A curve is formed by lines. If the length of these lines is reduced to zero, we get a very smooth curve.
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Although all lines have the relationship that defines slope, one can argue that not all lines do have one. The exception would be vertical lines. Slope is defined as the vertical rate of change divided by the horizontal rate of change. In the case of a vertical line, there is no horizontal rate of change, and calculating slope would cause division by zero. The closest you could come to expressing the slope of a vertical line would be ∞
The slope of the tangent line at the maximum point of the curve is zero. So we say that as a curve point approaches to the maximum point, the slope of the tangent line at that point approaches to zero.
By differentiating the answer and plugging in the x value along the curve, you are finding the exact slope of the curve at that point. In effect, this would be the slope of the tangent line, as a tangent line only intersects another at one point. To find the equation of a tangent line to a curve, use the point slope form (y-y1)=m(x-x1), m being the slope. Use the differential to find the slope and use the point on the curve to plug in for (x1, y1).
They don't, they are parallel to each other.
Horizontal lines have a slope of 0.