Determine the equation of the tangent line to the given path at the specified value of t. (sin(7t), cos(7t), 2t9/2); 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), 2t9/2)----->(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{2t9/2-1}
f'(t0) = (7cos7t, -7sin7t, 9t7/2}
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).
Determine the equation of the tangent line to the given path at the specified value of t. (sin(7t), cos(7t), 2t9/2); t = 1
Summary:
The equation of the tangent line to the given path at the specified value of t in (sin(7t), cos(7t), 2t9/2); t = 1 is P(t) = {sin7, cos7, 2} + (7cos7, -7sin7, 9)(t - 1).
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