Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = 1 + 8√t, y = t⁴− t, z = t⁴ + t; (9, 0, 2)

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asked Apr 11, 2024 144k views

1 Answer

Mir S Mehdi · Answer 1 · 2024-04-17T05:22:42+0000

To find the parametric equations for the tangent line to the curve, we need to find the derivative of each component of the parametric equations.

To find the parametric equations for the tangent line to the curve, we need to find the derivative of each component of the parametric equations.

First, we find the derivatives of x, y, and z with respect to t:

dx/dt = 4√t

dy/dt = 4t³ - 1

dz/dt = 4t³ + 1

Next, we substitute t = 25 into these derivatives to find the slopes:

slope of x = 4√25 = 20

slope of y = 4(25³) - 1 = 9999

slope of z = 4(25³) + 1 = 10001

The parametric equations for the tangent line are:

x = 1 + 20t

y = 0 + 9999t

z = 2 + 10001t