find-the-equation-of-the-line-tangent-to-y2sin-x-at-xpi-2-graph-the-function-and-the-tangent-line

Name: find the equation of the line tangent to y2sin x at xpi 2 graph the function and the tangent line
Uploaded: 2020-06-16
Duration: 6 min 10 s
Channel: Numerade
Description: Find the equation of the line tangent to $y=2^{\sin x}$ at $x=\pi / 2 .$ Graph the function and the tangent line.

Question

Find the equation of the line tangent to $y=2^{\sin x}$ at $x=\pi / 2 .$ Graph the function and the tangent line.

Philip Christensen · Accepted Answer

the equation of why equal to Mari equal to sign Moved to the sign of X. Excuse me. I need to find the tangent line the point X equal to hi over to. So to find the tangent line, we first need to find the point that the line we&#x27;ll cross work, meet with the initial graph. So we need to plug in hi over to to our equation. So sign of pi over two so we can get that to to the sign Hi were to It&#x27;s going to equal sign up high over two is One said this goes to a one equals two to the one which is equal to two So we need our tangent line to cross the point Hi or two to now we need to find the rate of change at that current point So we need to take the derivative. So to do this we need to use algorithmic differentiation. So we get natural log of y equal to sign of X natural log to take the derivative of both sides and we get why prime over Why? Because the derivative of natural walk. Why is why times the derivative of why my prime is equal to the derivative of sign of X times The natural look too. And the derivative sign of X is co sign axe Times match along too. Now we need to multiply both sides by Why So we get why prime equal to co sign X natural log of to times two to sign X Now we can plug in hi over to at in this equation because we have now found a derivative. And we can quickly do this because co sign it was a horrible see oh, sign hi over to zero. So we know that since that is equal to zero, we get zero times the natural water, too times to to the sign Hi over two. But since we have a zero multiplied by everything, this goes to zero. And when we have a one y prime equal to zero, that means our tangent line going to be horizontal. So are tangent line. We&#x27;re going to call this line t equal to using the equation and X plus be and I m is 00 and be equal to two as we solved above. So we get t is equal to to and that is the equation for a tangent line. But the pro also says we need to graft this. So I am going to drag in window for graphing calculator and so we can see quickly are graph of to Science X already And now I can turn on the plot for our tangent line. Why is equal to two? We can see that it just crosses and meats our initial graph, you know, just a little. And there we go. You need to plot with a raft calculator. Copy down this graph to to the sine x No, I equal to two. And this is Dismas is a free online graphing calculator for anyone to use as well. But I am going to switch back to our solution here. Zoom helps for all of the work. There we are.

Will Erickson · Answer

weather in this problem, we are going to find on equation of the tangent line, huh? For disc, Irv X squared plus sign Wine equals X Y squared plus one. And we&#x27;re going to find the equation of this line, which is tangent at the 0.1 comma zero. All right, Um, so, as always, we will use the point slope form at the very end for our Tangela and we have the point. We just need to find the slope, which means we need to take the derivative and then we can plug in the 0.1 comma zero. So let&#x27;s begin. As always, you can tell we&#x27;re gonna need implicit differentiation. So let&#x27;s take the derivative of both sides with respect to X and see what we get on left. The derivative X squared is to x the derivative of sign wise co sign y. But because why is a function of X? As always, we multiplied by y prime on the right. We&#x27;re going to need a product rule here since we have x Times y squared. So let&#x27;s follow the product rule. So the first part is X times the derivative of the second on the derivative of Y squared is too. Why and then there&#x27;s always be multiplied by y prime. Here comes the second half of the product rule. It&#x27;s the derivative of X is just one and then multiply that by y squared. And the derivative of that plus one is just zero. Okay, so we&#x27;ve taken the derivative. Now all that we have to do is solve for y prime since we want the derivative. So no matter what were we do, we&#x27;re gonna have to shuffle around a few terms. Let&#x27;s just decide that we will keep all the y prime terms on the left side of the equation. So, in other words, our plan will have to move this term with the white prime over to the left by subtracting it and we&#x27;ll want to move this to X over to the right by subtracting it. So just algebra here, what do we have? So, on the left side, we will still have co sign. Why times y prime, you will also. I&#x27;ll put it in red here so you can see where it came from. We&#x27;re going to subtract this term from the right since it had a white prime. Let&#x27;s rewrite a little and better orders of two x y y prime. All we did is copied from the right side, and we subtracted it from both sides. So it ends up on the left on the right side of the equation. We still have our y squared from before. And then we also subtracted that blue term from the left. So minus two X. Okay, now we&#x27;re ready to factor out the Y prime from the left. So that just leaves. Co sign why, minus two, Lex. Why? And then on the right, nothing changes the final step to get our derivative. Just divide both sides by that parentheses there and we have are derivative. And so there&#x27;s why prime at any point, I mean his plumbing the point, and that&#x27;ll give us the slope. Know specifically we care about the 0.1 common zero. Right? So that means way really want is the slope at that point. So we want why prime restricted to the 0.1 comma zero. All right. And that will give us No, it&#x27;s just plugging one for X and zero for wire. So why zero X is one? Why zero Texas one. And why zero So on top, you can see we have negative too. On bottom. Coastline of zero is one, Remember? So this here is just one. And then obviously this is zero. So the bottom is just one. And so our slope was negative too. And I finally remember how the point slope form works. Why? Minus our white coordinated zero. Um And then we get our slope. We said his negative too. Our X coordinate was one. And if we want, we can clean this up just a little bit to get negative two x plus two. And that&#x27;s the equation of archangel on. Hopefully this helps.

