guy. So that's integrate this and differentiate. This minus s multiplying. So this becomes e to the minus s sine of t evaluated between pi Over to two pi on the minus into go from minus pirate to two pi of plus s e to the minus Esty. Yes. Can come out Sign T DT, right. You ve minus V D u. Okay, And now let's simplify this. So sign of pi zero. So that's just gonna give us minus e to the minus pi the pie s remember where substituting for T values and then sign of Pirated is just one that's plus s. So notice that this is exactly I to again. So I two is equal to this. Okay, so let's just simplify, okay? And so I to into one plus s squared is equal to this right hand side. And so I two is equal to one plus s over one plus x squared times e to the minus pious. Okay? No, I made a mistake. And I hope you caught that pious over to right. Because I'm substituting teas equals pi over to the t easy with the pie thing. The teasing The pie part is zero. Because sign up. I zero. So let's track back. Okay on. And then this is just gonna be okay. So that's what I two is. So I want is this I've slightly simplified it, and I to is this. Now let's go back to original problem. Right? So we're trying to figure out I write, we split up into these two into girls. So So eyes equal to this. This is equal to two over pi times by I one. But I want is this business right here? So adding them together we get this the twos council. Well, I could bring the pie over here, so that's what I is equal to. And so our answer is going to be whatever we got for I that must miss expression times by this factor. Okay, so therefore the LaPlace transform of f is going to be 1/1, minus e to the minus. Pi s into into all of this business. All right, let me let me write it back so that you write it out again. If you want, you can combine the denominators. I didn't bother doing that. Okay, so that should be the LaPlace transform back

Okay. So for our first question, we've got enough of t zero to t squared between zero and two and everywhere else while its periodic. So this right here implies that periodic with period Big T is equal to two. And this is sort of what the graph might look like. Okay. And so on and so forth. Now, from the room 10.3 point three in the book, the LaPlace transform is going to be given by this. Okay? And so this is equal to Well, big T is equal to two. So this is that This is interval from 0 to 2 of its the minus s t t squared. DT. Now aside, let's figure out what this integral right here is. Well, this is clearly going to use integration by parts. We're gonna want to differentiate this polynomial and integrate this exponential. Okay, so this is going to be given by. So if you like you is equal t squared on D V by D. T is equal to eat the minus esty. So this is U times V, and I'm and and you evaluate that between the bounds and then that's minus V D u d t but the minuses cancel. So you get this. So evaluating this one, this is gonna be minus one over us. E to the minus. Two s times T squared two squared, which is that's four. So remember where evaluating TZ equals two and t Z equals zero No s. Now, finally this. What is this? So this is to over us times the integral from zero to t of tea times eats the minus s t d t. So again we identify you in DVD t like this. So on the other side, we have to do integration by parts again. That's going to be so minus one over S T E to the minus Esty. Serious too minus within the council of the minus one over this. Now let's think to go from 0 to 2. Me to the minus esty T t. So now this is minus one over s times two times eats the minus two s. Plus, I was just gonna be minus over a squared. Need to the minus two s minus one. Okay, so this is equal to this. All right, so what's that going to give us? And so now, combining this all on over. A common denominator of s cubed we get. So we get this when we combine it. And I just factored out into the monies to us and we get this. So therefore, this is what the integral from 0 to 2 of t squared e to the minus s t d t is. So if we go back, the plaice transform of F was given by this. And so if we plug everything in, we get eats the minus two s into minus forest squared minus four s minus two plus two K over s cubes. Times this and this is the answer the LaPlace transform.


Similar Solved Questions

5 answers
QSYM' S"' 13M-/ S-Ea 3.31 J/mol-K tokloudkSolue kc Ea : ;' plase show) detaled wovk
QSYM' S"' 13M-/ S- Ea 3.31 J/mol-K tok loudk Solue kc Ea : ;' plase show) detaled wovk...
