Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

### Parametric Equations

Parametric functions show a relationship between a variable and time

Often it’s used to show the position of an object or shape of a curve

The formula for a parametric equation is given by:

x = f(t)

y = g(t)

X & Y are

**dependent**and time is**independent**

### Arc Length of Curves

Arc length is the distance along a curve

We have to square the derivative, but because both X & Y are dependent, we square

**both**derivativesWe then take the square root and the integral

This time it’s the integral from t=a to t=b

### Vector-Valued Functions

These functions map numbers to vectors!

For parametric equations, they represent position, velocity, and acceleration!

To derive these functions we have to take the derivative of each component

**individually**

The same applies to integration- integrate each component

**individually**

### Polar Coordinates

This is a coordinate system where x,y pairs are replaced with r (the distance from the orgin) and θ (angle from x-axis)

To go from a set of polar coordinates to regular (cartesian) there is a conversion:

Remember that when we differentiate with polar coordinates we differentiate with respect to theta → dθ

Area between two curves in polar coordinates is given by A = (1/2) ∫(a,b) (R^2 - r^2) dθ

Instead of top - bottom we have inner - outer

# Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

### Parametric Equations

Parametric functions show a relationship between a variable and time

Often it’s used to show the position of an object or shape of a curve

The formula for a parametric equation is given by:

x = f(t)

y = g(t)

X & Y are

**dependent**and time is**independent**

### Arc Length of Curves

Arc length is the distance along a curve

We have to square the derivative, but because both X & Y are dependent, we square

**both**derivativesWe then take the square root and the integral

This time it’s the integral from t=a to t=b

### Vector-Valued Functions

These functions map numbers to vectors!

For parametric equations, they represent position, velocity, and acceleration!

To derive these functions we have to take the derivative of each component

**individually**

The same applies to integration- integrate each component

**individually**

### Polar Coordinates

This is a coordinate system where x,y pairs are replaced with r (the distance from the orgin) and θ (angle from x-axis)

To go from a set of polar coordinates to regular (cartesian) there is a conversion:

Remember that when we differentiate with polar coordinates we differentiate with respect to theta → dθ

Area between two curves in polar coordinates is given by A = (1/2) ∫(a,b) (R^2 - r^2) dθ

Instead of top - bottom we have inner - outer