for-the-following-exercises-find-the-slope-of-a-tangent-line-to-a-polar-curve-rftheta-let-xr-cos-t-9

Question

For the following exercises, find the slope of a tangent line to a polar curve $r=f(	heta) .$ Let $x=r \cos 	heta=f(	heta) \cos 	heta$ and $y=r \sin 	heta=f(	heta) \sin 	heta, \quad$ so the polar equation
$r=f(	heta)$ is now written in parametric form.
Use the definition of the derivative $\frac{d y}{d x}=\frac{d y / d 	heta}{d x / d 	heta}$ and the product rule to derive the derivative of a polar equation.

Benjamin Schreyer · Accepted Answer

So in this problem were asked to find D Y d x at a certain data for a polar function. So what is the slope of the tangent line for a polar function at some data. And we know that if we have some polar function, R equals F data than X equals. If time stayed effort data times co sign data and y equals FF data times signed data. So what we can do is find d x. Oh, I mean, find d Y d theta and divide that by d x dif data and then the d fate is in the denominators will cancel. So we get d Y d X, which is what we were looking for. And it will be in terms of theta because we had dif ada in both of our in our numerator and our denominator. So first we&#x27;ll find de y d theta by taking the derivative of why with respect to data. So we know by the product rule that&#x27;s going to give us af of state, uh, times and the derivative of sign is co sign data and then plus f derivative of they&#x27;d, uh, times signed data. So just the product rule being applied here and now we&#x27;ll find DX dy theta, which is going to be again product rule. But on X of data, which is f if data times negative sign data is the derivative of co sign data is negative. Sign data. You said a little glitch there. Times negative sign data. Plus And now we have f derivative of data Times Co sign data. So now we just need to divide the two by each other. So d y eat, they&#x27;d ah, over de X de fate. Ah, equals we say, actually, I&#x27;ll just bring these using the power of computer magic. I&#x27;ll just bring this where we need it now. So we&#x27;re taking our d y d theta and we&#x27;re dividing it by r d x d theta And we know that d y d faded, divided by D X teeth Data is also equal to d just d Y d axe. So that&#x27;s how you that is our definition of the derivative for a polar with respect to X and y

Benjamin Schreyer · Answer

so in this problem were given that we have a polar curve are that is equal to four times coasts on data. And we are asked, what is the slope or D y d x of our at polar coordinates to comma pi over three? And to answer that, we used the definition of the slope for a polar at some angle, which is this, and you can see we already have our And so the only thing we don&#x27;t know in this equation is drd Keita. So what we&#x27;ll do is we&#x27;ll say we&#x27;ll take our our formula and take d d data of both sides find the derivative of our respective data. So Sam on this side and four times co signed data and so the derivative of co sign is negative signs. We get negative for times. Signed data equals D R. De data. So now evaluating d r d theta and are and signing co signs evaluating. Do you Why d x at the angle pi over three, Because do I d actually going to see it takes data and its input, and that gives us the answer, which is Route three over three, which is a little more than 1/2

Benjamin Schreyer · Answer

All right. So in this video, we are asked to find the slope of the tangent to the polar curve. R equals two theta at the position Magnitude pi over two angle pi over four. So we have the formula for the slope of a polar at a given angle theta and the only thing that we don&#x27;t know inside of this formula is drd fate us. We&#x27;re gonna quickly find out what that is. So we just take the derivative of our with respect to data. So and this is quite simple because we&#x27;re taking the derivative of a linear function of data. So we just got drd Fate ah is just a constant to And then we use our d Y DX formula and now we know drd fate us so we can calculate a number using the formula and we get negative pi plus four over pai minus four as our d. Y d. X And let me write this a little better R d Y d x with an angle of pi over four. So I use a function notation. Do you i d x at high over four is equal to negative pi plus for over pi minus four

Bofan Zhu · Answer

the tips off the lives shows thanks from points off most interest. So the tips are where the upfront off our is natural maximum. So for Ari coachable two times science to receive him. There&#x27;s occurs once Science theater kind three Shit! Are you coach your to or minus you? So she&#x27;s high. He called you Hi overseas by over two off by over six on the brother over So well said I could you How overseas are equal to chew? We&#x27;ll see how you go to power over six. Are you coach? You minus two So we can know The intro is front You two pi over too so we can get the riddle. Geez, not this form And the less execute you are terms. Course I in C town and why he coached you our Tom size You talk and they used the products rules. We can&#x27;t know that for a while. Over d excess. You got to do what Albert, is it? Hop over the X over it. Is it? No. So it&#x27;s equal to G our course I see it, huh? Over this&#x27;d over the our science over. Is it mystical Chu two rules off three there is equal to minus two run off And I, uh, on the miners true to pyro virtue the dirty off X And why will be another seas? Just use Listen, used us and, um, process the production offered Roll Let&#x27;s g y over. D x is equal to what you want over D set up over DX over dish. It s equal. Chu, do you our science yet, huh? Over you see it, huh? Over tea Our course And he said over kiss It is Uncle Chu dear. Over two stories Secrets of General So we can know the you can know that slop is minus uh, your three the square root of after it adds two to pi over thinks and the sub is zero Mind that you over a pile of virtue. So what do you see? The front every doubt. So it

