Question

Find the parametric equations for the tangent line to the curve with the given parametric equations at the specified point. The parametric equations are x = 4 ln(t), y = 6t, z = t⁵. The specified point is (0, 6, 1).

Answer 1

To find the parametric equations for the tangent line to the curve with the given parametric equations at the specified point, we need to find the derivative of each equation with respect to t, and then substitute the value of t at the specified point.

Step-by-step explanation:

To find the parametric equations for the tangent line to the curve with the given parametric equations at the specified point, we need to find the derivative of each equation with respect to t, and then substitute the value of t at the specified point.

Step 1: Find the derivatives of the given equations:
x' = 4/t
y' = 6
z' = 5t^4

Step 2: Substitute t = 25 into the derivatives:
x'(25) = 4/25
y'(25) = 6
z'(25) = 5(25)^4

Step 3: Write the parametric equations for the tangent line:
x = x'(25)(t - 25) + x(25)
y = y'(25)(t - 25) + y(25)
z = z'(25)(t - 25) + z(25)

Therefore, the parametric equations for the tangent line to the curve at the specified point are:
x = (4/25)(t - 25)
y = 6(t - 25) + 6
z = 15625(t - 25) + 1