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Lesson: Slope of a Polar Curve

In this lesson, we will learn how to find the slope of a polar curve at a point and sketch this curve along with its tangent at that point.

Worksheet • 25 Questions

Q1:

Find the slope of the tangent line to the curve π‘Ÿ = 1 πœƒ at πœƒ = πœ‹ .

  • A βˆ’ πœ‹
  • B 1 πœ‹
  • C0
  • D πœ‹
  • E βˆ’ 1 πœ‹

Q2:

Find the slope of the tangent line to the polar curve π‘Ÿ = 2 πœƒ c o s at the point πœƒ = πœ‹ 6 .

  • A √ 3 7
  • B 7 √ 3 1 6
  • C0
  • D 7 √ 3 3

Q3:

Find the slope of the tangent line to the curve π‘Ÿ = ο€½ πœƒ 3  c o s at πœƒ = πœ‹ 2 .

  • A √ 3 9
  • B √ 3
  • C √ 3 1 2
  • D √ 3 3
  • E 3 √ 3

Q4:

Find the slope of the tangent line to the curve π‘Ÿ = πœƒ c o s at πœƒ = πœ‹ 6 .

  • A βˆ’ √ 3 3
  • B βˆ’ √ 3 4
  • C √ 3 3
  • D √ 3
  • E βˆ’ √ 3

Q5:

Find the slope of the tangent line to the curve π‘Ÿ = 2 πœƒ s i n at πœƒ = πœ‹ 6 .

  • A 5 √ 3 3
  • B 3 √ 3 5
  • C 5 √ 3 1 6
  • D0
  • E √ 3 5

Q6:

Find the slope of the tangent line to the curve π‘Ÿ = 2 βˆ’ 3 πœƒ s i n at πœƒ = 5 πœ‹ 4 .

  • A βˆ’ √ 2 βˆ’ 1 √ 2 + 2
  • B βˆ’ 2 βˆ’ √ 2
  • C βˆ’ √ 2
  • D βˆ’ √ 2 1 + √ 2
  • E βˆ’ √ 2 + 2 √ 2 βˆ’ 1

Q7:

Find the slope of the tangent line to the polar curve π‘Ÿ = 1 + πœƒ c o s at the point πœƒ = πœ‹ 4 .

  • A βˆ’ √ 2 + 1
  • B βˆ’ 1 βˆ’ √ 2 2
  • C βˆ’ √ 2 2 βˆ’ 1 2
  • D βˆ’ 2 + √ 2
  • E βˆ’ √ 2 βˆ’ 1

Q8:

Find the slope of the tangent line to the curve π‘Ÿ = 1 + πœƒ s i n at πœƒ = πœ‹ 4 .

  • A βˆ’ √ 2 βˆ’ 1
  • B 1 + √ 2
  • C βˆ’ √ 2 2 βˆ’ 1 2
  • D βˆ’ 1 + √ 2
  • E βˆ’ √ 2 + 1

Q9:

Find the slope of the tangent line to π‘Ÿ = 2 + 4 πœƒ c o s at πœƒ = πœ‹ 6 . Round your answer to 3 decimal places.

Q10:

Find the slope of a tangent line to π‘Ÿ = 6 + 3 πœƒ c o s at ( 3 , πœ‹ ) .

  • AThe slope is undefined at ( 3 , πœ‹ ) .
  • B1
  • C0
  • Dβˆ’1
  • E4

Q11:

Find the slope of a tangent line to π‘Ÿ = 4 πœƒ c o s at ( 2 , πœ‹ 3 ) .

  • A √ 3 3
  • B βˆ’ 2 √ 3
  • C 2 √ 3 3
  • D 3 + 2 √ 3 3
  • E βˆ’ 2 βˆ’ √ 3

Q12:

Find the slope of a tangent line to π‘Ÿ = 1 βˆ’ πœƒ s i n at ο€Ό 1 2 , πœ‹ 6  .

Q13:

Find the slope of the tangent line to π‘Ÿ = 4 + πœƒ s i n at the point ο€Ό 3 , 3 πœ‹ 2  .

Q14:

Find the slope of the tangent line to π‘Ÿ = πœƒ l n at πœƒ = 𝑒 . Give your answer to 3 decimal places.

Q15:

For the cardioid π‘Ÿ = 1 + πœƒ s i n , find the slope of the tangent line at πœƒ = πœ‹ 3 .

Q16:

Find the points at which π‘Ÿ = 4 πœƒ c o s has a horizontal or vertical tangent line.

  • AHorizontal tangents at ( 2 √ 2 , πœ‹ 4 ) and ( 2 √ 2 , βˆ’ πœ‹ 4 ) , vertical tangents at ( 4 , 0 ) and ( 0 , πœ‹ 2 )
  • BHorizontal tangents at ( 4 , 0 ) , no vertical tangents
  • CHorizontal tangents at ( 4 , 0 ) and ( 2 √ 2 , πœ‹ 4 ) , vertical tangents at ( 0 , πœ‹ 2 ) and ( 2 √ 2 , βˆ’ πœ‹ 4 )
  • DHorizontal tangents at ( 4 , 0 ) and ( 2 √ 2 , βˆ’ πœ‹ 4 ) , vertical tangents at ( 0 , πœ‹ 2 ) and ( 2 √ 2 , πœ‹ 4 )
  • ENo horizontal tangents, vertical tangents at ( 4 , 0 ) and ο€» 0 , πœ‹ 2 

Q17:

Given a polar curve defined by π‘Ÿ = 𝑓 ( πœƒ ) , form an expression for the slope of the curve d d 𝑦 π‘₯ in terms of πœƒ and 𝑓 .

