solve-the-given-problems-find-the-slope-of-a-line-tangent-to-the-curve-of-yfrac2-sin-3-xx-where-x0-3

Question

Solve the given problems.
Find the slope of a line tangent to the curve of $y=\frac{2 \sin 3 x}{x},$ where $x=0.15 .$ Verify the result by using the derivative-evaluating feature of a calculator.

Edward Downes · Accepted Answer

Okay. This question is asked us to find the slope of the tangent line with this equation at X equals 0.15 And then verify it using the derivative finding function on the calculated. Okay, so we need to take the derivative clearly, we&#x27;re going to use quotation role. So why prime equals X. The denominator times the derivative of the top. The derivative of the top will be six. Co sign three X. That three popping out in front minus the derivative. The bottom which is one times the original top which is to sign three X. All over the denominator squared. Ok. We don&#x27;t have to worry about simplifying this down too much because we&#x27;re substituting a number in here. We&#x27;re substituting in that 0.15 15 times three will be a 30.45 Make a little bit easier calculations. Okay, substituting in 0.15 Sorry about that everywhere we see an X. Yeah, we&#x27;re dividing by 0.5 square. Okay, we use that calculator for this number crunching So set up a fraction 0.15 six co sign 0.45 Okay, my ass to sign 0.45 divided by 0.15 squared gives us negative 2646 Oh. This is the slope of the tangent line. We can just check that. Using the derivative. Finding function on a calculator. Okay, on the T I 84 I&#x27;ll show you where that is. Second. Okay, we&#x27;re gonna go into math, we&#x27;re gonna go down here to end the riff. We&#x27;ll find us a numerical derivative. We&#x27;re looking for DDX we&#x27;re gonna put our function in here. We&#x27;ll set it up as a fraction is to sign three X over X. And we wanted the numerical derivative at 0.15 So we put that in here and there, it confirms it for us. So, slope of the tangent line at that point negative 2.646

Ernest Castorena · Answer

Teoh compute the slope of white defined by this function at this point, well, we could be let either implicitly over here are explicitly over here, so I&#x27;m physically we just differentiate with respect to X. You don&#x27;t know the river wires. So what is get why prime here next? Then we&#x27;re going to do the product rule for the second time. So take the derivative of X squared leave Why alone on leave like squared alone Take the dirt of y with respect X and then do the same thing for the right hand side. Then when you can collect our white prime terms so far it plug in the 0.0 comma one to find the slope of that point, which is a zero. And then there&#x27;s the explicit version where we can use the equation to solve for white, much well done here and then take the derivative of that I did. Product rule here will have negative sign of X or co sign. Leave the second factor Ireland. Then leave co sign a loan, take the route of the second factor over here. And now let&#x27;s plug in at zero comma one. So then why primal zero here will be well, actually signing 00 So this whole first term right here, get zeroed out and then we&#x27;ll have co sign of zero, which is just one, and then be multiplying by negative 1/1 minus zero squared, which is just Ah, negative one multiplied by negatives, XX. But that&#x27;s also as Europe. So we end up with zero plus zero, which is just a zero. All right, they agree.

Amy Jiang · Answer

Okay, so we look to find a spoke over tangent line. Given our xlu, what s Time X is given to be one that means are a value is also equipped to button So it&#x27;s find are following slope are using this equation here But we have other wrecks and then we need up great Well, we said that are a value is equal to one So we got one cube which is equal to one and then we also have X minus a which is equal to X minus 81 Um, now let&#x27;s put this into our equation here. So we said are a volley was one We have anthrax that&#x27;s X cubed minus a one over X minus one. So let&#x27;s recall doubts here leukemia are one as one cubed and then we have the difference of two cubes, so we can factor this accordingly. So that&#x27;s going to be X minus one times x squared plus x and n plus one all over X minus one. And now let&#x27;s cancel out our like terms and used Rex up to find a falling limit. So we have one squared plus one. What&#x27;s one. So we shoot out the slope of our following 10. Deadline is equal to three

Amy Jiang · Answer

Okay, so we want to find a slope of the tangent line, give her our X value. This time our X value is given to the one that means our A is it. It&#x27;s one. So it&#x27;s years are following equation two parts well, so we have that relax and now we need to find after Bay. But I&#x27;ve sequitur after one, so that&#x27;s going to be two times one to Part three, which is equal to two, and I was also fined explain. It&#x27;s a not able to explain this one. Now let&#x27;s put this into our full of equation to have the limit as X approaches one of off of X. That&#x27;s two X cubed minus F of a. That&#x27;s to all over X minus one. So it&#x27;s not in our numerator. You can actually factor out to that. We haven&#x27;t called him, but it&#x27;s two times. Excuse my just one. Now we can rewrite X cubed minus one, as wanted Part three. And this year is a difference of two cubes, so we can practiced into Not my Love it. That&#x27;s deliberate as experts is one of two times experts one times X squared plus X plus one all over X minus one. Now let&#x27;s cancel out our like tens. And I was in use Excel to find out falling. So for here we have two times one plus one plus one so that you put 22 times three. So we get that are slope of the tangent line is equal to six.

