Finding Horizontal Tangents of x^2+4y+22=y^2+10x

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In summary, the conversation discusses finding the point(s) at which the tangent line to a given equation is horizontal. It explains that a horizontal tangent line means that the slope of the line is zero, and this is determined by the first derivative of the equation. The process of finding these points involves differentiating the equation with respect to x and solving for x. The conversation also provides a brief explanation of the origin of the term "tangent" and its connection to the slope of the tangent line.
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Find the point(s) (x,Y) at which the tangent line to x^2+4y+22=y^2+10x is horizontal.
 
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What does it means when the tangent line is horizontal? what does the first derivative has to do with the "tangent"?
 
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Differentiating this equation implicitly with respect to x yields 2x-10/2y-4. Thus we see that 2x-10=0 (2y-4 can't equal zero, that would be undefined) From here x=5. So the point is P(5,y). set x=5 in the original equation, solve for y and there you got the y. This turns out that that y has two values for x=5, and those being 1 and 3.

Btw, tangent comes from the latin word tangens, which means touch. So a tangent line touches a curve at just one point. And the derivative of the curve at that point gives us the slope of the tangent line.
 

FAQ: Finding Horizontal Tangents of x^2+4y+22=y^2+10x

1. How do you find the horizontal tangents of a given equation?

To find the horizontal tangents of an equation, first set the equation equal to zero and then solve for the variable. The resulting value will be the x-coordinate of the point where the tangent is horizontal. Plug this value back into the original equation to find the corresponding y-coordinate.

2. What is the equation for a horizontal tangent?

The equation for a horizontal tangent is y = c, where c is a constant. This means that the tangent line is parallel to the x-axis and has a slope of 0.

3. Can a function have more than one horizontal tangent?

Yes, a function can have more than one horizontal tangent. This occurs when the function has a point of inflection, which is where the concavity changes from positive to negative or vice versa.

4. How do you determine if a point is a horizontal tangent?

If a point is a horizontal tangent, it means that the slope of the tangent line at that point is 0. To determine this, you can use the derivative of the function and set it equal to 0. If the derivative is 0 at a given point, then that point is a horizontal tangent.

5. What is the significance of finding horizontal tangents in a graph?

Finding horizontal tangents can help us understand the behavior of a function at a specific point. It can also help us identify points of inflection and critical points, which are important in determining the overall shape and behavior of a function.

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