What is the slope of #f(t) = (t-1,t^3)# at #t =1#?

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Answer 1 · 2017-07-28T15:59:18

Slope is #3#

The slope of #f(t)-(x(t),y(t))# is given by #((dy)/(dt))/((dx)/(dt))#

Here #y=t^3# hence #(dy)/(dt)=3t^2#

and #x=t-1# and hence #(dx)/(dt)=1#

Hence slope is #(3t^2)/1=3t^2#

and at #t=1#, it is #3xx1^2=3#

one can see it in the following graph as #t=1# means slope at #(0,1)#

graph{y=(x+1)^3 [-5, 5, -2.5, 2.5]}