Answered: 6. Locate all the points (if any) of…

6. Locate all the points (if any) of the horizontal and vertical tangents to the curve defined by the set of parametric equations x = t² – t, y = t³ - 3t.

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.6: Parametric Equations

Problem 1ECP: Sketch and describe the orientation of the curve given by the parametric equations...

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pleas show me the steps too, thank you!

6. Locate all the points (if any) of the horizontal and vertical tangents to the curve defined by the
set of parametric equations x =
t² – t, y = t³ - 3t.

Expert Solution

Step 1

$C o n s i d e r t h e g i v e n e q u a t i o n s, x = t^{2} - t, y = t^{3} - 3 t F i n d d e r i v a t i v e w i t h r e s p e c t t o t . \begin{matrix} \frac{d y}{d t} = \frac{d}{d t} (t^{2} - t) \\ = 2 t - 1 \end{matrix} A n d, \begin{matrix} \frac{d x}{d t} = \frac{d}{d t} (t^{3} - 3 t) \\ = 3 t^{2} - 3 \end{matrix}$

Step 2

$F o r t h e h o r i z o n t a l \tan g e n t, \begin{matrix} \frac{d y}{d x} = 0 \\ \frac{3 t^{2} - 3}{2 t - 1} = 0 \\ 3 t^{2} - 3 = 0 \\ t = - 1, 1 \end{matrix} A n d, \begin{matrix} 2 t - 1 = 0 \\ t = \frac{1}{2} \end{matrix}$

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