Worksheet: Slope of a Polar Curve

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In this worksheet, we will practice finding the derivatives of polar curves and the slope of a polar curve.

Q1:

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

  • A−1𝜋
  • B−𝜋
  • C0
  • D1𝜋
  • E𝜋

Q2:

Find the slope of the tangent line to the polar curve 𝑟=2𝜃cos at the point 𝜃=𝜋6.

  • A7√33
  • B0
  • C7√316
  • D√37

Q3:

Find the slope of the tangent line to the curve 𝑟=𝜃3cos at 𝜃=𝜋2.

  • A3√3
  • B√33
  • C√3
  • D√39
  • E√312

Q4:

Find the slope of the tangent line to the curve 𝑟=𝜃cos at 𝜃=𝜋6.

  • A−√33
  • B−√34
  • C−√3
  • D√33
  • E√3

Q5:

Find the slope of the tangent line to the curve 𝑟=2𝜃sin at 𝜃=𝜋6.

  • A√35
  • B0
  • C3√35
  • D5√316
  • E5√33

Q6:

Find the slope of the tangent line to the curve 𝑟=2−3𝜃sin at 𝜃=5𝜋4.

  • A−√2
  • B−√2+2√2−1
  • C−2−√2
  • D−√2−1√2+2
  • E−√21+√2

Q7:

Find the slope of the tangent line to the polar curve 𝑟=1+𝜃cos at the point 𝜃=𝜋4.

  • A−√2+1
  • B−√2−1
  • C−1−√22
  • D−2+√2
  • E−√22−12

Q8:

Find the slope of the tangent line to the curve 𝑟=1+𝜃sin at 𝜃=𝜋4.

  • A−1+√2
  • B−√2+1
  • C1+√2
  • D−√22−12
  • E−√2−1

Q9:

Find the slope of the tangent line to 𝑟=2+4𝜃cos at 𝜃=𝜋6. Round your answer to 3 decimal places.

Q10:

Find the slope of a tangent line to 𝑟=6+3𝜃cos at (3,𝜋).

  • A4
  • B0
  • C−1
  • D1
  • EThe slope is undefined at (3,𝜋).

Q11:

Find the slope of a tangent line to 𝑟=4𝜃cos at (2,𝜋3).

  • A3+2√33
  • B−2√3
  • C2√33
  • D√33
  • E−2−√3

Q12:

Find the slope of a tangent line to 𝑟=1−𝜃sin at 12,𝜋6.

Q13:

Find the slope of the tangent line to 𝑟=4+𝜃sin at the point 3,3𝜋2.

Q14:

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

Q15:

For the cardioid 𝑟=1+𝜃sin, find the slope of the tangent line at 𝜃=𝜋3.

Q16:

Find the points at which 𝑟=4𝜃cos has a horizontal or vertical tangent line.

  • AHorizontal tangents at (2√2,𝜋4) and (2√2,−𝜋4), vertical tangents at (4,0) and (0,𝜋2)
  • BHorizontal tangents at (4,0) and (2√2,𝜋4), vertical tangents at (0,𝜋2) and (2√2,−𝜋4)
  • CNo horizontal tangents, vertical tangents at (4,0) and 0,𝜋2
  • DHorizontal tangents at (4,0), no vertical tangents
  • EHorizontal tangents at (4,0) and (2√2,−𝜋4), vertical tangents at (0,𝜋2) and (2√2,𝜋4)

Q17:

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

  • Addsincoscossin𝑦𝑥=𝑓′(𝜃)𝜃+𝑓(𝜃)𝜃𝑓′(𝜃)𝜃+𝑓(𝜃)𝜃
  • Bddsincoscossin𝑦𝑥=𝑓′(𝜃)𝜃+𝑓(𝜃)𝜃𝑓′(𝜃)𝜃−𝑓(𝜃)𝜃
  • Cddsincoscossin𝑦𝑥=𝜃+𝜃𝜃−𝜃
  • Dddsincoscossin𝑦𝑥=𝑓(𝜃)𝜃+𝑓′(𝜃)𝜃𝑓(𝜃)𝜃−𝑓′(𝜃)𝜃
  • Eddcossinsincos𝑦𝑥=𝑓′(𝜃)𝜃−𝑓(𝜃)𝜃𝑓′(𝜃)𝜃+𝑓(𝜃)𝜃

Q18:

Find the slope of the tangent line to 𝑟=𝜃 at 𝜃=𝜋2.

  • A−2𝜋
  • B1
  • CThe slope of the tangent line is undefined.
  • D−𝜋2
  • E2𝜋

Q19:

Find the slope of the tangent line to 𝑟=8𝜃sin at the point 4,5𝜋6.

  • A−√3
  • B√3
  • C2−√3
  • D0
  • E3−2√33

Q20:

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

  • AThe slope is √3 at 2,𝜋6, −√3 at 2,5𝜋6, and 0 at −2,𝜋2.
  • BThe slope is √33 at 2,𝜋6, −√33 at 2,5𝜋6, and undefined at −2,𝜋2.
  • CThe slope is √3 at 0,𝜋3, −√3 at 0,2𝜋3, and 0 at (0,𝜋).
  • DThe slope is −√3 at 2,𝜋6, √3 at 2,5𝜋6, and 0 at −2,𝜋2.
  • EThe slope is 0 at the tips of all the leaves.

Q21:

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

  • AThe slope is 0 at the tips of all the leaves.
  • BThe slope is 0 at (4,0) and (4,𝜋) and the slope is undefined at −4,𝜋2 and −4,3𝜋2.
  • CThe slope is undefined at (4,0) and (4,𝜋) and the slope is 0 at −4,𝜋2 and −4,3𝜋2.
  • DThe slope is 1 at 0,𝜋4 and 0,5𝜋4 and the slope is −1 at 0,3𝜋4 and 0,7𝜋4.
  • EThe slope is undefined at the tips of all the leaves.

Q22:

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

  • A−12𝜋
  • B−2𝜋
  • C0
  • D12𝜋
  • E2𝜋

Q23:

Find the slope of the tangent line to the polar curve 𝑟=1+𝜃cos at the point 𝜃=3𝜋4.

  • A−√2−1
  • B−√2+1
  • C−√22+1
  • D√2+2
  • E−√22+12

Q24:

Find the slope of the tangent line to the curve 𝑟=1+𝜃sin at 𝜃=𝜋3.

  • A−√3+2
  • B1
  • C−1−√32
  • D−1

Q25:

Find the slope of the tangent line to the curve 𝑟=2𝜃sin at 𝜃=𝜋4.

  • A2
  • B−12
  • C−1
  • D1
  • E12

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