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

Question

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

b. Graph the tangent lines on the given graph.

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. Sketch the graph of the function X + 23 minus 8 Y is equal to 4, and its tangent lines at X equals 4 using a graphing utility. Awesome. So it appears for this particular problem we're asked to ultimately what we're trying to solve for is we're asked to create a graph of this function that we are given. And not only are we asked to graph this specific function, we're also asked to graph all of the tangent lines at X equals 4 and include them in our sketch of our function itself. So we're trying to graph the function curve, and we're trying to grab all the tangent line curves as well. Awesome. So with that said, our first step in order to solve for this particular problem is we need to, and let's write this down, our first step is we need to differentiate both sides of our 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 we need to take When we differentiate our expression, which we're told once again our function is X plus 2 3 minus 8 Y is equal to 4, so when we differentiate this expression, we will get 1 + 6Y to the power 2 multiplied by Y minus 8 Y is equal to 0. Fantastic. And now our next step. we need to rearrange our differentiated expression to isolate and solve for Y all by itself. So I'm gonna skip a few steps, but Using a little bit of algebra, we will find that Y is going to be equal to, once again, we rearrange our differentiated expression to isolate and solve for Y, we will find that Y is equal to negative 1 divided by 6 to the power of 2 minus 8. Fantastic. So now our second step that we need to solve for is we need to find the values of Y when X is equal to 4 by substituting the value of X equals 4 back into our original equation, which we're told once again is X + 2 multiplied by 3 or 23 minus 8 Y equals 4. So when we do that, when we plug in a value of 4 in for X, we'll get that 4. Plus 2 to the power of 3 minus 8 Y is equal to 4, and now we need to take this expression, and we're going to be factoring out like terms because we need to find what our values of Y are, since we're trying to ultimately figure out what the points of tangency are. So when we ultimately factor out like terms and we try to solve for the different values of Y, we will find that it's factored out form is gonna be 2 Y multiplied by Y plus 2. Multiplied by Y minus 2 is equal to 0, and once again we're taking our expression once we plugged in a value of x equals 4 into our original equation expression, and then we're going to be like rearranging it to factor out and solve for our different values of Y. So we're trying to factor out our factor out our expression so we can find what our values of Y are. So when we do that, we will find that it once again it's equal to 2y multiple. Y + 2 multiplied by Y minus 2 is equal to 0. So therefore, Y is going to be equal to 0, Y is going to be equal to -2, or Y is going to be equal to 2, and the points of tangency are going to be at parentheses 40, so parentheses 4.0. And parentheses 4.2 and 4.2. Fantastic. So once again, 4.0, 4.2, and 4.2. Awesome. So now our third step is we need to calculate the slope of the tangent lines at X equals 4 for each y value found in step 2 that we found together in step 2, and in order to do that, we need to substitute Y back into our derivative equation. Which once again our derivative equation is Y is equal to -1 divided by 6 Y2 minus 8. Fantastic. So with that said, we need to say for Y equals 0. The slope M is going to be equal to -1 divided by 6 multiplied by 0 squad minus 8 because once again we're plugging in a value of 0 in for our value for Y. So when we simplify and solve, we will find that once again the slope M for when Y equals 0 is going to be equal to 1/8. So instead of writing out our once again writing out this expression for our derivative equation every single time, I'm just going to give you the value for the slope. So for when Y is equal to -2, the slope M is going to be equal to -16, and once again, instead of plugging in a value of 0 in for Y, we're plugging in a value of -2, and for Y equals 2, the slope M is going to be equal to. -16, and once again instead of plugging in a value of -2 in for Y, we're plugging in a value of 2 in for Y. The sur slope M, fantastic. So now our fourth step. We need to find the equation of the tangent lines using our point slope form, which, as we should recall, the point slope form states that Y minus Y1 is equal to M slope, multiplied by X minus X1. Where once again, M is the slope that we found, or rather determined from step 3, and we also need to note that X1, Y1 is the point of tangency. So X1. Y1 is our point of tangency. So, with that said, for for the condition where, and let's make a note of this, so for the condition where M is equal to 1/8 and we have parentheses 40, that will mean when we plug in these values into our Point slope form, you will get Y minus 0 is equal to 1/8 multiplied by X minus 4. And when we rearrange this equation to isolate and solve for Y all by itself, we will find that Y is equal to 1/8 X minus 1/2. So similarly, For parenthesis 4.2 and where M is equal to -116. We will find, when we plug in these values into our point slope form, we will find that Y minus -2 is equal to -1 divided by 16 or 1/16 multiplied by X minus 4, and when we rearrange to isolate and solve for Y all by itself, Y is going to be equal to negative 1/16. X minus 7/4, fantastic. And finally for 4.2, and where the slope is equal to negative, 1 divided by 16. That will mean, without a doubt, since we're once again we're trying to solve for the point slope form, you'll find that Y minus 2 is equal to -16 because that's our slope, multiplied by X minus 4 because our value of X1 once again is going to be our value of 4 given to us by our point that we found. So, when we rearrange this expression to solve for Y, we will find that Y. It's going to be equal to -1, so -116 X plus 9 divided by 4. Fantastic. So therefore, without a doubt, the equations for all the lines tangent to X plus 2 3 minus 8 Y equals 4 X equals 4 are going to be the following Y equals 1/8 X minus 1/2, Y is equal to 1/16 X minus 7 divided by 4. And Y equals, and once again, our final one is going to be Y equals -116 X plus 9 divided by 4. Fantastic, we did it. We found our, once again, these are our lines that are tangent to our original expression at X equals 4. So now, We need to note, without a doubt that we need to note that our input function, so this is a function we're plugging into our graphing software of choice, you can use whatever software makes the most sense to you, but we need to note that our input function is X + 23 minus 8 Y is equal to 4, and we need to plug in all of our tang. lines into our graphing tools we need to plug in all of these equations for Y into our graphing tool. And when we do that, we will produce the following graph. We will find that our graph most definitely has to look like the following that that denotes all three of our tangent lines. Fantastic. So, as you could see, We have our different, we have our original curve, which looks like an elongated S. And it's represented in green, so that is our original input function, it's going to be denoted by this green curve. As you can see, and it looks like an elongated S shape, and then we have our three tangent lines represented in red, black and blue. And that's when we once again, when we plug in our different equations for Y into our graphing software. And that is it. That is our final answer. So that is our graph that we were asked to solve for. With all of the tangent curves or tangent lines graphed on the same graph as well. And that's 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>
b. Graph the tangent lines on the given graph.
x+y³−y=1; x=1

Key Concepts

Tangent Lines

Implicit Differentiation

Graphing Techniques

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