the-current-i-in-mathrma-as-a-function-of-the-time-t-in-mathrms-in-a-certain-electric-circuit-is-g-2

Name: the current i in mathrma as a function of the time t in mathrms in a certain electric circuit is g 2
Uploaded: 2021-06-12
Duration: 2 min 9 s
Channel: Numerade
Description: The current $i$ (in $\mathrm{A}$ ) as a function of the time $t$ (in $\mathrm{s}$ ) in a certain electric circuit is given by $i=(4 t+3) /\left(2 t^{2}+3 t+1\right) .$ Find the total charge that passes through a point during the first second.

Question

The current $i$ (in $\mathrm{A}$ ) as a function of the time $t$ (in $\mathrm{s}$ ) in a certain electric circuit is given by $i=(4 t+3) /\left(2 t^{2}+3 t+1\right) .$ Find the total charge that passes through a point during the first second.

Vikash Ranjan · Accepted Answer

in the problem the coupon did he. Which is equal to current. So this has given us 40 plus three A bomb duty Squire plus three d. Plus one. So here 40 plus three upon to t quiet plus three T plus one is written as airborne To T. Plus one. Blessed be upon D. Plus one so far there you can write this ass 40 plus three that equals to a. N. Two T. Less fun plus B. Into twice of T plus one. So after solving this we have the value of a equal two b equals two one. So now we have the accu integral that equals to duration two upon to T plus one glass one upon D. Plus one DT Ladies. Q. F. D. Equals two. Ellen mode 30 plus one plus Ellen mud D. Plus one plus the so we have finally U. F. D. Equal to Ellen moored to T. Plus one plus Ellen monte plus one plus the so this is the answer.

Cinsy Krehbiel · Answer

here we have a charge function Q of T. And we want to find out the current is a function of time current capital I of tea that&#x27;s equal to the first derivative of the charge function. So to do that, we&#x27;re gonna have to use the product rule in the chain rule. So let&#x27;s start by defining our products. The first is F of tea. It&#x27;s a fft better tea in their goodness. So fft it&#x27;s going to be eating the tea and this one&#x27;s ice goes F prime of tea is also equal to e to the T and then our other function GFT is going to equal three who sign of two t plus the sign of two t So Jean prime of tea I know how to use a but a chain rule The first derivative of the co sign of acts is equal to negative Sign of X and the first derivative of two tea is too. So we&#x27;re gonna have a negative two times the co sign of excuse me times the sign of two t and we multiply that by three. We end up with negative six times the sign of two t The first derivative of the sine function is the co sign, and the first derivative of two tea is, too. You would have plus to times the co sign of tune Tea and Our product. Roll says. If we have two functions F and G and we multiply them together and then what? The first derivative we&#x27;re gonna have to take f Times G Prime and add to that F prime times Jeep. So let&#x27;s go f is eating the tea times g prime. That&#x27;s negative. Six times the sign of two T plus two times the co sign of two t We&#x27;re gonna add to that F prime. So that&#x27;s again E to the T times G, which is three times the co sign of two t less the sign of two t and now we&#x27;re gonna combine up the like term. So let&#x27;s do our signs. First, we&#x27;re gonna have either than tee times. Negative six. Sign of two t and E to the T times. One sign of Tootie&#x27;s If you add those together, Negative six plus one. It&#x27;s negative. Five e to the t times the sign of two T, and we&#x27;ll do the same thing with our coastline terms, So we&#x27;re gonna have positive two times e to the T Times Co. Cited to t plus Positive three times eating the tee times coastline to t. So that&#x27;s plus five e groups. But instead of G five e to the t times the coastline of two t. So if we want a factor out either the tea and factness factor out of five e to the T and we&#x27;re going to swap the order. So we&#x27;re have a leading positive, so we&#x27;ll start with this guy five feet of the tea. What&#x27;s left behind is the co sign A to T and then minus the sign to t, and that is our current function.

