What are the parametric equations for the tangent line at t=3 for the motion of a particle given by x(t)=4t^2+3, y(t)=3t^3?

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Answer 1 · 2018-06-09T14:59:46

#bb l(lambda) = (39,81) + lambda (8 , 27)#

#bb r(t) = (4t^2 + 3, 3t^3)#

#bbr(3) = (39,81)#

#bb r'(t) = (8t , 9t^2)#

That is the tangent vector.

#bb r'(3) = (24 , 81)#

The tangent line is:

#bb l(lambda) = bb r(3) + lambda bb r'(3)#

#= (39,81) + lambda (24 , 81)#

We can factor the direction vector a little:

#bb l(lambda) = (39,81) + lambda (8 , 27)#