Nicholas Barvinok · Answer

we have the function, Why equals sine X plus sign Square X and we want to find the equation of the tangent line to that function at the 0.0 Now the first thing we should check is that 00 is actually a point on the graph of dysfunction. By plugging X equals zero, I get sign of zero, which is zero plus sine squared of zero, which is zero times 00 So I get zero and the 00.0 is indeed on the graph of this of this function. So that&#x27;s good. So we want the equation of the tent of some tangent line to find the equation of a line in general, it is sufficient to know the slope of the line and a point on the line so that we can then used this standard formula for the equation of a line. Why, minus y one equals and the times x minus X one where X one y one is a point on our line and em is the slope. Well, we already have a point on our line. The tangent line to this function at 00 passes through 00 So we have a point on our line. The only thing we need to find is M the slope of our tangent line and the slope of our tangent line at 00 will be the derivative of this function at X equals zero. So what we should do next is take the derivative of dysfunction. So the derivative of Sine X is co sign X. And now we use the chain rule. We have an inter function sign of X and an outer function and input raised to the power of two by the chain rule. We start with taking the derivative of the outermost layer, which is an input squared. Take the derivative of that by the power we bring the two down, keep the input the same and subtract one from the original exponents that becomes exponents of one and then multiply by the derivative of the input. The inter function sign of X. The derivative of that is co sign next and this is a one in fact, not a prime. So here&#x27;s our why Prime, We can now plug in zero to get the slope of the tangent line at that point co sign of zero is one plus two times sign of zero is zero and then times one. So this term is zero. So this whole thing equals once our slope is one. So now we just plug in the relevant data into our equation, and it simplifies to y equals X, and that&#x27;s it.

Gregory Higby · Answer

eso these tangent line problems you typically need the X coordinate Negative pi over two. It typically need to like, well, you definitely the white core. And you need the slope. While the slope is always the derivative at a point, Um, this is usually the notation that we write it as And the reason why these are the things you need to know is I like to write my answers and point slope form expanse your X coordinate, plus your wide corners. So maybe best fire like X one y one. Because we&#x27;re you know, this x coordinate. Um, and they also tell you what the equation is, which is the seeking cubed of pirate two minus x. Um, So the first thing I would do is plug in pi over two into this problem. Well, as you do that, um, and be pirate two plus high over to, um, which is two Beira, too. So this is basically the same thing is I would write it this way. One of her co sign cubed of pie. Um well is you&#x27;re looking in a circle Pi is over here. Uh, that&#x27;s the order. Para native 10 So CO sign a pious negative one when you cubit is still negative on one over naked one is negative one. So when I get to this point, you know we&#x27;re gonna have a plus neighbor on r minus one here. And the other thing you need to do then is find the derivative. Well, that&#x27;s definitely a chain roll. It&#x27;s actually change role inside of a chain role. But we&#x27;ll get to that. You bring the three in front. It&#x27;s now seeking squared of pirate two minus X. They had to take the drift of of seeking, which is seeking tangents will make the seeking cubed tangent of pirate to money. Sex. I kind of ruined things for us times the drift over the inside, which is negative one. So I&#x27;m just going to take a negative for the reason I say this actually ruined stuff for you is right. This piece right here because tangent is signed over co sign Well, if signing zero over negative one will zero divided by anything is still zero. I don&#x27;t care what this is. I don&#x27;t care. This is zero times something times something is still zero. So the drift of actually call zero. So the long answer is, if you want to your right. Like will&#x27;s zero x minus the X coordinate. So plus pirate too. Plus your white corner, which is negative one. But what? Zero times? Anything at zero. No need to write That you know is right. The horse on the line y equals. Think of one is a tangelo on here?

