Determine the equation of the tangent line to the given path at the specified value of t. (sin(7t), cos(7t), 2t^9/2); t = 1 

Solution:

We know that the equation of the tangent line to the given path at the specified value t is written as

P(t) = f(t₀) + f'(t₀)(t - t₀)

f(t₀) = (sin(7t), cos(7t), 2t^9/2)----->(1)

When t₀ = 1

f(t₀) = {sin7(1), cos7(1), 2(1)^9/2}

f(t₀) = {sin7, cos7, 2}

Now differentiate (1) with respect to t

f'(t₀) = (7cos7t, -7sin7t, 9/2{2t^9/2-1}

f'(t₀) = (7cos7t, -7sin7t, 9t^7/2}

When t₀ = 1

f'(1) = (7cos7(1), -7sin7(1), 9(1)^7/2)

f'(1) =(7cos7, -7sin7, 9)

By substituting the given function

P(t) = f(t₀) + f'(t₀)(t - t₀)

P(t) = {sin7, cos7, 2} + (7cos7, -7sin7, 9)(t - 1)

Therefore, the equation of the tangent line is P(t) = {sin7, cos7, 2} + (7cos7, -7sin7, 9)(t - 1).

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Determine the equation of the tangent line to the given path at the specified value of t. (sin(7t), cos(7t), 2t^9/2); t = 1 

Summary:

The equation of the tangent line to the given path at the specified value of t in (sin(7t), cos(7t), 2t^9/2); t = 1 is P(t) = {sin7, cos7, 2} + (7cos7, -7sin7, 9)(t - 1).