Question

Consider the parametric equations
x=te^(2t) y=t^2e^(2t)

a) Find all points (x,y) on the parametric curve where the tangent lines are vertical

b) Find all points (x,y) on the parametric curve where the tangent lines are horizontal

c) Find the interval(s) where the parametric curve is concave upward

d)Find the interval(s) where the parametric curve is concave downward

Answer 1

\[x=te ^{2t} \]
\[y=t^{2}e^{2t}\]

Answer 2

Try the following:
\[ \Large \frac{dx}{dt}=e^{2t}+2te^{2t} \]
And same for:
\[\Large \frac{dy}{dt}=2te^{2t}+2t^2e^{2t} \]
Do you know how to get dy/dx from these?

Answer 3

i got what you have but didn't know what to do from there

Answer 4

\[\frac{ dy }{dx }=\frac{ 2t^{2}e^{2t}+2te^{2t} }{ 2te^{2t}+e^{2t} }\]

Answer 5

well now you have \[m=\frac{dy}{dx} \]
You can even clean this equation up a bit. And then it becomes pretty is, for vertical tangents, the slope has to be infinite, so you have to divide by zero, or see where the denominator is equal to zero.