Question

Can someone help me with this? he didn't lecture on finding points other than when t=0, so i am very, very lost. thanks. don't know where to start.

(1 pt) Use a graph to estimate the coordinates of the highest point and the leftmost point on the curve
x=4te^t,y=4te^_t.
Then find the exact coordinates.
Highest point: .
Leftmost point: .
The curve has one horizontal asymptote and one vertical asymptote. Find them.
Horizontal asymptote: y= .
Vertical asymptote: x= .

Answer 1

firstly, we must calculate dx/dt ,
dx/dt = 4te^t + 4e^t=(4t + 4)e^t
dx/dt=0 implies (4t + 4)e^t=0 so 4t + 4 = 0 because e^t>0, and t= -1,
fort t = -1, x = 4 (-1)e^-1, and y=4(-1)e^1,
x= - 4 / e, and y= - 4 e
Then find the exact coordinates.
Highest point:- 4/e
Leftmost point: - 4e
Horizontal asymptote: lim 4te^_t
lim 4te^_t = lim 4/ e^t /t = 0, (x=o H.A), because e^t /t = infinity when t= infinity, so
t=infinity t=infinity
lim 4te^t =4 limte^t = 4 x o =0
t= - infinity t= - infinity

The curve has one horizontal asymptote and one vertical asymptote. Find them.
Horizontal asymptote: y= . 0
Vertical asymptote: x= 0