Benjamin Schreyer · Answer

So in this problem, were given the polar curve r equals four times co sign tooth data and were asked, What is the slope or D y d x at each, um, angle theta that is the tip of a leaf. So if we were to plot if you remember, co sign with a multiple inside will give you some kind of flower. And if you have a two or uneven number, then you&#x27;ll have two times two, which is four pedals, and they will be evenly spaced around the origin. So our graph is gonna look like this four paddles starting at, um zero angle and increasing 90 because they&#x27;re evenly spaced. So we know the tips are gonna be at We have one at zero angle, one at pi over two angle, one at pi angle and one at three pi over two angle. So so solving. For we need to find the d. Y d X of this curve at a given angles that we can plug in these angles we found. So we know everything in this formula except for d r d theta so quickly we&#x27;re gonna take the derivative of, um, are with respect to data by just taking the derivative on both sides of the equation d d theta And that is going to give us, obviously, on the left d r Do you think, uh, and on the right four times or negative four times signed data. So now that we no are de are de Feydeau weaken usar d y d ax here on all of our angles And so we&#x27;re gonna do D y d x ah zero equals And it&#x27;s gonna be undefined we&#x27;re gonna find, because there&#x27;s gonna be a divide by zero in our formula. The denominator here will be zero in the case of zero degrees and then, well, do do I d x for pi over two. Where will get zero? Because the numerator will be zero and the denominator will be non zero. And then one of our last ones D Y d x at high is well, we know that co sign of pie is just negative. Ah co sign of Well, we know that co sign of pie, which is in our our is going to just be the negative version of itself and sign of pie is going to be zero So it&#x27;s gonna work out to be that once again, our d y d X is the undefined plugging in to our do I d x formula after finding d r d theta c and c d r d theta Do you i d axe That&#x27;s what we used to find our slope each time. And then finally the deep the slope d y d axe at three pi over two which is our last leaf is once again zero. So we found what our graph looked like, which we know is going to be a symmetrical leaf rotation Aly symmetrical leaf with since our coefficient on data was even, it&#x27;s gonna be two times the coefficient, which was two leaves on our graph equally spaced starting here. So we know that there are before leaves starting at zero pi over two apart. So we generated are angles where the leaves will be and then we found Do I d. X at each of those angles using the formula for D y dx of a polar and as on extra, we need to define d r d theta, which was just simple. There derivation are not deprivation, but taking the

Ben Brown · Answer

the um slope of the tangent line is to the curve given by our is equal to a woman a sign vater at the point where already go to 1/2 and vater is equal to pi over six. So today this one of our not want d y t x is he would too. But this is also too. Do you want my defense, sir? Despite de um t x de fainter we take bm invest minus one of them So let us find out What do you want by defense? The wind by d&#x27;you Fator It&#x27;s like a gun. Why eagle to our science later Where are is all one minus sign factor here So we put it in to get sign feet on the sign. Oops. Sewing squared fatal. So we different Jamis expression. Do you worry about the fatal We get co sign later minus two. Um sign Fator Because just give in. Now we want to put in, um faces. He was part of a six. Well, that is so sign sign. Why have the six you want to off and they clear up. We&#x27;ll carry a sign. When I was six. Sequel to square it well free too. This, um so expression. So we&#x27;re gonna get So do you want by dif a toe? I&#x27;m going to get a square roots of free number two minus two times. Um, so it&#x27;s time for you know. So it&#x27;s a heart times scourge for free. I would too. Now, yeah, Ms is equal to So we keep there square roof forever too. It&#x27;s not minus. Um Well, it was too, and this half gets killed off. So it was just left with this. It&#x27;s great. A free of a cheap walk minus it. So it&#x27;s Scrabble free of it, too. Minus squire. It&#x27;ll free, too is equal to zero. And we then therefore because Because, um, the numerator here. Yeah, I&#x27;m very sorry because the numerator is derived by de fazer, but that&#x27;s equal to zero. We don&#x27;t even need to care about what, Deep exploiting favorites, because, um, well, because the numerous zero therefore the whole expression zero Therefore, this slope or off a ton of line is see right

Problem

For the following exercises, find the slope of a …

Problem 235 Easy Difficulty

Answer

we use product rule to derive the derivative of a polar equation.

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we use product rule to derive the derivative of a polar equation.

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