Central topic

Parametric Functions

Chain Rule for Parametric equations

Sometimes, it is hard to differentiate equations in terms of x and y. Thus, instead we differentiate parametrically. That is, both x and y are expressed in terms of t.

and similarly,

such that,

Hence, we take advantage of chain rule to find

It is also useful to know how to convert a parametric equation to Cartesian form.

Steps to convert Parametric to Cartesian

Firstly, we make t as the subject topic for both x and y equations.

Secondly, we equate them.

astly, simplify the equation and make x or y the suibject or you can even leave it in any form you want

Arc Length of Curves

Condition

Read Notes

r

1. The segment of the curve m<t<n is only traced once.2. Both dx/dt and dy/dt are continuous functions for m<t<n

Formula

For Parametric equation

For Cartesian Equation

Tangent Lines to Parametric equations

Find the expression for dy/dx

Substitute t=t0

Evalutate the expression

Special Cases

1. If dy/dt = 0, but dx/dt != 0, then it is a horzeontal tangent line.2. If dy/dt != 0, but dx/dt = 0, then it is a vertical tangent line.3. If dy/dt = 0 and dx/dt = 0, then dy/dx is of inderterminate form. Such points are known as singular points and no gerneral statements can be made about it until further probe.

Polar Functions

Converting between polar and cartesian form

Convert from polar to cartesian

Convert from cartesian to polar

Finding the Derivative

Method

1. Express x and y in terms of theta.2. Simplify the equation3. Evaluate dy/dx

formulas

Tangent/Normal Lines to Polar Curve

Method

. Find the expressing for dy/dx.2. Evalutet it at \theta = \theta03. Find the corresponding value of x and y4. Write down the equation for tangentNote: Normal lines are found the same way but the gradient is -1/m, where m is the gradient of the tangent.

Arc Length of a Polar Curve

Conditions

The segment of polar curve from alpha to beta is not traced more than once

dr/d\theta is continuous from alpha to beta.

Formulas