A tangent line is a line that touches a curve at exactly one point, without intersecting the curve at that point.
In order to find the points on a curve with a tangent line, you must first identify the slope of the curve at that point. This can be done by taking the derivative of the curve. Then, using the slope and the coordinates of the point, you can use the point-slope formula to find the equation of the tangent line.
Finding points on a curve with a tangent line allows us to determine the rate of change of the curve at a specific point. This is useful in many fields, such as physics, engineering, and economics, where understanding the rate of change is important in predicting future behavior.
No, a curve can only have one tangent line at a point. This is because a tangent line must only touch the curve at that point and not intersect it.
Calculus provides the tools necessary to calculate the slope of a curve at a specific point, which is essential in finding the equation of a tangent line. Calculus also helps us understand the behavior of curves and how they change over time, which is crucial in determining the rate of change at a specific point.