Louisa Benatovich · Answer

here we have. Why equals f of X at X equals a ever backs equals X science and it equals pi. Over two were asked to find the equation. Police attendant Line four. This function underlined in green. At this point, a equals pi over two. Well, what is the line Look like? A line is y equals MX plus B. We know this formula and what is the director? Derivative shrinks pills he derivative slope park attentions function. So the derivative of a function is the slope of the tangent of that function. And then you couldn&#x27;t plug in any given point like high over to to get a slope at a particular point. So, yes, we shall start where it makes the most sense. Finding the derivative of the function f of X equals X sine X equals X sign. Next. Yes, we&#x27;ve got some product rule action here X times signing. So remember, it&#x27;s the first times the derivative of the second plus second time for the derivative of the first. That&#x27;s always a tongue twister for me with that being you in that being vite. So without mantra, analyze The first is X times the derivative of the second, The derivative of Sine X is co sign. It&#x27;s plus the second sine x times, the derivative her first, which is one. So you know, that&#x27;s what that prime of excess. This is the general formula for the derivative, the general formula for the tension. But we wanted at a specific point a equals, however to, But we can&#x27;t just plug a equals pi over too into eggs. They&#x27;re not the same variable, but we were told over here that X equals a so that&#x27;s basically like saying X equals pi over two so we can happily plug it in. F Crime of pi over to my pies. Just get her says the problem goes on pi over two times. Co sign Pi over two plus sign high over too well, Coastline pi over two is zero. This is zero. So zero times anything is zero that&#x27;s gone. Sign Pi over to think about the unit circle in your mind. Sign Pi over two is one. So remember that the derivative is the slope left basically saying that M equals one. This is our slow, but we&#x27;re not quite done. We look back at y equals mx plus B, we have em. We don&#x27;t have beat that. We can&#x27;t directly find Pete. But what we do have is X. We figured out earlier that X was pi over two. If we plug X in to the original equation F of X equals X, I next. Then we can find why. Because y equals f of X. It&#x27;s all coming together. So down here, half of X equals X side X Come out when we plug in X equals pi over two. Probably right. Better. What professional color F, uh, high over to just pride Overshoot Time signed by over to sign pot over two is one. So one times pi over two is just piratey. That&#x27;s a lot of high over twos, so f of X is signed pi over two. We know this is actually why. So we have executed over two Weigel over to, which gives us a great coordinate point of pi over two. Pilot Jim. So now we have everything we need to plug into our y equals MX plus a B base equation. I&#x27;ll be right to slope. We&#x27;re slope with one Michael&#x27;s. I&#x27;m explicit. E We know why it&#x27;s pi over two. You know I am. That&#x27;s that one we know. Except also pi over two plus speak are variable When we saw for B be nickel zero. So when we rewrite that with everything, we know em being right in the middle, Why equals ex? Ah, lovely diagonal lying. Let&#x27;s rewrite. That&#x27;s why to make it better. And that, my friends, is the answer y equals X.

Gregory Higby · Answer

eso There&#x27;s tangent. Um, attended line problems. You need the X coordinate the y coordinate through on Comex with my one and the slope. Well, the slope is equal to the derivative at that x coordinate. Um, and they tell you that X is the square root of pi squared of high on. They also tell you that why is equal to tangent of four x squared? Uh, yeah. So if you were to plug in the square root of pi in right here kind of overshot that Anyway, the square to pie when you square it, um, this will tell you that you have the tangent of four pie because the square in the square would cancel on the unit circle is just too full revolutions. That&#x27;s your repair. 10 in tangent signed, Divided by coast. Final zero divided by one is zero. Um, and then we need to figure out the derivative of that. Well, this is definitely a chain rule where the derivative of tangent is seeking squared. You leave that for X squared alone times a drift of of the inside, which is, um, eight X. So now you need to plug in route pi into this. Um, so we have seeking squared of four pi times Ate fruit pie? Well, just like what I mentioned before will seek it is one of her co sign will co sign is the X coordinate. So it&#x27;s one, um So seeking a four pies, one squared is just one. So this is equal to me. All right. Over here. Eight route I So the three pieces we need we&#x27;ve found and I use points up for Michael&#x27;s em X minus your X coordinate, plus your y coordinate. So as you look at this, you just say OK, well, the AM is a true pie X minus root pie and plus zero. So there&#x27;s no need to write that. I would just leave my answer like this. You might see somebody distribute this in like a troop I X minus group I times replies. His pie so ate pie. This could be acceptable as well

Problem

The graph of $y=(\cos x) \ln \cos ^{2} x$ has se…

Problem 95 Hard Difficulty

Find the equation of the line tangent to $y=2^{\sin x}$ at $x=\pi / 2 .$ Graph the function and the tangent line.

Answer

y = 2

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