A tangent curve is a curve that touches a function or a graph at only one point, without crossing or intersecting it. It represents the instantaneous rate of change or slope of the function at that point.
To find the equation of a tangent curve, you need to first determine the slope of the curve at the given point. This can be done by taking the derivative of the function at that point. Then, you can use the point-slope form y-y1 = m(x-x1) to find the equation, where m is the slope and (x1, y1) is the given point on the curve.
A parametric curve is a curve that is defined by a set of parametric equations, where the coordinates of points on the curve are expressed in terms of one or more parameters. These curves are often used to represent motion or changing values over time.
To graph a parametric curve, you need to plot points by substituting different values for the parameters in the parametric equations. These points will then form a curve when connected. Depending on the complexity of the equations, you may need to use a computer program or graphing calculator to plot the points accurately.
Tangent and parametric curves have various applications in fields such as physics, engineering, and computer graphics. For example, they can be used to model the trajectory of a projectile, the motion of a pendulum, or the movement of a car on a race track. They are also used in designing roller coasters, creating computer animations, and analyzing the behavior of complex systems.