What Parametric Differentiation Means | CAIE 9709 Pure Mathematics 2 Notes

Chapters

Pure Mathematics 2

Mathematics9709

The shared parameter

Definition

What is meant by parametric equations?

Model answer: Equations $x=f(t)$ and $y=g(t)$ that describe a curve using the shared parameter $t$ .

For example, if $x=t^2+1$ and $y=t^3-3t$ , then $t=2$ gives the point $(5,2)$ .

The parameter is the shared clock

x(t)

→ dx/dt

y(t)

→ dy/dt

dy/dx = (dy/dt) / (dx/dt)

tangent = dy/dx

normal = −1/(dy/dx)

Differentiate both equations with respect to the same parameter before dividing.

Divide the rates

Both derivatives must be rates against the same parameter. The chain rule gives $dy/dt=(dy/dx)(dx/dt)$ . Therefore, when $dx/dt\ne0$ ,

\frac{dy}{dx}=\frac{dy/dt}{dx/dt}

Parameter t0

Point: (1, 0)
dx/dt: 0
dy/dt: -3
dy/dx: not finite

The tangent is vertical.

The blue curve is sampled from x = t² + 1 and y = t³ − 3t. The dashed line is the tangent at the selected parameter value.

Your turnIndependent check [5 marks]

A curve has $x=t^2$ and $y=5-2t$ . Find $dy/dx$ in terms of $t$ , then find its tangent at $t=2$ .

Show worked answer

\frac{dx}{dt}=2t,\qquad\frac{dy}{dt}=-2

\frac{dy}{dx}=-\frac1t

Answer

y=-\frac12x+3