5 answers
Iding Taylor Series In Exercises 1-12, use the iinition of Taylor series to find the Taylor series, centered at €, the function_
Iding Taylor Series In Exercises 1-12, use the iinition of Taylor series to find the Taylor series, centered at €, the function_...
5 answers
You will need to generate a set of graphs to answer this question:Concentration vs time data for the following decomposition reaction was collected at 300 K The data is shown in the table below:What is the rate constant for the reaction at this temperature (in units of min-1)?HzOz(aq) 5 HzO(l) + 1/2 Oz(g)time (min)[H2O2] (mol/L)00.043735000.0286810000.0187315000.0122620000.00802
You will need to generate a set of graphs to answer this question: Concentration vs time data for the following decomposition reaction was collected at 300 K The data is shown in the table below: What is the rate constant for the reaction at this temperature (in units of min-1)? HzOz(aq) 5 HzO(l) + ...
5 answers
For the following data set: (use megastat)Apply the diagnostics technique for the predictor variable and comment, Find the residuals. Find the standard residuals, Draw the sequence plot ofthe standard residuals and comment. Test the autoconelation of the etrors Using un wppropriate test at 5"0 signifieance level, Estimate the Variance of the model' etton,
For the following data set: (use megastat) Apply the diagnostics technique for the predictor variable and comment, Find the residuals. Find the standard residuals, Draw the sequence plot ofthe standard residuals and comment. Test the autoconelation of the etrors Using un wppropriate test at 5"0...
5 answers
Suppose that the population P(t) of country satisfies the differential equation =kP(400 - P) with k constant Its population in 1960 was 100 million and was then growing at the rate of million per year Predict this country's population for the year 2010.This country's population in 2010 will be million. (Type an integer or decimal rounded to one decimal place as needed )
Suppose that the population P(t) of country satisfies the differential equation =kP(400 - P) with k constant Its population in 1960 was 100 million and was then growing at the rate of million per year Predict this country's population for the year 2010. This country's population in 2010 wi...
5 answers
For the systemhemoglobin $cdot mathrm{O}_{2}(a q)+mathrm{CO}(g) ightleftharpoons$ hemoglobin $cdot mathrm{CO}(a q)+mathrm{O}_{2}(g)$$K=2.0 imes 10^{2} .$ What must be the ratio of $P_{mathrm{Co}} / P_{mathrm{O}_{2}}$ if $12.0 %$ of the hemoglobin in the bloodstream is converted to the CO complex?
For the system hemoglobin $cdot mathrm{O}_{2}(a q)+mathrm{CO}(g) ightleftharpoons$ hemoglobin $cdot mathrm{CO}(a q)+mathrm{O}_{2}(g)$ $K=2.0 imes 10^{2} .$ What must be the ratio of $P_{mathrm{Co}} / P_{mathrm{O}_{2}}$ if $12.0 %$ of the hemoglobin in the bloodstream is converted to the CO complex...
5 answers
[email protected] 3Find the following given the information above:J 29 (2) dzQuestion 4Find the following given the information aboveJ5 [29 (2) #3] d2Questio
[email protected] Question 3 Find the following given the information above: J 29 (2) dz Question 4 Find the following given the information above J5 [29 (2) #3] d2 Questio...
5 answers
In January, a car-maker performed a safety test on 1000 new cars and 92% passed the test: In March another 1000 cars were tested and 89% passed. Report the Z statistic you would use to test the null hypothesis that the percentage of cars that passed the safety test in the two samples are the same_Z-1.14Z-2.27Z-1.17Z-0
In January, a car-maker performed a safety test on 1000 new cars and 92% passed the test: In March another 1000 cars were tested and 89% passed. Report the Z statistic you would use to test the null hypothesis that the percentage of cars that passed the safety test in the two samples are the same_ Z...
5 answers
Please show Step by step solutions with every single step#1 let y-3x^2+5x+3. if delta X-0.3 at x-5, use linear approximation to estimate delta y ? #2 let y-8x^2+5x+3. if delta X-0.3 at X-5, use linear approximation to estimate delta y ?
