A tangent line is a line that touches a curve at only one point and has the same slope as the curve at that point. It can be thought of as the instantaneous direction of the curve at that specific point.
To find the equation of a tangent line to a curve, you need to first find the slope of the curve at the point of tangency. This can be done by taking the derivative of the curve at that point. Then, using the point-slope formula, plug in the point of tangency and the slope to find the equation of the tangent line.
No, a curve can only have one tangent line at a given point. This is because the tangent line represents the instantaneous direction of the curve at that point, and if there were more than one, the curve would not have a unique direction at that point.
The slope of the tangent line at a point is equal to the value of the derivative of the curve at that same point. This is because the derivative represents the rate of change of the curve, and the slope of the tangent line represents the instantaneous change of the curve at that point.
A line is tangent to a curve if it only intersects the curve at one point and has the same slope as the curve at that point. If a line intersects the curve at more than one point, it is not a tangent line.