Worksheet: Slope of a Polar Curve

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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 𝑟 = 1 𝜃 at 𝜃 = 𝜋 .

  • A − 1 𝜋
  • B0
  • C 𝜋
  • D − 𝜋
  • E 1 𝜋

Q2:

Find the slope of the tangent line to the polar curve 𝑟 = 2 𝜃 c o s at the point 𝜃 = 𝜋 6 .

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

Q3:

Find the slope of the tangent line to the curve 𝑟 = 𝜃 3 c o s at 𝜃 = 𝜋 2 .

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

Q4:

Find the slope of the tangent line to the curve 𝑟 = 𝜃 c o s at 𝜃 = 𝜋 6 .

  • A − √ 3
  • B √ 3 3
  • C √ 3
  • D − √ 3 3
  • E − √ 3 4

Q5:

Find the slope of the tangent line to the curve 𝑟 = 2 𝜃 s i n at 𝜃 = 𝜋 6 .

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

Q6:

Find the slope of the tangent line to the curve 𝑟 = 2 − 3 𝜃 s i n at 𝜃 = 5 𝜋 4 .

  • A − √ 2 + 2 √ 2 − 1
  • B − √ 2
  • C − √ 2 1 + √ 2
  • D − √ 2 − 1 √ 2 + 2
  • E − 2 − √ 2

Q7:

Find the slope of the tangent line to the polar curve 𝑟 = 1 + 𝜃 c o s at the point 𝜃 = 𝜋 4 .

  • A − √ 2 − 1
  • B − √ 2 2 − 1 2
  • C − 2 + √ 2
  • D − √ 2 + 1
  • E − 1 − √ 2 2

Q8:

Find the slope of the tangent line to the curve 𝑟 = 1 + 𝜃 s i n at 𝜃 = 𝜋 4 .

  • A − √ 2 + 1
  • B − √ 2 2 − 1 2
  • C − 1 + √ 2
  • D − √ 2 − 1
  • E 1 + √ 2

Q9:

Find the slope of the tangent line to 𝑟 = 2 + 4 𝜃 c o s at 𝜃 = 𝜋 6 . Round your answer to 3 decimal places.

Q10:

Find the slope of a tangent line to 𝑟 = 6 + 3 𝜃 c o s at ( 3 , 𝜋 ) .

  • A4
  • B0
  • C−1
  • DThe slope is undefined at ( 3 , 𝜋 ) .
  • E1

Q11:

Find the slope of a tangent line to 𝑟 = 4 𝜃 c o s at ( 2 , 𝜋 3 ) .

  • A − 2 − √ 3
  • B 2 √ 3 3
  • C 3 + 2 √ 3 3
  • D √ 3 3
  • E − 2 √ 3

Q12:

Find the slope of a tangent line to 𝑟 = 1 − 𝜃 s i n at 1 2 , 𝜋 6 .

Q13:

Find the slope of the tangent line to 𝑟 = 4 + 𝜃 s i n at the point 3 , 3 𝜋 2 .

Q14:

Find the slope of the tangent line to 𝑟 = 𝜃 l n at 𝜃 = 𝑒 . Give your answer to 3 decimal places.

Q15:

For the cardioid 𝑟 = 1 + 𝜃 s i n , find the slope of the tangent line at 𝜃 = 𝜋 3 .

Q16:

Find the points at which 𝑟 = 4 𝜃 c o s has a horizontal or vertical tangent line.

  • ANo horizontal tangents, vertical tangents at ( 4 , 0 ) and 0 , 𝜋 2
  • BHorizontal tangents at ( 4 , 0 ) and ( 2 √ 2 , 𝜋 4 ) , vertical tangents at ( 0 , 𝜋 2 ) and ( 2 √ 2 , − 𝜋 4 )
  • CHorizontal tangents at ( 4 , 0 ) and ( 2 √ 2 , − 𝜋 4 ) , vertical tangents at ( 0 , 𝜋 2 ) and ( 2 √ 2 , 𝜋 4 )
  • DHorizontal tangents at ( 2 √ 2 , 𝜋 4 ) and ( 2 √ 2 , − 𝜋 4 ) , vertical tangents at ( 4 , 0 ) and ( 0 , 𝜋 2 )
  • EHorizontal tangents at ( 4 , 0 ) , no vertical tangents

Q17:

Given a polar curve defined by 𝑟 = 𝑓 ( 𝜃 ) , form an expression for the slope of the curve d d 𝑦 𝑥 in terms of 𝜃 and 𝑓 .

