Parametric Equation

    First we discuss the chain rule and then we illustrate how to find the slope of the tangent line to the graph of a curve given by parametric equations. We finish by giving the parametric equations for the cycloid.

(1) Proposition (Chain Rule) If the derivatives and both exist, and is the composite function defined by then exists and is given by the product



In Libyans notation, if and are both differentiable functions, then

(2) Example (Chain Rule) Let be a function for which If what is Also, if what is

    Solution. By the chain rule



Also, by the chain rule

(3) Example (Chain Rule) Let be a function for which and If what is

    Solution. Using the chain rule and the product rule we determine,





Therefore,

(4) Example (Chain Rule) Assuming that the following derivatives exists, find



    Solution. Using the chain rule,
    


which is the second derivative evaluated at the function multiplied by the first derivative; while,



is the first derivative evaluated at the first derivative multiplied by the second derivative. When will these derivatives be the same?

(5) Example (Chain Rule) Show that if a particle moves along a straight line with position and velocity then its acceleration satisfies Use this formula to find in the case where

    Solution. By the chain rule,



In the case where we determine,



Thus, What does this rate of change represent?

(6) Example (Using the Chain Rule) Find an equation of the tangent line to the graph of the function at the point

    Solution. By using the chain rule we determine,



and so Therefore, an equation of the tangent line is which implies to   Here is a plot of both and the tangent line to at

(7) Example (Using the Chain Rule) Find an equation of the tangent line to the graph of the function at the point

    Solution. By using the chain rule we determine,



and so Therefore, an equation of the tangent line is which implies to   Here is a plot of both and the tangent line to at



(8) Example (Using the Chain Rule) Determine the point(s) at which the graph of has a horizontal tangent.

    Solution. By using the chain rule we determine,



which simplifies to Thus the only point where has a horizontal tangent line is and these can be seen in the following sketch.

(9) Definition (Parametrized Curve)  Let be a curve defined by where and are defined on the interval The equations and for are parametric equations for with parameter

(10) Definition (Orientation of a Curve)  The orientation of a parameterized curve is the direction determined by increasing values of the parameter.

(11) Example (Orientation of a Parametrized Curve) Sketch the graph of the curve given by the following parametric equations by plotting values for and then check your graph by finding an equation in and only and then graphing. Show the orientation of the curve.

(a) on

    Solution. The graph starts at the point and follows the line until it reaches the other endpoint at
    



(b) on

    Solution. The graph starts at the point and follows the line until it reaches the other endpoint at
    


Comparing (a) and (b) we see that the graph of a curve may have more than one parametrization. Can you think of another set of parametric equations that gives the same graph?  

(c) on

    Solution. The curve starts at and follows the upper part of the unit circle until it reaches the other endpoint of
    
    

Can you think of another set of parametric equations that give the same graph?

(12) Example (Parametric Equations for the Cycloid) The curve traced out by a fixed point on the circumference of a circle as the circle rolls along the top of the -axis is called a cycloid. Find parametric equations for a cycloid.

    Solution. Suppose the circle rolls along the -axis in the positive direction and has radius Here are some examples of the position that a circle will take as the circle rolls along on top of the -axis.
    
    


In order to obtain parametric equations consider the following diagram.



    Let represent the angle in radians. Thus, the distance the circle has rolled at time is where is the radius of the circle, by definition of arc length. So the coordinates at this point in time for is If we introduce the coordinate system and then and If represents an angle in standard position in the -plane, then Thus,





Now substitution into and yields





for any

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