Tangent Lines

The Derivative of Parametric Equations

Suppose that 

        x  =  x(t)     and     y  =  y(t) 

then as long as dx/dt is nonzero


Find dy/dx for 

        x(t)  =  2 cos t     and     y(t)  =  2 sin t


We have 

        dx/dt  =  -2 sin t     and     dy/dt  =  2 cos t 


            dy             dy / dt
            dx             dx / dt

                  2 cos t
         =                       =  -cot t
                 -2 sin t

The Second Derivative of Parametric Equations

To calculate the second derivative we use the chain rule twice. 

Hence to find the second derivative, we find the derivative with respect to t of the first derivative and then divide by the derivative of x with respect to t.




        x(t) = t3         y(t) = t4  


           dy             4t3           4
                     =              =          t
           dx             3t2           3


              d2y              d/dt (4/3 t)            4/3             4
                          =                           =              =            
              dx2                 dx / dt                3t2            9t2

Arc Length

We can find the arc length of a curve by cutting it up into tiny pieces and adding up the length of each of the pieces.  If the pieces are small and the curve is differentiable then each piece will be approximately linear.  

We can use the distance formula to find the length of each piece


Multiplying and dividing by D gives


Adding up all the lengths and taking the limit as Dt approaches 0 gives the formula




Find the arc length of the curve defined parametrically by

        x(t) = t2 + 4t,     y(t) = 1 - t2,     0 < t < 2


We calculate

        x '  =  2t + 4,     y '  =  -2t


The integral of this


Is quite difficult (but not impossible) to do by hand.  Either by hand or by computer we get


Surface Area of Revolution

If we revolve a curve around the x-axis what is surface area of the region that is formed?  If we cut the curve into tiny pieces, then each piece is approximately a line segment, which when revolved around the x-axis will have area

        2p (radius) (length)  =  2p y(t) D

Similarly, if the piece of the curve is revolved about the y-axis, then the resulting surface are is

        2p (radius) (length)  =  2p x(t) D

Adding up all the pieces and taking the limit as Dt approaches 0 gives

          Definition of Surface Area 





Set up the integral that gives the surface area of the solid formed by revolving the curve

        x(t) = t2 ,    y(t) = t3

About the y-axis


We compute

        x' = 2t,    y' = 3t2

The formula gives



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