10.2 – 10.3 Parametric Equations

sin 2

2cos 5

x t t

y t t

There are times when we need to describe motion (or a curve) that is not a function.

We can do this by writing equations for the x and y coordinates in terms of a third variable (usually t or ).

x f t y g t These are calledparametric equations.

“t” is the parameter. (It is also the independent variable)

Parametric Equations

Example

Sketch the curve described by the parametric equations.

3 , 2 3 , 1 4x t y t t

When t = -1, we have x =4 and y = -5.

The point (4,-5) is called the initial point.

When t = 4, we have x = -1 and y = 5.

The point (-1,5) is called the terminal point.

Example Sketch the curve defined by parametric equations

2 1x t y t

Eliminate the parameter to write in rectangular equation.

2( 1)x y

Examples

21 , 4 , 0 9x t y t t t

Eliminate the parameter to find a Cartesian equation of the curve. Then sketch the curve defined by parametric equations.

0x t y t t

20,cos4,sin4 ttytx

4cos 4sinx t y t

3sin 4cosx t y t

Parametric Form of the Derivative

Example: Find the slope of the tangent line to the curve given by

at the point (5, -3).

If a smooth curve is given by the equations x = f (t) and y = g(t), then the slope of the tangent line to the curve at (x,y) is:

dy

dy dtdxdxdt

This makes sense if we think about canceling dt.

1 4cos 3 2sinx t y t

To find the second derivative of a curve, we find the derivative of the first derivative:

dydtdxdt

2

2

d y

dx d

ydx

1. Find the first derivative (dy/dx).2. Find the derivative of dy/dx with respect to t.

3. Divide by dx/dt.

The Second Derivative

2 3 and x t t y t t

a. Is the parametric curve concave up or down at the origin?

Example

b. Find the equation of the tangent line at the origin.

1. Find dy/dx.

dy

dy dtydxdxdt

21 3

1 2

t

t

2. Find the derivative of dy/dx with respect to t.

21 3

1 2

dy d t

dt dt t

2

2

2 6 6

1 2

t t

t

Quotient Rule

3. Divide by dx/dt.

2

2

d y

dx

dxdt

dydt

2

2

2 6 6

1 2

1 2

t t

t

t

2

3

2 6 6

1 2

t t

t

Example2 3 and x t t y t t

If a smooth curve does not intersect itself on an interval [a, b] (except possibly at the endpoints), then the arc length of the curve over the interval is:

This formula can be derived from

2 2b

a

dx dyL dt

dt dt

Arc Length in Parametric Form

2

1d

c

dyL dx

dx

Revolution about the -axis 0x y 2 2

2b

a

dx dyS y dt

dt dt

Revolution about the -axis 0y x

2 2

2b

a

dx dyS x dt

dt dt

If a smooth curve C does not cross itself on an interval [a, b], then the area of the surface formed by revolving C about the coordinated axes is given by

Surface Area in Parametric Form

Examples

1. Write the equation in parametric form.2. Find the circumference of the circle.3. Find the area of the surface formed by revolving the

arc of the circle from (5,-1) to (2,2) about the y-axis.

2 2( 2) ( 1) 9x y