Worksheet: Slope of a Polar Curve

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.

  • A1𝜋
  • B𝜋
  • C0
  • D1𝜋
  • E𝜋

Q2:

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

  • A733
  • B0
  • C7316
  • D37

Q3:

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

  • A33
  • B33
  • C3
  • D39
  • E312

Q4:

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

  • A33
  • B34
  • C3
  • D33
  • E3

Q5:

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

  • A35
  • B0
  • C335
  • D5316
  • E533

Q6:

Find the slope of the tangent line to the curve 𝑟=23𝜃sin at 𝜃=5𝜋4.

  • A2
  • B2+221
  • C22
  • D212+2
  • E21+2

Q7:

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

  • A2+1
  • B21
  • C122
  • D2+2
  • E2212

Q8:

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

  • A1+2
  • B2+1
  • C1+2
  • D2212
  • E21

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+233
  • B23
  • C233
  • D33
  • E23

Q12:

Find the slope of a tangent line to 𝑟=1𝜃sin 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 (22,𝜋4) and (22,𝜋4), vertical tangents at (4,0) and (0,𝜋2)
  • BHorizontal tangents at (4,0) and (22,𝜋4), vertical tangents at (0,𝜋2) and (22,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 (22,𝜋4), vertical tangents at (0,𝜋2) and (22,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)𝜃𝑓(𝜃)𝜃
  • Cddsincoscossin𝑦𝑥=𝜃+𝜃𝜃𝜃
  • Dddsincoscossin𝑦𝑥=𝑓(𝜃)𝜃+𝑓(𝜃)𝜃𝑓(𝜃)𝜃𝑓(𝜃)𝜃
  • Eddcossinsincos𝑦𝑥=𝑓(𝜃)𝜃𝑓(𝜃)𝜃𝑓(𝜃)𝜃+𝑓(𝜃)𝜃

Q18:

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

  • A2𝜋
  • B1
  • CThe slope of the tangent line is undefined.
  • D𝜋2
  • E2𝜋

Q19:

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

  • A3
  • B3
  • C23
  • D0
  • E3233

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.

  • A12𝜋
  • B2𝜋
  • C0
  • D12𝜋
  • E2𝜋

Q23:

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

  • A21
  • B2+1
  • C22+1
  • D2+2
  • E22+12

Q24:

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

  • A3+2
  • B1
  • C132
  • D1

Q25:

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

  • A2
  • B12
  • C1
  • D1
  • E12

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