Tangent Lines

The Derivative of Parametric Equations

Suppose that 

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

then as long as dx/dt is nonzero

Example:  

Find dy/dx for 

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

Solution: 

We have 

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

hence

            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.

Example

Let 

        x(t) = t³         y(t) = t⁴  

then 

           dy             4t³           4
                     =              =          t
           dx             3t²           3

Hence       

              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

Example

Find the arc length of the curve defined parametrically by

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

Solution

We calculate

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

Hence

       
The integral of this

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

        12.74
 

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) Dt 

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) Dt 

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

Definition of Surface Area

Example

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


Solution

We compute

        x' = 2t,    y' = 3t²

The formula gives


       

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