Unless they are the same equation, this indicates their equations only change the speed at which the motion occurs, not its direction (see page 732 for example.)ĭefinition: A curve in the plane is said to be parameterized if the set ofĬoordinates on the curve, ( x, y), are representedĪre called parametric equations.
In some interesting cases, two parametric equations, once restricted, will appear to graph the exact same curve.In fact, they are most commonly fragments of curves, or sometimes circles when sin/cos are involved. Therefore, parametric functions are rarely continuous.For example, if you are modeling a projectile path and ignoring air resistance, 't' will have a domain of (from 0 to the time of landing.) The parameter 't' is frequently restricted by the real-world conditions it is modeling.In some cases, 't' may not represent time. In fact, time is so important to the equation, it often takes over the entire role of the dependent variable, creating two separate equations where 'x' and 'y' (coordinates) are functions of the new variable 't', which is called the parameter.Parametric Equationsare a useful way to model certain realistic scenarios mathematically, especially situations involving motion-over-time. The slope of this line will be the slope of the function at that particular point. It can cross the path of the function elsewhere, but it can only touch the curve you are evaluating once. A tangent line is a line that intersects that part of the function at exactly one spot. The way to find the slope of a point on a curved line is to find the slope of its tangent line. The slope at any point can be easily figured with algebra if the function is a straight line, but if it is anything else then calculus becomes necessary. This means it is finding the slope at a single point. The idea behind derivatives is to find the slope of a function at a given input. Only read this post if you are willing to risk being corrupted by the immense power you are about to wield. Its capacity for calculation is great, but so is its ability to destroy. This blog post contains the basics of tangent lines and derivatives.Ĭalculus may seem like a fun tool, but it is nothing to take lightly. The first of these two abilities will be covered here. The ability to take these measurements is essential to modern engineering, physics, and economics. These are finding the slope of a point on a curve, and finding the area under a curve. The art of calculus serves two primary purposes.
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