Find differentiable parametric equations x=x(t), y=y(t) for the unit circle on xy-plane which ...

Question:

Find differentiable parametric equations {eq}x=x(t), y=y(t) {/eq}for the unit circle on xy-plane which

are not smooth at {eq}t=0 \: and\: (x,y)=(1,0) {/eq}

Parametric Equation:

Let {eq}C = \left( {x,y} \right) {/eq} be a curve. Then, the parametric representation of the equation of a curve is {eq}C = \left( {x\left( t \right),y\left( t \right)} \right) {/eq}. So, the parametric equations are represented by {eq}x = x\left( t \right) {/eq} and {eq}y = y\left( t \right) {/eq}. Here, t is the parameter. The general parametric equations for a unit circle can be written as {eq}x = \cos t {/eq} and {eq}y = \sin t {/eq}.

Answer and Explanation:

Given

  • The given circle is unit circle.

Consider the parametric equations: {eq}x = \cos t {/eq} and {eq}y = \sin t {/eq} for the unit circle.

These equations are in the form {eq}x = x\left( t \right) {/eq} and {eq}y = y\left( t \right) {/eq}.

Differentiate the equations {eq}x = \cos\left( t \right) {/eq} and {eq}y = \sin\left( t \right) {/eq} with respect to {eq}t {/eq}.

{eq}\begin{align*} \dfrac{d}{{dt}}[x] &= \dfrac{d}{{dt}}[\cos t]\\ \dfrac{{dx}}{{dt}} &= - \sin t \end{align*} {/eq}

And

{eq}\begin{align*} \dfrac{d}{{dt}}[y] &= \dfrac{d}{{dt}}[\sin t]\\ \dfrac{{dy}}{{dt}} &= \cos t \end{align*} {/eq}

Find the differentiable parametric equation.

{eq}\begin{align*} \dfrac{{dy}}{{dx}} &= \dfrac{{dy/dt}}{{dx/dt}}\\ &= \dfrac{{ - \cos t}}{{\sin t}}\\ &= - \cot t \end{align*} {/eq}

Here, {eq}\dfrac{{dx}}{{dt}} \ne 0 {/eq} or {eq}\sin t \ne 0 {/eq}. So, {eq}t \ne 0 {/eq}.

So, the differentiable parametric equation {eq}\dfrac{{dy}}{{dx}} = - \cot t {/eq} for the unit circle on xy lane that is not a smooth at {eq}t = 0 {/eq} and {eq}\left( {x,y} \right) = \left( {1,0} \right) {/eq}.


Learn more about this topic:

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Evaluating Parametric Equations: Process & Examples

from Precalculus: High School

Chapter 24 / Lesson 3
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