Find dy/dx and d2y/dx2

Math Jan 01, 2021

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.)
?

Expert Solution

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

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