Find the parametric equations for the tangent line to the curve with the given parametric equations at specified point.
x= e^t
y=te^t
z=te^(t^2)
(1,0,0)
Try a circle with parameter t=θ
x=cos(t)
y=sin(t)
dx/dt=-sin(t)
dy/dt=cos(t)
at t=0, (dx/dt,dy/dt)=(0,1)
at t=π/4, (dx/dt,dy/dt)=(-√2/2,√2/2)
...etc
Here:
x= e^t
y=te^t
z=te^(t^2)
(1,0,0)
Solve for t at the point (1,0,0) gives t=0 (the only way to get y=z=0).
dx/dt=e^t
dy/dt=(1+t)e^t
dz/dt=t(2+t)e^t
Substitute t=0 and find the equation of the line passing through (1,0,0) with the given slopes.
To find the parametric equations for the tangent line to the curve at the point (1, 0, 0), we need to perform the following steps:
Step 1: Determine the derivative of each component function with respect to the parameter.
For the given parametric equations:
x = e^t,
y = t * e^t,
z = t * e^(t^2).
Differentiating each equation with respect to t:
dx/dt = d/dt(e^t) = e^t,
dy/dt = d/dt(t * e^t) = e^t + t * e^t,
dz/dt = d/dt(t * e^(t^2))
Step 2: Plug in the value of t that corresponds to the specified point (1, 0, 0).
Since we are looking for the tangent line at the point (1, 0, 0), we need to find the value of t that satisfies x = 1, y = 0, and z = 0.
From the first equation, x = e^t, we can solve for t:
e^t = 1
t = ln(1)
t = 0.
Therefore, at t = 0, x = 1, y = 0, and z = 0.
Step 3: Find the derivative values at t = 0:
Substituting t = 0 into the derivatives we found earlier:
dx/dt = e^t = e^0 = 1,
dy/dt = e^t + t * e^t = e^0 + 0 * e^0 = 1,
dz/dt = d/dt(t * e^(t^2)) = d/dt(0 * e^(0^2)) = d/dt(0) = 0.
Step 4: Determine the parametric equations of the tangent line.
The tangent line has the following form:
x = x_0 + (dx/dt) * t,
y = y_0 + (dy/dt) * t,
z = z_0 + (dz/dt) * t,
where (x_0, y_0, z_0) is the point on the curve where the tangent line passes through, and t is the parameter.
Using the values we obtained in Step 3:
x = 1 + 1 * t,
y = 0 + 1 * t,
z = 0 + 0 * t.
Simplifying the equations, we get the parametric equations for the tangent line:
x = 1 + t,
y = t,
z = 0.
Therefore, the parametric equations for the tangent line to the curve at the point (1, 0, 0) are:
x = 1 + t,
y = t,
z = 0.