Determine the equation of the tangent line to the given path at the specified value of t

vaixellause

vaixellause

Answered question

2022-12-20

Determine the equation of the tangent line to the given path at the specified value of t. (Enter your answer as a comma-separated list of equations in (x, y, z) coordinates.) ( sin ( 3 t ) , cos ( 3 t ) , 2 t 7 / 2 ); t=1

Answer & Explanation

Ramiro Brooks

Ramiro Brooks

Beginner2022-12-21Added 7 answers

For a path r(t), the general equation k(t) of its tangent line at a specified point r ( t 0 ) is given by;
k ( t ) = r ( t 0 ) + r ( t 0 ) [ t t 0 ] -----------------(i)
Where
r'(t) is the first derivative of the path r(t) at a given value of t.
From the question:
r ( t ) = ( sin 3 t , cos 3 t , 2 )  and  t 0 = 1 r ( 1 ) = ( sin 3 , cos 3 , 2 )  at  t 0 = 1
Find the first derivative component-wise of r(t) to get r'(t)
r ( t ) = ( cos 3 t , 3 sin 3 t , 7 ) r ( 1 ) = ( cos 3 , 3 sin 3 , 7 )
Now, at t 0 = 1, equation (i) becomes;
k ( t ) = r ( 1 ) + [ r ( 1 ) ( t 1 ) ] [substitute the necessary values]
k ( t ) = ( sin 3 , cos 3 , 2 ) + [ ( t 1 ) ( cos 3 , 3 sin 3 , 7 ) ]

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