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In this lesson, we will learn how to find the tangent and normal to a curve represented in parametric form.

Q1:

Find the slope of the tangent to the astroid π₯ = π π c o s 3 , π¦ = π π s i n 3 in terms of π .

Q2:

Find an equation of the tangent to the curve π₯ = 1 + π‘ l n , π¦ = π‘ + 2 2 at the point ( 1 , 3 ) .

Q3:

Find an equation of the tangent to the curve π₯ = π‘ β π‘ 2 , π¦ = π‘ + π‘ + 1 2 at the point ( 0 , 3 ) .

Q4:

Find an equation of the tangent to the curve π₯ = 1 + β π‘ , π¦ = π π‘ 2 at the point ( 2 , π ) .

Q5:

Find an equation of the tangent to the curve π₯ = β π‘ , π¦ = π‘ β 2 π‘ 2 at the point corresponding to the value π‘ = 4 .

Q6:

Find the equation of the tangent to the curve π₯ = π‘ + 1 3 , π¦ = π‘ + π‘ 4 at the point corresponding to the value π‘ = β 1 .

Q7:

Find an equation of the tangent to the curve π₯ = π‘ π‘ c o s , π¦ = π‘ π‘ s i n at the point corresponding to the value π‘ = π .

Q8:

Find an equation of the tangent to the curve π₯ = π π π‘ π‘ s i n , π¦ = π 2 π‘ at the point corresponding to the value π‘ = 0 .

Q9:

Given that π₯ = 2 π‘ β 9 3 , and π¦ = β 7 π‘ + 8 3 , determine the equation of the tangent to the curve at π‘ = β 1 .

Q10:

Determine the equation of the normal to the curve π₯ = β 4 π + 3 c o t , π¦ = 3 π + β 2 π s i n s e c 2 at π = π 4 .

Q11:

Find the equation of the normal to the curves π₯ = 2 π s e c and π¦ = 4 π t a n at π = π 6 .

Q12:

A curve πΆ is defined by the parametric equations π₯ = π‘ 2 and π¦ = π‘ β 3 π‘ 3 .

Find the two equations of the tangents to curve πΆ on the point at ( 3 , 0 ) .

Find all possible points on πΆ where the tangent is horizontal.

Find all possible points on πΆ where the tangent is vertical.

Q13:

Consider the curve π₯ = π‘ β 3 π‘ 3 , π¦ = π‘ β 3 π‘ 3 2 .

Find all points on this curve where the tangent is horizontal.

Find all points on this curve where the tangent is vertical.

Q14:

Consider the curve π₯ = π s i n π , π¦ = π c o s π .

Q15:

A cycloid curve is given by the equations π₯ = π ( π‘ β π‘ ) s i n and π¦ = π ( 1 β π‘ ) c o s .

Find the tangent to the cycloid at the point where π‘ = π 3 .

Find all points on the curve where the tangent is horizontal.

Find all points on the curve where the tangent is vertical.

Q16:

Find all possible equations of the tangents to the curve π₯ = 3 π‘ + 1 2 , π¦ = 2 π‘ + 1 3 that pass through the point ( 4 , 3 ) .

Q17:

Find the value of π at which the curve π₯ = 8 π + 5 π + π β 1 3 2 , π¦ = 5 π β π + 2 2 has a vertical tangent.

Q18:

Determine the equation of the tangent to the curve π₯ = π + 9 π 2 , π¦ = π 2 at π = β 6 .

Q19:

Consider the curve π₯ = π‘ β 3 π‘ 3 , π¦ = π‘ β 3 2 .

Q20:

Consider the curve π₯ = π c o s , π¦ = 3 π c o s .

Q21:

Find an equation of the tangent to the curve π₯ = π π‘ s i n , π¦ = π‘ + π‘ 2 at the point ( 0 , 2 ) .

Q22:

Find the equation of the normal to the curve π₯ = 3 π c o s , π¦ = β 2 + π s i n at π = 3 π 4 .

Q23:

Find the equation of the tangent to the curve π₯ = β 9 π c s c , π¦ = 4 π β 5 π c o t c s c at π = 5 π 3 .

Q24:

Find the equation of the tangent to the curve π₯ = 5 π β 2 c o t , π¦ = 4 π + β 2 π s i n s e c 2 at the point π = π 4 .

Q25:

Find the equation of the tangent to the curve π₯ = 9 π c o s , π¦ = β 2 + 3 π s i n at π = 3 π 4 .

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