Find the points on the curve y = 2x³ - 3x² - 12x + 20 where th...

Question

Find the points on the curve y = 2x³ - 3x² - 12x + 20 where the tangent line is

a. perpendicular to the line y = 1 - (x/24).

_

b. parallel to the line y = √2 - 12x.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to find the points on the curve, Y is equal to x cubed minus 6 X2 + 11 x minus 6, where the tangent line is parallel to the line, Y is equal to 2 x + 9, and perpendicular to the line Y is equal to -1 divided by 26 multiplied by X + 5. Now, the first step we need to do is determine the slope of our tangentine at any point on our curve. So, to do this, we'll need to take a derivative of our function for a curve. So we'll need to calculate Y, which is equal to the derivative with respect to X, of the quantity of X cubed minus 6 X squad. Plus 11 X minus 6 in quantity. So to take this derivative, we'll need to utilize our power rule, constant multiple rule, and constant rule for derivatives. So utilizing all three of those rules, we'll get for our derivative as 3 X2 minus 12 X plus 11. So this will give us the slope of the tangent line at any point on our curve. So we're gonna start off by looking for the tangent lines that are parallel. To the line Y is equal to 2 X plus 9. Now recall that if two lines are parallel, that means that they have the same slope. And if we look at the line Y is equal to 2 X plus 9, it has a slope of 2. So that means we need to set our first derivative here equal to 2. So, we'll set 3 X 2 minus 12 X. Plus 11 equal to 2. And so now we need to solve this for X. So our first step is to subtract 2 from both sides of the equation. So we'll get 3 X squared minus 12 X. Plus 9 is equal to 0. I can simplify this expression by dividing both sides of the equation by 3, so I'll get X squared minus 4 X. Plus 3 is equal to 0. And the quadra on the left hand side I can actually factor. And so for one factor, I'll have the quantity of X minus 1 in quantity, and for the other factor, I'll be multiplying that by the quantity of X minus 3 in quantity, and that has to be equal to 0. So now we'll do is set each factor equal to 0 and solve for X. So we'll have X minus 1 is equal to 0, and that means that X is going to be equal to 1. And for the other factor, we'll have X minus 3 is equal to 0, and so therefore X is equal to 3. So now I need to determine the points. Or X equals 1 and X equals 3. So I need to determine the Y values. So we need to calculate Y of 1. And this is going to be equal to one cubed. Minus 6 multiplied by 1 squared. Plus 11 multiplied by 1. -6. And when we evaluate this expression, this is going to be equal to 0. The other Y value that we need to calculate is when X is equal to 3, so we'll have Y of 3. And this is going to be equal to 3 cubed. Minus 6 multiplied by 3 squad. Plus 11, multiplied by 3. -6. And when we evaluate this expression, this is also going to be equal to 0. So, the two points that we're interested in are the points of 1.0. And 3.0. So that is the first half of our question answer. So these are the two points. Or the line for the 10 when the tangent line is parallel to Y is equal to 2 X plus 9. So now we need to look at the case of when our tangent line is going to be Perpendicular To the line. Of Y is equal to minus 1 divided by 26, multiplied by X plus 5. Now recall that. In order for two lines to be perpendicular, their slopes have to be the negative reciprocals of each other. And so, what I want to do now is set our first derivative, which is 3 X 2d. Minus 12 X. Plus 11. equal to the negative reciprocal of the slope of Y is equal to -1 divided by 26 multiplied by X plus 5, which is equal to 26. And so now I need to solve this expression for X. So the first thing I'll do is subtract 26 from both sides of the equation. And so we'll have 3 X squared, minus 12 X minus 15 is equal to 0. Again, we can simplify this by dividing both sides of the equation by 3, so we'll get X2 minus 4 X. Minus 5 is equal to 0. And again, we can factor the quadraticle on the left hand side, so we'll have the quantity of X plus 1. In quantity, multiplied by the quantity of X minus 5. In quantity, and this has to be equal to 0. So again, I'll set each factor equal to 0 and solve for x. So I'll have X plus 1 equal to 0, therefore, X is equal to -1, and for the other factor, we'll have X minus 5 is equal to 0, and therefore X is equal to 5. So now I need to find the Y value for each one of these um values of X. And so we'll calculate Y of -1. And that's going to be equal to. -1 cubed. -6 multiplied by -1 squared. Plus 11, multiplied by -1. -6. And when we evaluate this expression, this is going to be equal to -24. Now we need to calculate Y of 5. Which is going to be equal to 5 cubed. Minus 6 multiplied by 5 squad. Plus 11, multiplied by 5. -6. And when we evaluate this expression, this is going to be equal to 24. So, the two points that we're looking for here are gonna be the point of -1, -24. And the other point is going to be 5.24. So those are the two points at which our tangent line is perpendicular. To the line of Y is equal to -1 divided by 26, multiplied by X plus 5. So that is the second half of our problem answered. So I hope that this explanation has been useful, and I'll see you in the next video.

Find the points on the curve y = 2x³ - 3x² - 12x + 20 where the tangent line is

a. perpendicular to the line y = 1 - (x/24).
_
b. parallel to the line y = √2 - 12x.

Key Concepts

Derivative

Slope of a Line

Critical Points

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