Write an equation of each horizontal tangent line to the curve.

In summary, the conversation discusses finding an equation for each horizontal tangent line to a given curve. The participant uses the derivative to find the points where the derivative equals zero, and then solves for y to find the equations. However, there is some confusion about the specific values to use and the possibility of multiple solutions.
  • #1
lude1
34
0

Homework Statement



Write an equation of each horizontal tangent line to the curve.

Homework Equations



y = 2y^3 + 6x^2y - 12x + 6y = 1
y' = (4x - 2xy) / x^2 + y^2 + 1)

The Attempt at a Solution



Well, horizontal tangent line means the derivative equals zero. Thus,

4x - 2xy = 0
2x(2-y) = 0
x = 0, y = 2​

Since I need an equation, I plug x = 0 back into the original function and end up with

2y^3 + 6y = 1​

Then I need to solve for y. I guess this is more of an algebra question than a calculus question? But nevertheless, I don't know how to solve for y.. I could do 2y(y^2 + 3) = 1 but that doesn't help me.
 
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  • #2
Hi lude1! :smile:

(try using the X2 tag just above the Reply box :wink:)
lude1 said:
2y^3 + 6x^2y - 12x + 6y = 1
y' = (4x - 2xy) / x^2 + y^2 + 1)

Where does the 4x come from? :confused:

(Or should the 12x be 12x2 ?)
 
  • #3
Ah, yes, you are correct! It should be 12x2

Thus:

y = 2y3 + 6x2y - 12x2 + 6y = 1

and

y' = (4x - 2xy) / (x2 + y2 + 1)
 
  • #4
lude1 said:
Ah, yes, you are correct! It should be 12x2

Thus:

y = 2y3 + 6x2y - 12x2 + 6y = 1

(why do you keep starting with "y =" ? :frown:)

hmm … then, sorry, I don't think your 2y3 + 6y = 1 has a straightforward answer

(unless the = 1 should be = 8)

oh, and don't forget you still have to deal with the other case, y = 2. :wink:
 

1. What is a horizontal tangent line?

A horizontal tangent line is a line that touches a curve at only one point and has a slope of 0. This means that the line is parallel to the x-axis and does not rise or fall as it moves along the curve.

2. How do you find the equation of a horizontal tangent line?

To find the equation of a horizontal tangent line, you need to determine the x-coordinate of the point where the line touches the curve. Once you have this point, you can write the equation in the form y = c, where c is the y-coordinate of the point.

3. What information do you need to write the equation of a horizontal tangent line?

You need to know the x-coordinate of the point where the line touches the curve. This can be found by setting the derivative of the curve equal to 0 and solving for x. Once you have the x-coordinate, you can write the equation in the form y = c, where c is the y-coordinate of the point.

4. Can there be more than one horizontal tangent line to a curve?

Yes, it is possible for a curve to have more than one horizontal tangent line. This can occur when the curve has a point of inflection or when the curve is a horizontal line itself.

5. How do you graph a horizontal tangent line?

To graph a horizontal tangent line, plot the point where the line touches the curve and then draw a horizontal line through that point. This line will have a slope of 0 and will be parallel to the x-axis.

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