4. For which values of t is the curve concave up

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x = et,    y = tet


For which values of t is the curve concave upward? (Enter your answer using interval notation.)
 
Apr 4th, 2015

Parametric equations: x = et; y = t e–t

Find the first derivative of y by x: y'x = y't / x't = ( e–t – te–t ) / et = e–2t – te–2t

Find the second derivative of y by x: y''xx = (y'x )'t / x't = ( –2e–2t – e–2t +2te–2t ) / et =

(–3e–2t + 2te–2t )/et = –3e–3t + 2te–3t = (2t – 3)e–3t

y''xx = (2t – 3)e–3t > 0 if t > 3/2 = 1.5 . The curve is concave upward if t is in the interval (1.5, +∞).


Apr 4th, 2015

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