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 = t, y = e⁽⁻³ᵗ⁾, z = 4t - t⁴. Find the tangent line at the point (0, 1, 0).

Answer 1

The parametric equations for the tangent line to the given curve at the specified point are x = t, y = -3t + 1, z = 4t.

Step-by-step explanation:

To find the parametric equations for the tangent line to the curve at a specified point, we need to find the derivatives of the parametric equations. Let's start by finding the derivatives of x(t), y(t), and z(t) with respect to t:

x'(t) = 1, y'(t) = -3e^(-3t), z'(t) = 4 - 4t^3

Next, we substitute the value of t at the specified point (0) into the derivatives to find the slopes of the tangent line:

x'(0) = 1, y'(0) = -3, z'(0) = 4

Using the point-slope form of a line, the tangent line can be written as:

x - x₀ = x'(0)(t - t₀), y - y₀ = y'(0)(t - t₀), z - z₀ = z'(0)(t - t₀)

Substituting the values from the specified point and the derivatives, we get the parametric equations for the tangent line: x = t, y = -3t + 1, z = 4t