  • A d d s i n c o s c o s s i n 𝑦 𝑥 = 𝜃 + 𝜃 𝜃 − 𝜃
  • B d d s i n c o s c o s s i n 𝑦 𝑥 = 𝑓 ′ ( 𝜃 ) 𝜃 + 𝑓 ( 𝜃 ) 𝜃 𝑓 ′ ( 𝜃 ) 𝜃 + 𝑓 ( 𝜃 ) 𝜃
  • C d d c o s s i n s i n c o s 𝑦 𝑥 = 𝑓 ′ ( 𝜃 ) 𝜃 − 𝑓 ( 𝜃 ) 𝜃 𝑓 ′ ( 𝜃 ) 𝜃 + 𝑓 ( 𝜃 ) 𝜃
  • D d d s i n c o s c o s s i n 𝑦 𝑥 = 𝑓 ′ ( 𝜃 ) 𝜃 + 𝑓 ( 𝜃 ) 𝜃 𝑓 ′ ( 𝜃 ) 𝜃 − 𝑓 ( 𝜃 ) 𝜃
  • E d d s i n c o s c o s s i n 𝑦 𝑥 = 𝑓 ( 𝜃 ) 𝜃 + 𝑓 ′ ( 𝜃 ) 𝜃 𝑓 ( 𝜃 ) 𝜃 − 𝑓 ′ ( 𝜃 ) 𝜃

Q18:

Find the slope of the tangent line to 𝑟 = 𝜃 at 𝜃 = 𝜋 2 .

  • A − 𝜋 2
  • B 2 𝜋
  • C1
  • D − 2 𝜋
  • EThe slope of the tangent line is undefined.

Q19:

Find the slope of the tangent line to 𝑟 = 8 𝜃 s i n at the point 4 , 5 𝜋 6 .

  • A √ 3
  • B0
  • C 2 − √ 3
  • D − √ 3
  • E 3 − 2 √ 3 3

Q20:

Find the slopes of the tangent lines to 𝑟 = 2 ( 3 𝜃 ) s i n at the tips of the leaves.

  • AThe slope is √ 3 3 at 2 , 𝜋 6 , − √ 3 3 at 2 , 5 𝜋 6 , and undefined at − 2 , 𝜋 2 .
  • BThe slope is 0 at the tips of all the leaves.
  • CThe slope is √ 3 at 0 , 𝜋 3 , − √ 3 at 0 , 2 𝜋 3 , and 0 at ( 0 , 𝜋 ) .
  • DThe slope is − √ 3 at 2 , 𝜋 6 , √ 3 at 2 , 5 𝜋 6 , and 0 at − 2 , 𝜋 2 .
  • EThe slope is √ 3 at 2 , 𝜋 6 , − √ 3 at 2 , 5 𝜋 6 , and 0 at − 2 , 𝜋 2 .

Q21:

Find the slopes of the tangent lines to 𝑟 = 4 ( 2 𝜃 ) c o s at the tips of the leaves.

  • AThe slope is 1 at 0 , 𝜋 4 and 0 , 5 𝜋 4 and the slope is − 1 at 0 , 3 𝜋 4 and 0 , 7 𝜋 4 .
  • BThe slope is 0 at ( 4 , 0 ) and ( 4 , 𝜋 ) and the slope is undefined at − 4 , 𝜋 2 and − 4 , 3 𝜋 2 .
  • CThe slope is 0 at the tips of all the leaves.
  • DThe slope is undefined at ( 4 , 0 ) and ( 4 , 𝜋 ) and the slope is 0 at − 4 , 𝜋 2 and − 4 , 3 𝜋 2 .
  • EThe slope is undefined at the tips of all the leaves.

Q22:

Find the slope of the tangent line to the curve 𝑟 = 1 𝜃 at 𝜃 = 2 𝜋 .

  • A − 1 2 𝜋
  • B0
  • C 2 𝜋
  • D − 2 𝜋
  • E 1 2 𝜋

Q23:

Find the slope of the tangent line to the polar curve 𝑟 = 1 + 𝜃 c o s at the point 𝜃 = 3 𝜋 4 .

  • A − √ 2 + 1
  • B − √ 2 2 + 1 2
  • C √ 2 + 2
  • D − √ 2 − 1
  • E − √ 2 2 + 1

Q24:

Find the slope of the tangent line to the curve 𝑟 = 1 + 𝜃 s i n at 𝜃 = 𝜋 3 .

  • A − √ 3 + 2
  • B − 1 − √ 3 2
  • C1
  • D − 1

Q25:

Find the slope of the tangent line to the curve 𝑟 = 2 𝜃 s i n at 𝜃 = 𝜋 4 .

  • A2
  • B1
  • C − 1 2
  • D 1 2
  • E − 1

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