# Calculus with Parametric Curves

## Calculus can be applied to parametric equations as well.

#### Key Points

• Parametric equations are equations that depend on a single parameter.

• A common example comes from physics. The trajectory of an object is well represented by parametric equations.

• Writing the horizontal and vertical displacement in terms of the time parameter makes finding the velocity a simple matter of differentiating by the parameter time. Parameterizing makes this kind of analysis straight-forward.

#### Terms

• the change of velocity with respect to time (can include deceleration or changing direction)

• a vector quantity which denotes distance with a directional component

• the path of a body as it travels through space

#### Figures

1. ##### Trajectories

A trajectory is a useful place to use parametric equations because it relates the horizontal and vertical distance to the time.

Parametric equations are equations which depend on a single parameter. You can rewrite y=x such that x=t and y=t where t is the parameter.

A common example occurs in physics, where it is necessary to follow the trajectory of a moving object. The position of the object is given by x and y, signifying horizontal and vertical displacement, respectively. As time goes on the object flies through its path and x and y change. Therefore, we can say that both x and y depend on a parameter t, which is time (Figure 1).

This way of expressing curves is practical as well as efficient; for example, one can integrate and differentiate such curves term-wise. Thus, one can describe the velocity of a particle following such a parametrized path as:

$v(t)=r'(t)=(x'(t),y'(t),z'(t))=(-a\sin(t),a \cos(t),b)$

where v is the velocity, r is the distance, and x, y, and z are the coordinates. The apostrophe represents the derivative with respect to the parameter.

The acceleration can be written as follows with the double apostrophe signifying the second derivative:

$a(t)=r''(t)=(x''(t),y''(t),z''(t))=(-a\cos(t),-a \sin(t),b)$

Writing these equations in parametric form gives a common parameter for both equations to depend on. This makes integration and differentiation easier to carry out as they rely on the same variable. Writing x and y explicitly in terms of t enables one to differentiate and integrate with respect to t. The horizontal velocity is the time rate of change of the x value, and the vertical velocity is the time rate of change of the y value. Writing in parametric form makes this easier to do.

#### Key Term Glossary

acceleration
the change of velocity with respect to time (can include deceleration or changing direction)
##### Appears in these related concepts:
calculus
Any formal system in which symbolic expressions are manipulated according to fixed rules.
##### Appears in these related concepts:
coordinate
a number representing the position of a point along a line, arc, or similar one-dimensional figure
##### Appears in these related concepts:
curve
a simple figure containing no straight portions and no angles
##### Appears in these related concepts:
derivative
a measure of how a function changes as its input changes
##### Appears in these related concepts:
differentiation
the process of determining the derived function of a function
##### Appears in these related concepts:
displacement
a vector quantity which denotes distance with a directional component
##### Appears in these related concepts:
integration
the operation of finding the region in the x-y plane bound by the function
##### Appears in these related concepts:
parametric
of, relating to, or defined using parameters
##### Appears in these related concepts:
trajectory
the path of a body as it travels through space
##### Appears in these related concepts:
variable
a quantity that may assume any one of a set of values
##### Appears in these related concepts:
velocity
a vector quantity that denotes the rate of change of position with respect to time, or a speed with the directional component