10.1 Parametric functions

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008

Mark Twain’s Boyhood HomeHannibal, Missouri

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008

Mark Twain’s HomeHartford, Connecticut

In chapter 1, we talked about parametric equations.Parametric equations can be used to describe motion that is not a function.

x f t y g t

If f and g have derivatives at t, then the parametrized curve also has a derivative at t.

The formula for finding the slope of a parametrized curve is:

dy

dy dtdxdxdt

This makes sense if we think about canceling dt.

The formula for finding the slope of a parametrized curve is:

dy

dy dtdxdxdt

We assume that the denominator is not zero.

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

dydtdxdt

2

2

d y

dx dy

dx

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

3. Divide by dx/dt.

Example:2

2 32

Find as a function of if and .d y

t x t t y t tdx

Example:2

2 32

Find as a function of if and .d y

t x t t y t tdx

1. Find the first derivative (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

The equation for the length of a parametrized curve is similar to our previous “length of curve” equation:

(Notice the use of the Pythagorean Theorem.)

2 2dx dy

L dtdt dt

Likewise, the equations for the surface area of a parametrized curve are similar to our previous “surface area” equations:

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

Revolution about the -axis 0x y 2 2

2b

a

dx dyS y dt

dt dt

This curve is:

sin 2

2cos 5

x t t t

y t t t