find the tangent to the curve given by the parametric equations:
x=1/t y=sqrt(t)e^(-t)
dy/dx = (dy/dt) / (dx/dt) = [(1-2t)e^-t / 2√t] / (-1/t^2)
= (2t-1)/2 t^(3/2) e^-t
To find the tangent to a curve given by parametric equations, we need to find the derivatives of both x and y with respect to the parameter, and then form the equation for the tangent line.
Let's start by finding the derivatives of x and y with respect to t:
Given:
x = 1/t
y = sqrt(t) * e^(-t)
To find dx/dt, we differentiate x with respect to t:
dx/dt = d(1/t)/dt
To differentiate 1/t with respect to t, we use the quotient rule:
dx/dt = (t * 0 - 1 * 1) / (t^2)
= -1 / t^2
Next, let's find dy/dt. We differentiate y with respect to t:
dy/dt = d(sqrt(t) * e^(-t))/dt
To differentiate sqrt(t) * e^(-t) with respect to t, we use the product rule:
dy/dt = (sqrt(t) * d(e^(-t))/dt) + (e^(-t) * d(sqrt(t))/dt)
Differentiating e^(-t) with respect to t gives us:
d(e^(-t))/dt = -e^(-t)
To differentiate sqrt(t) with respect to t, we use the power rule:
d(sqrt(t))/dt = (1/2) * t^(-1/2)
Substituting these values back into dy/dt, we get:
dy/dt = (sqrt(t) * -e^(-t)) + (e^(-t) * (1/2) * t^(-1/2))
= -sqrt(t) * e^(-t) + (1/2) * e^(-t) * t^(-1/2)
Now that we have dx/dt and dy/dt, we can find the slope of the tangent line using the formula:
m = dy/dx = (dy/dt) / (dx/dt)
Substituting the derivatives we found, we have:
m = ((-sqrt(t) * e^(-t) + (1/2) * e^(-t) * t^(-1/2))) / (-1 / t^2)
= (-sqrt(t) * e^(-t) + (1/2) * e^(-t) * t^(-1/2)) * (-t^2)
Simplifying further:
m = sqrt(t) * e^(-t) - (1/2) * e^(-t) / t^(3/2)
Now we have the slope of the tangent line, and to find the equation for the tangent line, we need a point on the curve. For that, we can substitute a specific value of t into the original parametric equations for x and y.
For example, if we choose t = 1, then we have:
x = 1/t = 1/1 = 1
y = sqrt(t) * e^(-t) = sqrt(1) * e^(-1) = 1/e
So, with the point (1, 1/e) on the curve, and the slope from above, we can use the point-slope form to find the equation of the tangent line:
y - y1 = m(x - x1)
where (x1, y1) is the point on the curve.
Plugging in the values:
y - (1/e) = (sqrt(t) * e^(-t) - (1/2) * e^(-t) / t^(3/2))(x - 1)
This is the equation of the tangent line to the curve given by the parametric equations.