Amy Jiang · Answer

Okay, so we want to find a slope of our change in line. Given the X value this time, we&#x27;re given that our X value is equal to one. So we know that this a here is one. So far, Not so. Let&#x27;s use this equation here. So we have sex, and we need to find out. Okay, We said that our values equal to one. So we have three times disc reverts of one. So that&#x27;s equal to three. And then we have X minus A. That&#x27;s you. Put two X minus one. Okay, so it&#x27;s put that into our equation here. And so we have the limits as X approaches one of us Rex, that&#x27;s minus three square roots of X minus off way. That&#x27;s three all over X minus one. Okay, so I&#x27;m gonna factor out that negative dream that we have in common Inaugurated. So we&#x27;re gonna have a minus three and the square roots of X plus one over X minus one. So if you try to use directs up here, we&#x27;re gonna have division my cell. So instead I&#x27;m gonna multiply and divide by our con trick. It&#x27;s in our numerator and our denominator Okay, so this is going to give us the limit as X approaches one of negative three in the numerator. We have the difference of two squares. That is, we&#x27;re gonna rewrite this as the square root of X squared minus one squared. So this is going to give us a X minus one. Okay, so that&#x27;s times experts one. And this is all over. Experts one at times are conjugated. Okay, so now we can cancel out, Um, are like terms. What? Is that going to help? Um, this is going to give us a square root of one minus one. So we&#x27;re gonna have division by zero once again. So what should we do instead? Actually, I see that we made a mistake here, so let&#x27;s notice that we have a negative here, but I didn&#x27;t put one here, So here, we should actually have a negative and a negative. So let&#x27;s erase what we have here since we&#x27;re gonna have a non opposite sign. So yes, Sorry about that. So they should actually be a plus. Great. Here, So, no, we contract The outs are, um are three. So we have three times negative square roots of X plus one so we can rewrite this Oz one plus the square root of X over X minus one another. It&#x27;s multiply and divide Prior contact. It said it&#x27;s one minus is great root of X. Um, we&#x27;re actually gonna have a minus here about that. Give him for forgetting about that sign. So we&#x27;re gonna have actually one plus the square root of X times one plus the square root of X. So this is going to give us the limits as experts is one. So again, we have, um, the difference of two squares in our numerator. So we&#x27;re gonna end up with In this case, we&#x27;re gonna have one squared minus the square root of X road. So that&#x27;s going to give us, um, one minus X. Well, one minus X is the opposite of this value. So we can rewrite this as, um, you have a constant times a negative on an X minus, one all over X minus one times the square root of X plus one. Okay, so now we can counter. All are like terms, and I let&#x27;s use decks up. So we&#x27;re gonna get negative DRI over one plus one, which is equal to negative green over to

Gregory Higby · Answer

Okay, so, uh, given that X is negative three just feels weird in this problem. But that&#x27;s okay. What we need also find is why And we need to figure out the slope a derivative when X equals negative three. The reason why these are the three things we need is when we find the equation. But tangent line, we want the slope, which is what I mentioned before the derivative at that point X minus the x coordinate, which they gave you, plus the wide coordinate. Um, yeah. So when they tell you why equals sign of one plus X cubed. The first thing I would do is figure out what that white corn it is. And that&#x27;s plugging negative three right here. Um, so negative three cubed is negative. 27. So why is gonna be sign of negative 26? So now what you need is the derivative, which again, that&#x27;s gonna give me the slope of the equation line on. That&#x27;s definitely the changer or the drift of a Sinus co sign. You leave one plus X cubed alone times the derivative of the inside. While the derivative of one is zero. The derivative of X cubed is three x squared. Um, you bring the three in front. Subtract one. So now you need a plug in native three into that problem. Well, that looks eerily similar. It&#x27;s gonna be co sign of one minus 27 lessening of 26 But negative three. When you square, it is positive. Nine times story would be positive. 27. So your equation of the tangent line? Um, I wouldn&#x27;t actually mess with this. I just plug in your slope and leave it as X plus three and then plus your y coordinate sign of negative 26. But if your teacher wants you to simplify this, I would probably just go to a calculator and type in. You know, coast. I make sure your Arabians, um, otherwise, I think they would accept this.

Problem

Solve the given problems. An object is oscillatin…

Problem 51 Easy Difficulty

Solve the given problems.
Find the slope of a line tangent to the curve of $y=\frac{2 \sin 3 x}{x},$ where $x=0.15 .$ Verify the result by using the derivative-evaluating feature of a calculator.

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Basic Technical Mathematics with Calculus

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