Cinsy Krehbiel · Answer

were given a charge function. Q. Of T. And we want to determine what is the current current capital. I of T is the first derivative of the charge function with respect to times Q Prime of tea. And for this it&#x27;s going to get a little bit complicated. We want to be careful. We&#x27;re gonna use the product rule and the chain role. So let&#x27;s define FFT as e to the negative to t It&#x27;s a FFT equals E to the negative to t and then f prime of tea use the chain role and say, Well, the first derivative of E to the negative to t would be either the negative to tee times the first derivative of negative to T, which is negative two. So we&#x27;ll get negative too e to the negative to t that are other function GFT That&#x27;s gonna be this part right here. It&#x27;s a GFT is going to be co sign of three T minus two times the sign of three T, and the first derivative of that again will have to use a little bit of the chain role. First derivative of co sign is negative sign and the first derivative of three. Tia&#x27;s three. So we&#x27;re gonna end up with negative three times the sign of three T and for our second term, the first derivative of Sinus co side in the first derivative of three. T is three, so we&#x27;d have three co sign of three tee times. Negative two is negative six times the co sign of three T and our product rule tells us that if you have f times G and you want to take the first derivative, you&#x27;d say times g prime plus g times f prime. So let&#x27;s go ahead and do that F That was each of the negative to t times g prime. That was negative three times the sign. But keep that need. It&#x27;s gonna be hard enough not to make any mistakes. Alright, negative three times the sign of three T minus six times the co sign of three t and we&#x27;re gonna add to that g Times f prime. So we&#x27;ll put it in the other order. Well put f prime first. So it&#x27;s going to be negative, too no to the negative to t Times G, which is co sign of three T minus to sign of three T Now we could clean this stop, and there are several ways to go about it. But let&#x27;s go ahead and do it sort of step by step so we don&#x27;t mess anything up, so we&#x27;ll distribute those eat of the negative to tease. We eat with the negative to t times Negative three Sign. Excuse me. Negative three sign of three t so we&#x27;ll have negative three e to the minus two T sign of three T That&#x27;s our first term. Minus six e to the negative to T co sign of three t What&#x27;s that term minus two e to the negative to t times the co sign of three T that&#x27;s that term negative. Two times negative two is positive four plus four e to the minus two T times this sign of three T Now that we can go ahead and combine up our like terms so we&#x27;ll go ahead and take our to sign terms and combine them and we get for you to the minus two T minus three e to the minus. Two tea leaves us e to the minus two T times the sign of three T, and they will also combine up our our co sign of three teas. We get negative six feet of the minus two T minus two. So that&#x27;s minus eight E to the minus two t times the co sign of three t We&#x27;re almost there. One more step. It&#x27;s factor out that e to the negative to t So we end up with E to the negative to t times the sign of three T minus eight times the co sign of three t and then we take a quick moment to pat ourselves on the back and be super proud cause we got it right.

Louisa Benatovich · Answer

we are given an equation. Q of t equals three sign to t plus t plus four and we&#x27;re told that this is the charge. Then we&#x27;re asked to find the current when t equals zero and t equals one. So a little tidbit of information that we should know before we get started is that the current is the derivative charge with respect to time with Back T, which is basically what saying I am t ive. He is a commonly new symbols. Her eyes common five piece equals Q priority. So yet first up derivative my lovely friend here because here will be like Q T equals three. Sign to t plus let&#x27;s do some mapping que perimeter t So the derivative of three. It sign duty. This three co sign to tee times the derivative of the inside change your old times two plus the derivative of tea, which is one and then the derivative of a constant is zero. So rewriting that I&#x27;ve t equal six co sign t t plus one. Then we were asked to compare it. At T equals zero and tables, so hi zero way have six coastline two times zero plus one coastline of zero is 16 times one plus one equals seven already. And then we have one That would be six co sign do terms. One plus coastline of to is at some sort of decimal number that we don&#x27;t know if the top of her head from the unit circle so you could write it as six co sign, too. And if you&#x27;d like to plug it into your calculator, you make. But here we go. You know the answer. Seventh only the answer at one six goes into.

