# 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(t_{0}) + f'(t_{0})(t - t_{0})

f(t_{0}) = (sin(7t), cos(7t), 2t^{9/2})----->(1)

When t_{0} = 1

f(t_{0}) = {sin7(1), cos7(1), 2(1)^{9/2}}

f(t_{0}) = {sin7, cos7, 2}

Now differentiate (1) with respect to t

f'(t_{0}) = (7cos7t, -7sin7t, 9/2{2t^{9/2-1}}

f'(t_{0}) = (7cos7t, -7sin7t, 9t^{7/2}}

When t_{0} = 1

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

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

By substituting the given function

P(t) = f(t_{0}) + f'(t_{0})(t - t_{0})

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).

## 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).

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