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Worksheet: Slope of a Polar Curve

In this worksheet, we will practice finding the slope of a polar curve at a point and sketching this curve along with its tangent at that point.

Q1:

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

  • A βˆ’ 1 πœ‹
  • B0
  • C πœ‹
  • D βˆ’ πœ‹
  • E 1 πœ‹

Q2:

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

  • A βˆ’ 1 2 πœ‹
  • B0
  • C 2 πœ‹
  • D βˆ’ 2 πœ‹
  • E 1 2 πœ‹

Q3:

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

Q4:

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

Q5:

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

Q6:

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

Q7:

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.

Q8:

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

  • A4
  • B0
  • Cβˆ’1
  • DThe slope is undefined at ( 3 , πœ‹ ) .
  • E1

Q9:

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

Q10:

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

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

Q11:

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  .

Q12:

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  .

Q13:

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.

Q14:

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 √ 3 5
  • B 5 √ 3 1 6
  • C0
  • D 5 √ 3 3
  • E 3 √ 3 5

Q15:

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

  • A2
  • B1
  • C βˆ’ 1 2
  • D 1 2
  • E βˆ’ 1

Q16:

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

Q17:

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

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

Q18:

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‘₯ = πœƒ + πœƒ πœƒ βˆ’ πœƒ
  • B 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
  • C 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
  • D 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
  • E 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

Q19:

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

  • A βˆ’ πœ‹ 2
  • B 2 πœ‹
  • C1
  • D βˆ’ 2 πœ‹
  • EThe slope of the tangent line is undefined.

Q20:

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

Q21:

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

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

Q22:

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

Q23:

Find the slope of the tangent line to the curve π‘Ÿ = ο€½ πœƒ 3  c o s at πœƒ = 3 πœ‹ 4 .

  • A 1 2
  • B 2 9
  • C0
  • D2
  • E βˆ’ 1

Q24:

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

Q25:

Find the slope of the tangent line to the polar curve π‘Ÿ = 2 πœƒ c o s at the point πœƒ = 5 πœ‹ 6 .

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

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