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Homework answers / question archive / Find dy/dx and d2y/dx2
Find
dy/dx and d2y/dx2.
x = et, y = te?t
dy/dx = ?
d2y/dx2 = ?
For which values of t is the curve concave upward? (Enter your answer using interval notation.)
?
dx/dt = e^t
dy/dt = -te^-t + e^-t
so dy/dt X dt/dx = dy/dx= -te^-t + e^-t /e^t = -t e^(-2t) + e^(-2t)
d2y/dx2 = -dt/dx e^(-2t) +2 t e^(-2t) dt/dx -2e^(-2t) dt/dx
= -e^(-3t) +2te^(-3t) -2e^(-3t)
for concave dy/dx >0 => -t e^(-2t) + e^(-2t) >0
=> t<0 the curve is concave
