How do you find the equation of the tangent line to the curve given by parametric equations: <ms

aniawnua

aniawnua

Answered question

2022-05-21

How do you find the equation of the tangent line to the curve given by parametric equations: x = 1 + ( 1 t 2 ) , y = 1 - ( 3 t ) at the point when t=2?

Answer & Explanation

Travis Fernandez

Travis Fernandez

Beginner2022-05-22Added 10 answers

Step 1
To find the slope of the tangent line, you need d y d x evaluated at a specified point.
You are given x(t) and y(t). You can use these to compute x'(t) and y'(t) which can be written using the notation d x d t and d y d t , respectively.
Step 2
d x d t = - 2 t - 3
d y d t = 3 t - 2
From the chain rule, we know that:
d y d t = ( d y d x ) ( d x d t )
Solve for d y d x :
d y d x = d y d t d x d t
d y d x = 3 t - 2 - 2 t - 3
d y d x = ( - 3 2 ) t
To obtain the slope, m, you evaluate the above at t=2:
m = ( - 3 2 ) ( 2 ) = - 3
Evaluate x and y at t = 2:
x = 1 + 1 2 2 = 5 4
y = 1 - 3 2 = - 1 2
This is the point ( 5 4 , - 1 2 )
Use the point-slope form of the equation of a line:
y - y 1 = m ( x - x 1 )
y - - 1 2 = - 3 ( x - 5 4 )
y = - 3 x - 17 4

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