60–62. {Use of Tech} Multiple tangent lines Complete the follo...

Question

60–62. {Use of Tech} Multiple tangent lines Complete the following steps.

a. Find equations of all lines tangent to the curve at the given value of x.

x+y³−y=1; x=1

Accepted Answer

Below there, today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the equations of all lines tangent 22 X plus the power of 3 minus 4 Y is equal to 6 at X equals 3. Awesome. So it appears for this particular problem we're trying to figure out, or rather determine all the equations of the lines that are tangent to this expression, which is 2 X plus y to the 3 minus 4 Y is equal to 6 at x equals 3. Awesome. So now that we know that we're trying to figure out what the tangent lines are to this expression. Our first step in order to solve for this problem is we need to differentiate both sides of the given equation with respect to X in order to help us find the slope of the tangent line at any point on the curve. So when we differentiate our expression, we will find that it's equal to 2. Plus 3 Y 2 multiplied by Y minus 4 Y is equal to 0. So now what we're trying to do is we're going to be taking this expression that we currently have when we differentiate both sides of our given expression or a given equation with respect to X. Our next step now is we're going to be rearranging our current expression to isolate and solve for Y all by itself, and when we do that, we will find, using a little bit of algebra I'm going to skip a few steps. We will find when we rearrange. Our expression to solve for Y, we will find that Y is equal to -2 divided by 3 Y2 minus 4. Fantastic. And now our second step now is we need to find the values of Y when X equals 3 by substituting in a value of X equals 3 into our original equation. So we're taking and we're playing in a value of X equals 3 indoor expression for 2 X plus Y 3 minus 4 Y. is equal to 6. So when we do that, we will find that 2 multiplied by 3, and once again, we're plugging in a value of 3 in for our value of X. So plus the 3 minus 4 Y is equal to 6. So now, What we're gonna be trying to do is we're going to be trying to rearrange this expression. To pull out our wise we're trying to we're trying to write it. In a more simple form. So we're gonna be taking this expression and we're gonna simplify it a bit. So when we start to simplify, we will find that 6 plus the power of 3 minus 4 Y is equal to 6. Which we could do a little bit more simplifying, we could say that to the 3 minus 4 Y is equal to 0, and when we pull out our common term, you will find out that we can simplify this to write that Y multiplied by Y squared minus 4 is going to be equal to 0, fantastic. So then we can finally write, when we factor out our inner parenthesis, we can also go ahead and write that Y multiplied by Y + 2 multiplied by Y minus 2 is equal to 0. Fantastic. So now, we will determine that When we use our expression, we will determine that Y equals 0, Y equals -2. Or Y equals 2. So that will mean, without a doubt, the points of tangent C are 3.0 in parentheses and in parentheses 3.2, and finally in parentheses 3.2. Fantastic. So now our third step is we need to calculate the slope of the tangent lines at X equals 3 for each y value found in our step 2 by substituting our value for Y into the derivative equation, which once again that equation is Y is equal to -2 divided by 3 Y2 minus 4. fantastic. So now, with that said, We need to note that for when Y is equal to 0, the slope M is going to be -2. Or rather negative, 2 divided by 3 multiplied by 0 squad minus 4, and once again we're plugging in a value of 0 in for a Y. So when we do that, we'll find that 2 divided by 3 multiplied by 0 to the power 2 minus 4 is going to be equal to -2 divided by -4, which is going to be equal to. One half and similarly, instead of writing it out each time, when we instead of writing out our expression for the slope each time and plugging in our value for Y, I'll just give you the answer. So when we use Y is equal to -2, so instead of plugging in a value of 0 in for Y or plugging in a value of -2, that will mean that our M, our slope, is going to be equal to negative. 14th, and for when Y is equal to 2, our slope M is going to be equal to -14. And once again, we're plugging in a value of 2 in for Y instead of 0 or -2. Into our expression. Fantastic. So now, Our next step, which is our 4th step, our 4th step is we need to find the equation of the tangent lines using the point slope form, which, as we should recall, point slope form states that Y minus Y1 is equal to M the slope multiplied by X minus X1. Awesome. So with that said, we need to note once again that M is the slope, and we found this value for slope from our step 3, and we need to also note and recall that X1, and Y1 are our points of tangency, or rather B point of tangency. It's better to be specific. So X1.com Y1 is our point of tangency. Fantastic. So with that said, for the 0.3.0 in parentheses where M are slope equals 12, that will mean that Y minus 0 is equal to 12 multiplied by X minus 3 because once again we're plugging in a value for 3 in for X and a value of 0 in for Y. And when we simplify to get Y all by itself, we will find that Y is equal to 1/2 X minus 3 divided by 2. And our 3 halves, and that is our one of our final answers. Hooray, we did it. So, Y will be equal to 1/2 X minus 3 halves, and 4. Parentheses 3.2, where M is equal to -14, that will mean that Y minus 2 is equal to -14 multiplied by X minus 3. Fantastic. And when we simplify this expression just to get Y all by itself, we will find that Y is going to be equal to -1/4. X minus 5/4, and that is our second answer. For one of our lines that are tangent to. Our expression that we had were given above by the problem itself. Fantastic. And finally, for our last point, which is going to be at parentheses 3.2, where the slope is equal to -14, that will mean that Y. -2 is going to be equal to 1/4 multiplied by X minus 3, so we rearrange this expression to isolate and solve for Y. All by itself, we will find that Y is equal to negative 1/4. X + 11/4, and that's it. That is our 3rd and final answer. Hooray, we did it. So Therefore, let's scroll up to the top to review the problem itself and to write down our final results. So it appears without a doubt, our final three answers are going to be. Y is equal to 1/2 X minus 3 halves, Y is equal to 1/4 X minus 5/4 and Y is equal to 1/4 X. Plus 11/4, and that's it. Those are our three expressions for the lines that are tangent +22 X plus y to the 3 minus 4 Y equals 6 at X equals 3. Fantastic, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
a. Find equations of all lines tangent to the curve at the given value of x.
x+y³−y=1; x=1

Key Concepts

Implicit Differentiation

Tangent Line Equation

Finding Points on the Curve

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