Suppose a cat is chasing a ball around on the floor, and its position is describ

Question

Suppose a cat is chasing a ball around on the floor, and its position is described by the parametric equations (x(t), y(t)) = (t2-1, t-t3). (a) The cat is following one of the paths from the previous problem. Which path does the cat follow? A (b) At which times t does the cat run through the point (0,0)? (negative times are okay) Earlier time -1 , Later time 1 (c) Remember that it wasn't possible to find dy/dx at (0,0) using the method on problem 1. But now that the graph has been parametrized, you can do it. What are the tangent lines to the parametrized curve (x(t),y(t)) at (0,0)? Line with negative slope: y = x + Line with positive slope: y = x+

Explanation / Answer

x=t2-1 ,y =t-t3

dx/dt =2t ,dy/dt =1-3t2

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

dy/dx =(1-3t2)/2t

at t=-1 slope of tangent

dy/dx =(1-3(-1)2)/2(-1)

dy/dx =(1-3)/(-2)

dy/dx =1

at t=1 slope of tangent

dy/dx =(1-3(1)2)/2(1)

dy/dx =(1-3)/(2)

dy/dx =-1

equation of tangent with slope -1, passing through (0,0) is y-0 =-1(x-0)

equation of tangent with slope -1, passing through (0,0) is y=-1x +0

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equation of tangent with slope 1, passing through (0,0) is y-0 =1(x-0)

equation of tangent with slope 1, passing through (0,0) is y=1x +0

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