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> the derivative (tangent) of the minimum (lowest point) of any curve is zero (horizontal). > holding their pencil up to it at the lowest point and laying it al

by est 6y ago

> the derivative (tangent) of the minimum (lowest point) of any curve is zero (horizontal). > holding their pencil up to it at the lowest point and laying it along, and discovering that, sure enough, the tangent is horizontal. Any gifs showing that? just really curious and can't picture that in my head.

piinbinary 6y ago
Try holding your mouse* upside-down on the surface of your desk to get the same effect. The point at which the mouse meets the desk will always be the point where the tangent of the curve is horizontal. *Assuming your mouse is curved on top.
rexer 6y ago
It might help to think of a specific curve. An example would be a simple parabola: f(x) = x^2. At the bottom of the curve (x = 0), the slope (aka tangent, aka derivative) is 0, or a horizontal line.
jldugger 6y ago
Lets use the letter U as your curve: U One way to imagine the the horizontal tangent is that it's where the curve stops going down (negative slope) and starts going up (positive slope). Well, that is also basically the definition of a local minima, and one of those minima is also the global minima.
marcosdumay 6y ago
If the curve was inclined at that point, it would have to be lower either on the left or the right of it. Just draw any curve in a piece of paper, you'll see it happening.
imheretolearn 6y ago
This was my "ah-ha" moment as well. I always knew that derivative of the lowest point is 0 but I didn't really understand what it meant. Take any arbitrary curve. Tangent is a line that touches a single point on the curve. Draw a tangent aka pencil touching the lowest point of the curve. It's derivative ie dy/dx apply the formulae to the curve's formulae. It will always be 0 which means it will have no "slope" ie parallel to the x axis
thedirt0115 6y ago
Here is a GIF demonstrating the tangent line as a point traverses a curve: https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/1862/2017/06/23162553/of-sliding-derivative-line.gif https://s3-us-west-2.amazonaws.com/courses-images/wp-content... The "pencil" (the short line representing the tangent) is green when positive, black when zero, and red when negative. Note that when it is black/zero, the "pencil" line is flat. This occurs at both local maxima and minima.

by est 6y ago

piinbinary 6y ago

Try holding your mouse* upside-down on the surface of your desk to get the same effect. The point at which the mouse meets the desk will always be the point where the tangent of the curve is horizontal. *Assuming your mouse is curved on top.

rexer 6y ago

It might help to think of a specific curve. An example would be a simple parabola: f(x) = x^2. At the bottom of the curve (x = 0), the slope (aka tangent, aka derivative) is 0, or a horizontal line.

jldugger 6y ago

Lets use the letter U as your curve: U One way to imagine the the horizontal tangent is that it's where the curve stops going down (negative slope) and starts going up (positive slope). Well, that is also basically the definition of a local minima, and one of those minima is also the global minima.

marcosdumay 6y ago

If the curve was inclined at that point, it would have to be lower either on the left or the right of it. Just draw any curve in a piece of paper, you'll see it happening.

imheretolearn 6y ago

This was my "ah-ha" moment as well. I always knew that derivative of the lowest point is 0 but I didn't really understand what it meant. Take any arbitrary curve. Tangent is a line that touches a single point on the curve. Draw a tangent aka pencil touching the lowest point of the curve. It's derivative ie dy/dx apply the formulae to the curve's formulae. It will always be 0 which means it will have no "slope" ie parallel to the x axis

thedirt0115 6y ago

Here is a GIF demonstrating the tangent line as a point traverses a curve: https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/1862/2017/06/23162553/of-sliding-derivative-line.gif https://s3-us-west-2.amazonaws.com/courses-images/wp-content... The "pencil" (the short line representing the tangent) is green when positive, black when zero, and red when negative. Note that when it is black/zero, the "pencil" line is flat. This occurs at both local maxima and minima.