Need help with question please show work. 1 find all Doints lifpaints of tangencynysof horizontal and verticalto the curve36て+きーコーX

Given that: x=t2-t+2, y=t3-3t To calculate dx/dt, ddtx=ddtt2-t+2=ddtt2-ddtt+ddt2=2t-1+0dxdt=2t-1…

Answered: 1 find all points lif any) of hxizantal…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Need help with question please show work.

1 find all Doints lif
paints of tangency
nysof horizontal and vertical
to the curve
36
て+きーコーX

Expert Solution

Step 1

Given that:

$x = t^{2} - t + 2, y = t^{3} - 3 t$

To calculate dx/dt,

$\begin{matrix} \frac{d}{d t} x = \frac{d}{d t} (t^{2} - t + 2) \\ = \frac{d}{d t} t^{2} - \frac{d}{d t} t + \frac{d}{d t} 2 \\ = 2 t - 1 + 0 \\ \frac{d x}{d t} = 2 t - 1 . . . . . . . . (1) \end{matrix}$

To calculate dy/dt,

$\begin{matrix} \frac{d}{d t} y = \frac{d}{d t} (t^{3} - 3 t) \\ = \frac{d}{d t} t^{3} - 3 \frac{d}{d t} t \\ = 3 t^{2} - 3 (1) \\ \frac{d y}{d t} = 3 t^{2} - 3 . . . . . . . . (2) \end{matrix}$

Step 2

To calculate dy/dx,

$\begin{matrix} \frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}} \\ \frac{d y}{d x} = \frac{3 t^{2} - 3}{2 t - 1} \\ F o r h o r i z o n t a l tangent, \frac{d y}{d x} = 0 \\ \frac{3 t^{2} - 3}{2 t - 1} = 0 \\ 3 (t^{2} - 1) = 0 \\ t^{2} - 1 = 0 \\ t^{2} = 1 \\ t = \pm 1 \end{matrix}$

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