For Parametric Equations, we will start to work with our usual variables (or parameters) like x and y. However, now we will use these variables/parameters where their value is dependent on an extra third variable/parameter. Often we use t, for time, as this third parameter.
Parametric Equations can be used for situations where for example, the x coordinate and the y coordinate are both based on time, t. Instead of writing the coordinates like (x, y), they can be written as x and y both as functions of t, like (x(t), y(t)) each of these parameters still plot on a graph the same way (x, y) would, it’s just the values of the x and y both depend on the value of t.
If we have a parametric equation, where we have the parameters that are differentiable to t, of x = x(t) and y = y(t). And dx/dt does not equal 0, then,
We have the parametric equation (x, y) = (t, t2 + 1)
First, for Parametric equations, recognize that we have our x & y terms as functions of t; x(t) & y(t).
(x, y) = (t, t2 + 1)
x(t) = t
y(t) = t2 + 1
So differentiate x(t) to get dx/dt
And y(t) to get dy/dt
Differentiate the parametric equation (x, y) = (3t2 + 2t, sin(t))
We have (x, y) = (3t2 + 2t, sin(t))
We have x(t) = 3t2 + 2t
dx/dt = (2)3t2-1 + 2t0
y(t) = sin(t)
dy/dt = cos(t)
The example below show how to find the slope of a tangent by differentiating.
Below is another worked examples showing how to find the slope of a tangent by differentiating.
For a graph that gives us the position of something based off time, we can use a method of differentiation of that graph to calculate its speed (or velocity)
If we have the variables x = x(t) and y = y(t) that give the position of something with regards to the time, t, and these variable functions of t are differentiable of t, then,
Below a worked example showing how to find the speed of an object at given location and time.
Below is a worked examples showing how to determine at which location an object has greater speed.