Please show Step by step solutions with every single step #1 let y-3x^2+5x+3. if delta X-0.3 at x-5, use linear approximation to estimate delta y ? #2 let y-8x^2+5x+3. if delta X-0.3 at X-5, use linear approximation to estimate delta y ?...
2 answers
) Evaluate the commutators [A,B] of the following pairs ofoperators A and B.(a) A = q2 and B = d2 /dq2(b) A = q (the position operator) and B = p (the momentumoperator)
) Evaluate the commutators [A,B] of the following pairs of operators A and B. (a) A = q2 and B = d2 /dq2 (b) A = q (the position operator) and B = p (the momentum operator)...
5 answers
Asystem of ice and water at 0 C weight 3 gr. Thesystem is converted to ice at 0C by removal of 668 Jof heat. Howmuch ice was injtially present. Heat of fusion of ice 334 J/g1g2g2.5g3.5g3g
Asystem of ice and water at 0 C weight 3 gr. The system is converted to ice at 0C by removal of 668 Jof heat. How much ice was injtially present. Heat of fusion of ice 334 J/g 1g 2g 2.5g 3.5g 3g...
5 answers
Point) Write the complex number z = -4 8i in polar form: z = r(cos 0 + i sin 0) where sart(80 and @ arctan(2+pi) The angle should satisfy 0 < 0 2t _
point) Write the complex number z = -4 8i in polar form: z = r(cos 0 + i sin 0) where sart(80 and @ arctan(2+pi) The angle should satisfy 0 < 0 2t _...
5 answers
Problem 2) Consider the following LP:Min 2x1+4x2+8x3+x4s.t. x1 +x2 +x3+x4 >= 102x1 -x2 >= 3 x1, x2, x3, x4 >=0a) Write the dual LP. (2 points)b) Writecomplementary-slacknessconditionsfortheprimalanddualLP:(2points)c) Given that the optimal solution to the primal LP is(x1,x2,x3,x4)=(1.5,0,0,8.5), use theStrong Duality Theorem to find the optimal solution to the dualLP. (2 points)
Problem 2) Consider the following LP: Min 2x1+4x2+8x3+x4 s.t. x1 +x2 +x3+x4 >= 10 2x1 -x2 >= 3 x1, x2, x3, x4 >=0 a) Write the dual LP. (2 points) b) Writecomplementary-slacknessconditionsfortheprimalanddualLP:(2points) c) Given that the optimal solution to the primal LP is (x1,x2,x3,x4)...
1 answers
If $ f $ is continuous on $ (-\infty, \infty) $, what can you say about its graph?
If $ f $ is continuous on $ (-\infty, \infty) $, what can you say about its graph?...
5 answers
Find d2yldx2 in terms of x and y. Y3 = xdyldxz
Find d2yldx2 in terms of x and y. Y3 = x dyldxz...
5 answers
Atwhat speed would an electron's mass be measured as 4 times its rest mass?0.88c0.97c0.94c0.25c0.063c
Atwhat speed would an electron's mass be measured as 4 times its rest mass? 0.88c 0.97c 0.94c 0.25c 0.063c...
5 answers
(b) Consider the helicoidal surface represented parametrically byr(u,v) = u cos(v)i + u sin(v)j + vkwhere (u,v) € D and D = {(u,v)l 0 < u <1,0 < v < 2w}. Find a normal vector N to the helicoid using the formula dr dr N = du dv[4 marks] ii_ Caleulate the eurved surface area ol the helicoid using the formladr" A(S) fLIo dudv dv[4 marks]
(b) Consider the helicoidal surface represented parametrically by r(u,v) = u cos(v)i + u sin(v)j + vk where (u,v) € D and D = {(u,v)l 0 < u <1,0 < v < 2w}. Find a normal vector N to the helicoid using the formula dr dr N = du dv [4 marks] ii_ Caleulate the eurved surface area ol th...

-- 0.018903--