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Consider the equation22 = €e(a) Sketch the functions in this equation and then use this to explain why there are two solutions and describe where they are loc...

Question

Consider the equation22 = €e(a) Sketch the functions in this equation and then use this to explain why there are two solutions and describe where they are located for small values of (b)Find tWo-term asymptotic expansion; for small &, of each solution. Find three-term asymptotic expansion, for small &, of each solution_

Consider the equation 22 = €e (a) Sketch the functions in this equation and then use this to explain why there are two solutions and describe where they are located for small values of (b)Find tWo-term asymptotic expansion; for small &, of each solution. Find three-term asymptotic expansion, for small &, of each solution_



Answers

Finding an Exponential Function In Exercises $21-24$ , find the exponential function $y=C e^{k}$ that passes through the two given points.

In this problem. Why is it going to C to the power Katie as our function? So the graph passes to 0.0 full. So y zero is equal to fool is equal to see to deposit zero. So see the equal to food. Therefore, why is equal to four into the power? Katie, Since the grab passes through a pipe and won by two points So why five is equal to one by two is equal to 48 to the power five k going forward. I can write the value. Age eight of the power five k is equal to one by eight. So five k physical too long. One by eight. So K is equal to one by five long one by which is equal to minus 0.41 5 19. Now I will write the equation. So the question of why you become 42. The power minus 0.415 19

Hello. Welcome to this lesson. In this lesson we have two points open, exponential kev. And we would want to know which car it is. So the why is the dependent variable? And the tea is the independent variable? So, uh, task will be to find the C N A K. So that we can mother the whole thing. Okay, So what we do at this point is to find the covered the point. So at one and five we have the five us, the Y and the T. As the one. So C. E. Okay, so okay, times one that is key. Okay, so this becomes a question one. Also at five and two at this point we have the Y. That is the to the C. E. Then we have the tea, that is the the five. So then we said five key and this is the push into. So from me question one, we can make see the subject by dividing by eat to the park. E for both sides. The whole of these councils that only fc is in 4 to 5. E key. Okay, so we put this into equation two. Okay, so you have to is equal to in place of C. We put in five. Eat the parquet. Then you have E. The about five K. Okay, so this becomes too in the sport to five. This becomes okay. Eat the bar negative K. Eat the bar five K. Okay, So here, according to the loss of indices, we have positive five minus A K. Positive five K minus care. That is positive for key. Yeah. Five. That is E. And we have four. Okay, Okay. Now we can take the five to contact the five to divide both sides so that we live the four K. Together with the E alone. So this councils that we have zero 0.4, that is two or five. Let's go to eat the power for Cape. Now we can take the lane of both sides so that we can leave the key together with the four alone stepping down from the E. Yeah so this becomes the land of the lane observe 0.4 that is negative 0.91 6 to 90 7319 in Mexico. So the land will step down the four key. Okay so here Kay would become negative reserve point the whole of that. All over the four so that the key becomes mhm negative 0.2 291 for the smart places. Now that we have the care. Remember we made a supposition that she was called to. Mhm. The substitution will see is a call to E. To the power yeah five over E. To the park. E The walls for the question too. The side. Mhm. So all that will do is that we would put in the value of K. The found value. Mhm. Okay. And this is approximately 6.2873 and four this month places. So the whole question why is it called to six point to gauge 73? It to the partner negative 0.2291 T. Okay. So this is a question them with those that cat transfer time. This the end of the lesson.

This lesson in this lesson we have a curve that is an exponential curve. And that is modeled by why is equal to see then the experiment. Okay and T. Okay so our duty is to find the sea in the uh okay that would model the wire based on the graph wall. What we do is that we find the why are the points that was given in a question? So at four and 5. Yeah we have the Y. S. Five and the T. S. Four. So that is C. E. Then you have four key. So let's make this pushing one. Yeah. Then we have at 32 at this point. Also we have the Y. Us half C. E. Then the T. S. Tray. So that is three p. This is Dick Ocean too. Yeah. Okay so from a question too We can make see the subject. So she becomes one over to E. three key. That is you can divide both sides by Eat the Path three Key C. would be the subject. So we put this into equation one. The Question one is already five physical to see E four K. So this becomes five. That is he called. So in place of the sea, put in one over to E. Three K. Then times e. four key. So that is five is equal to Yeah, one of the two then times we are 14. We have we are four K. Here and three kids there so that three key goes up Subtracted from the four keys. You know, you have just a key. Okay, so here we multiply both sides by two. So you have five times to support 1/2 then times to again. So that this causes out that neon can is equal to E key. Now we'll take the lane on both sides that will be able to drop the case. And this becomes laying 10. Is he called to keep because the lane and the exponential uh mhm. Yeah. They negate themselves okay. Or they they allow themselves right? Yeah. So K. Is equal to yeah. Uh huh. Yeah. 2.30 259. That is the value of key. So now that we have Okay, let's look at where we substituted earlier. That is this bad? Yeah. Uh huh. Oh yeah. No. Mm hmm. Okay. So we have the K. That is not one of us to and three 2.30- 59. So that C is equal to 0.00 05. Okay. So we can model the cool Wyatt, 0.0005 E. then 2.30 to six. That is in for the smart places. T all right. So that is what will give us this line. All right. So thanks for your time. This the end of the lesson. Yeah.

Uh huh. This problem, we wish to buy the taylor series expansion for the folks at Quebec equals E X at c equals one. This question challenges our understanding of taylor series. The taylor series expansion at X equals C is given by equation one below. We see that in order to solve what we need to do is identify the truth is an F Plug into those derivatives see or x equals equals one. And then identify the given formula. So first final derivatives we have F prime equals E acceptable problems E X and so on. S N is by exponential differentiation. Always going to be either the X. So playing in C equals one to our functional conservatives gives FFC equals equals E. F prime of Z equals E. One equals E and so on. That's our series is F x equals E plus E x minus one plus over two. Factorial X minus one or the n over n. Factorial X minus one of the end. That's a series of some K equals +02 infinity over K. Factorial X minus one to the cat.


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