  • A d d s i n c o s c o s s i n 𝑦 π‘₯ = 𝑓 β€² ( πœƒ ) πœƒ + 𝑓 ( πœƒ ) πœƒ 𝑓 β€² ( πœƒ ) πœƒ βˆ’ 𝑓 ( πœƒ ) πœƒ
  • B d d s i n c o s c o s s i n 𝑦 π‘₯ = 𝑓 ( πœƒ ) πœƒ + 𝑓 β€² ( πœƒ ) πœƒ 𝑓 ( πœƒ ) πœƒ βˆ’ 𝑓 β€² ( πœƒ ) πœƒ
  • C d d s i n c o s c o s s i n 𝑦 π‘₯ = 𝑓 β€² ( πœƒ ) πœƒ + 𝑓 ( πœƒ ) πœƒ 𝑓 β€² ( πœƒ ) πœƒ + 𝑓 ( πœƒ ) πœƒ
  • D d d c o s s i n s i n c o s 𝑦 π‘₯ = 𝑓 β€² ( πœƒ ) πœƒ βˆ’ 𝑓 ( πœƒ ) πœƒ 𝑓 β€² ( πœƒ ) πœƒ + 𝑓 ( πœƒ ) πœƒ
  • E d d s i n c o s c o s s i n 𝑦 π‘₯ = πœƒ + πœƒ πœƒ βˆ’ πœƒ

Q18:

Find the slope of the tangent line to π‘Ÿ = πœƒ at πœƒ = πœ‹ 2 .

  • A βˆ’ 2 πœ‹
  • BThe slope of the tangent line is undefined.
  • C 2 πœ‹
  • D1
  • E βˆ’ πœ‹ 2

Q19:

Find the slope of the tangent line to π‘Ÿ = 8 πœƒ s i n at the point ο€Ό 4 , 5 πœ‹ 6  .

  • A βˆ’ √ 3
  • B 3 βˆ’ 2 √ 3 3
  • C0
  • D 2 βˆ’ √ 3
  • E √ 3

Q20:

Find the slopes of the tangent lines to π‘Ÿ = 2 ( 3 πœƒ ) s i n at the tips of the leaves.

  • AThe slope is βˆ’ √ 3 at ο€» 2 , πœ‹ 6  , √ 3 at ο€Ό 2 , 5 πœ‹ 6  , and 0 at ο€» βˆ’ 2 , πœ‹ 2  .
  • BThe slope is √ 3 at ο€» 2 , πœ‹ 6  , βˆ’ √ 3 at ο€Ό 2 , 5 πœ‹ 6  , and 0 at ο€» βˆ’ 2 , πœ‹ 2  .
  • CThe slope is 0 at the tips of all the leaves.
  • DThe slope is √ 3 at ο€» 0 , πœ‹ 3  , βˆ’ √ 3 at ο€Ό 0 , 2 πœ‹ 3  , and 0 at ( 0 , πœ‹ ) .
  • EThe slope is √ 3 3 at ο€» 2 , πœ‹ 6  , βˆ’ √ 3 3 at ο€Ό 2 , 5 πœ‹ 6  , and undefined at ο€» βˆ’ 2 , πœ‹ 2  .

Q21:

Find the slopes of the tangent lines to π‘Ÿ = 4 ( 2 πœƒ ) c o s at the tips of the leaves.

  • AThe slope is undefined at ( 4 , 0 ) and ( 4 , πœ‹ ) and the slope is 0 at ο€» βˆ’ 4 , πœ‹ 2  and ο€Ό βˆ’ 4 , 3 πœ‹ 2  .
  • BThe slope is undefined at the tips of all the leaves.
  • CThe slope is 0 at ( 4 , 0 ) and ( 4 , πœ‹ ) and the slope is undefined at ο€» βˆ’ 4 , πœ‹ 2  and ο€Ό βˆ’ 4 , 3 πœ‹ 2  .
  • DThe slope is 0 at the tips of all the leaves.
  • EThe slope is 1 at ο€» 0 , πœ‹ 4  and ο€Ό 0 , 5 πœ‹ 4  and the slope is βˆ’ 1 at ο€Ό 0 , 3 πœ‹ 4  and ο€Ό 0 , 7 πœ‹ 4  .

Q22:

Find the slope of the tangent line to the curve π‘Ÿ = 1 πœƒ at πœƒ = 2 πœ‹ .

  • A βˆ’ 2 πœ‹
  • B 1 2 πœ‹
  • C0
  • D 2 πœ‹
  • E βˆ’ 1 2 πœ‹

Q23:

Find the slope of the tangent line to the polar curve π‘Ÿ = 1 + πœƒ c o s at the point πœƒ = 3 πœ‹ 4 .

  • A βˆ’ √ 2 βˆ’ 1
  • B βˆ’ √ 2 2 + 1
  • C βˆ’ √ 2 2 + 1 2
  • D √ 2 + 2
  • E βˆ’ √ 2 + 1

Q24:

Find the slope of the tangent line to the curve π‘Ÿ = 1 + πœƒ s i n at πœƒ = πœ‹ 3 .

  • A βˆ’ 1
  • B βˆ’ 1 βˆ’ √ 3 2
  • C1
  • D βˆ’ √ 3 + 2

Q25:

Find the slope of the tangent line to the curve π‘Ÿ = 2 πœƒ s i n at πœƒ = πœ‹ 4 .

  • A 1 2
  • B βˆ’ 1
  • C1
  • D βˆ’ 1 2
  • E2
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