Question
Asked Oct 22, 2019

x = t3 − 12t,    y = t2 − 2

find

d2y
dx2

and for which values of t is the curve concave upward? (use interval notation).

check_circle

Expert Answer

Step 1

We find dx/dt and dy/dt. 

x=t 3-12t
dx
(t 3-12t)
dt
dt
dx
=3t 2-12
dt
y=t?-2
dy
(t 2-2)
dt
dt
dy
=2t
dt
help_outline

Image Transcriptionclose

x=t 3-12t dx (t 3-12t) dt dt dx =3t 2-12 dt y=t?-2 dy (t 2-2) dt dt dy =2t dt

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Step 2

Using dx/dt and dy/dt we find dy/dx

dy
dt
2t
dy
dx
3t 2-12
dx
dt
help_outline

Image Transcriptionclose

dy dt 2t dy dx 3t 2-12 dx dt

fullscreen
Step 3

then we find the second derivative. Apply ...

d2y
2t
dx2
3t 2-12
dx
dy
2t
dt
3t2-12
dx2
dt
dx
help_outline

Image Transcriptionclose

d2y 2t dx2 3t 2-12 dx dy 2t dt 3t2-12 dx2 dt dx

fullscreen

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