Express x=e^(-2t), y=4e^(3t) in the form y = f(x) by eliminating the parameter.
x = e^(-2t)
ln x = -2t
t = (-1/2)lnx
y = 4e^3t
ln y = ln4 + 3t
t = (lny - ln4)/3
so (-1/2)lnx = (lny - ln4)/3
times 6
-3lnx = 2lny - 2ln4
3lnx + 2lny = 2ln4
ln x^3 + ln y^2 = ln16
ln( (x^3)(y^2) ) = ln 16
x^3 y^2 = 16
y^2 = 16/x^3
y = ± 4/√(x^3)
f(x) = ± 4/√(x^3)
or, note that
e^3t = (e^(-2t))^(-3/2) = x^(-3/2)
so, you have 4x^(-3/2)
I think the -4/√(x^3) is an extraneous solution, since y>0 for all t
To eliminate the parameter and express the given equations in the form y = f(x), we need to solve for t in terms of x in each equation and substitute the solutions in the other equation.
For the equation x = e^(-2t):
Take the natural logarithm (ln) on both sides of the equation:
ln(x) = ln(e^(-2t))
Using the logarithmic property ln(a^b) = b * ln(a):
ln(x) = -2t * ln(e)
Since ln(e) = 1:
ln(x) = -2t
Now divide both sides by -2:
t = -ln(x)/2
For the equation y = 4e^(3t):
Substitute the value of t we found above:
y = 4e^(3*(-ln(x)/2))
Simplify by using the power property of exponents e^(a*b) = (e^a)^b:
y = 4(e^(-3ln(x)/2))
Now apply the logarithmic property ln(a^b) = b * ln(a) to simplify further:
y = 4(x^(-3/2))
Hence, the expression y = f(x) for the given equations x = e^(-2t) and y = 4e^(3t) after eliminating the parameter is:
y = 4(x^(-3/2))
To eliminate the parameter and express the given parametric equations in the form y = f(x), we need to solve one of the equations for t in terms of x (or vice versa) and substitute it into the other equation.
Let's start by solving the first equation, x = e^(-2t), for t.
Take the natural logarithm (ln) of both sides to get:
ln(x) = ln(e^(-2t))
Using the property of logarithms that ln(e^a) = a, we can simplify this to:
ln(x) = -2t
Divide both sides by -2 to isolate t:
t = -(1/2)ln(x)
Now, substitute this expression for t in the second equation, y = 4e^(3t):
y = 4e^(3(-(1/2)ln(x)))
Using the property of exponentials that e^(a⋅b) = (e^a)^b, we can rewrite this as:
y = 4(e^(-3/2ln(x)))
Simplifying further:
y = 4(x^(-3/2))
Therefore, the parametric equations x = e^(-2t) and y = 4e^(3t) can be expressed in the desired form as y = 4(x^(-3/2)).