Louisa Benatovich · Answer

were given an equation. Q. Of T that represents the charge of a circuit, and it&#x27;s equal to four coastline 40 minus three d plus but and were asked to compare the current AT T equals zero anti equals one. But we&#x27;re given the charge here that won&#x27;t get us the current, but we do know that the derivative of Q T is equal to the current with respect to sign so respected. That&#x27;s like writing. Q. Prime of tea is equal to pi of Tea eyes letter typically used to represent card. So our first Matthey step is to find the derivative of this equation that we see at the top official. Rewrite it at the bottom. I can t the ankles or co sign for a team. 31st 1 first step Q. Prime of tea que priority equals for times Negative. Sign 14. Using the chain General Times Another floor minus three arrested er, votive of three teat and then the derivative of a constant is you know if we will be right, that a little nicer. That&#x27;s five t pulls negative 16. Sign for good. Remember at the top, we were asked to find and compare t equals zero and T equals once will do some plugging five t her. I have zero equals negative. 16 sign for zero minus three before time. 00 Sign of zero is zero. Remember that the unit for current, his lamps. So we&#x27;re amps three camps. Then we have to find I had one once equals negative. 16 sign Hello. Four times like that. I just eliminated that sign. The floor isn&#x27;t something we know from the unit. Circle off the top of our heads. And because most cut classes in college don&#x27;t require calculators, you won&#x27;t be able to figure this out so you could just write it as it is. Negative 16 Sign for minus three camps. Those my friends are your two answers.

Yiming Zhang · Answer

Okay, so now we have ah, initial value problem. And the way of solving it seems to take up a transformer off both sides. So the question res double derivative. If I turns four, I equals through GFT. So the love was transformed for it will be lf lf second through two of I you cold plus alibi four times alibi. So here we apply the linearity. And, uh lf tree of tea so but a theory and five in the textbook we know for the double derivative, it&#x27;s a legal to the S squared times. I guess mine is s comes out zero and then minus I f zero I prime of zero. So this is equals two. We&#x27;re plugging all the initial value. It&#x27;s miss score times by a vast minus Fast miss three. So now we have this sum love transform. And, uh, it&#x27;s gonna be so on the right hand side, we have a square times I minus ASA minus three plus four I and the only Roy Insight we haven&#x27;t CGT is actually on three time sci fi time sworn months. You team on this to pie, and they love has transformed for this function will be three terms one once eat the naked to Pi s over as squared close one. So here we applied. Ah, therm six 38 from the textbook So now we have everything and let&#x27;s just rewrite everything together We know that I have a pass he calls as plus three over a squared Plus four Who us three times one Muncie to make it through pious over escort force one times x squared plus four. And, uh, we&#x27;re gonna take the English Let us transform to this equation one. Of course, we&#x27;re gonna apply linearity again. And also the through mate from the textbook. So for the 1st 1st part is actually a sober A squared plus four plus three over a squared plus four is the second part we have. So we on through the partial fraction decomposition It will become one minus e to the t to buy us times when the West Square pulls one minus one the West score plus four. Okay, so by that, apply the inverse X transform into each term. We&#x27;re gonna end up with co sign to a T minus three over to sign Tootie plus 70 minus 1/2 of Sai into itty, then minus U of T minus two pi times sign of T minus two pi minus 1/2. Some sign of two comes T months to pie. So that&#x27;s the final results in the simplified er&#x27;s Let&#x27;s see what we got. Society plus sign Tootie Poor schools into a tree. Pull us U of t minus two pi times. That&#x27;s it was 1/2 times tongue sign to t minus sign of tea, and that&#x27;s the final result.

Problem

Solve the given problems by integration. The forc…

Problem 34 Easy Difficulty

The current $i$ (in $\mathrm{A}$ ) as a function of the time $t$ (in $\mathrm{s}$ ) in a certain electric circuit is given by $i=(4 t+3) /\left(2 t^{2}+3 t+1\right) .$ Find the total charge that passes through a point during the first second.

Answer

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