x(t) = e^-t
x^-2 = (e^-t)^-2 = e^2t
3 x^-2 = 3 e^2t = 3/x^2 = y
Rewrite the following parametric equations by solving for y
x(t) = e^-t
y(t) = 3e^2t
a) y=1/3x^2,x>0
b)3/x^2, x>0
c) y= 3e^t, x>0
d) y=4e^t, x>0
2 answers
x(t) = e^-t
ln x = - t
t = - ln x
y = 3 e ^ ( 2 t )
y = 3 e ^ [ 2 ∙ ( - ln x ) ]
y = 3 e ^ [ - ( 2 ∙ ln x ) ]
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Remark:
e ^ [ ( 2 ∙ ln x ) ] = x ^ 2
e ^ [ ( - 2 ∙ ln x ) ] = 1 / x ^ 2
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So:
y = 3 e ^ [ - ( 2 ∙ ln x ) ]
y = 3 / x ^ 2
x > 0
ln x = - t
t = - ln x
y = 3 e ^ ( 2 t )
y = 3 e ^ [ 2 ∙ ( - ln x ) ]
y = 3 e ^ [ - ( 2 ∙ ln x ) ]
______________________
Remark:
e ^ [ ( 2 ∙ ln x ) ] = x ^ 2
e ^ [ ( - 2 ∙ ln x ) ] = 1 / x ^ 2
______________________
So:
y = 3 e ^ [ - ( 2 ∙ ln x ) ]
y = 3 / x ^ 2
x > 0