Slopes and Tangent Lines

a. Horizontal tangen...

Question

Slopes and Tangent Lines

a. Horizontal tangent lines Find equations for the horizontal tangent lines to the curve y = x³ − 3x − 2. Also find equations for the lines that are perpendicular to these tangent lines at the points of tangency.

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says determine the equations for the horizontal tangent lines to the curve Y is equal to 2 X cubed plus 3X2 minus 12 X plus 1. Also identify the equation for lines perpendicular to these lines at the points where they touch the curve. So the first part of this question, we need to determine the equations for the horizon, tangent lines to our curve Y. So to do this, we'll need to actually determine the slope of our tangent lines at any point when our curve Y. To do this, we'll need to take a derivative of Y with respect to X. So we'll need to calculate Y. That's going to be equal to the derivative with respect to X. Of the quantity of 2x cubed. Plus 3 x 2. Minus 12 X. Plus one. So taking this derivative, this is going to be equal to 6 X squad. Plus 6 X minus 12. So this would give us the slope of our tangent line at any point on our curve for Y. So what we want to do is determine the X values where that slope is equal to 0, which would give us a horizontal tangent line. So we're going to set 6 x 2 plus 6 X minus 12 equal to 0 and solve for X. So the first step is to divide both sides of this equation by 6 just to simplify our expression. So we'll get X2 plus x minus 2 is equal to 0. We can actually factor the left-hand side of this equation, and so we'll have the quantity of X minus 1 in quantity. Multiplied by the quantity of X plus 2 in quantity, and that is going to be equal to 0. So now we just need to set each one of these factors equal to 0 and solve for x. So for the first factor we'll have X minus 1 is equal to 0. That will give us X is equal to 1. And for the second factor, we'll have X plus 2 is equal to 0, which will give us X equals to minus 2. So these are the X coordinates for the points where our tangent line hits our curve that are horizontal. So now we just need to figure out the Y values of these two points. So we'll calculate Y of 1. As our first point, and that's gonna be equal to you. 2 multiplied by 1 cubed. Plus 3 multiplied by 1 squared. -12 multiplied by 1. Plus one. So, evaluating this expression, this is going to be equal to -6. Now, for the other point, we'll have Y of minus 2. That's gonna be equal to 2 multiplied by minus 2 cubed. Plus 3, multiplied by minus 2 squared. -12, multiplied by -2. Plus one. And if we evaluate this expression, this is going to be equal to 21. So now we just need to use this information to determine the equations for our horizontal tangent lines. So, for our case when X is equal to 1, the horizontal tangent line is going to have an equation of Y is equal to -6. And the other horizontal tangent line equation is going to be Y is equal to 21, which is going to occur when X is equal to -2. So those are two horizontal equation horizontal tangent line equations. So now we need to determine the equations for the lines that are perpendicular to these tangent lines at the point where they touch the curve. Well, that's just going to be our X coordinate. So, when Y is equal to -6 for a tangent line, We're going to have X equal to -1 as the line that is perpendicular to it. And or horizontal tangent line Y is equal to 21, the line is going to be perpendicular to it, it's gonna be X equal to minus 2. So those will both X equal to 1 and X equal to -2 will be vertical lines. And so this is going to be the answer to our problem, that we have a horizontal tangent line at Y is equal to -6. And the line perpendicular to it is going to be X equal to 1, and we also have another horizontal tangent line at Y is equal to 21, and the line perpendicular to it, it's going to be X equal to -2. So I hope that this explanation has been useful, and I'll see you in the next video.

Slopes and Tangent Lines

a. Horizontal tangent lines Find equations for the horizontal tangent lines to the curve y = x³ − 3x − 2. Also find equations for the lines that are perpendicular to these tangent lines at the points of tangency.

Key Concepts

Derivative

Tangent Line

Perpendicular Lines

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