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.

  • A 𝜋
  • B 1 𝜋
  • C0
  • D 1 𝜋
  • E 𝜋

Q2:

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

  • A 7 3 3
  • B0
  • C 7 3 1 6
  • D 3 7

Q3:

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

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

Q4:

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

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

Q5:

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

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

Q6:

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

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

Q8:

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

  • A 2 2 1 2
  • B 2 + 1
  • C 1 + 2
  • D 1 + 2
  • 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).

  • A1
  • B−1
  • CThe slope is undefined at (3,𝜋).
  • D0
  • E4

Q11:

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

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

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)
  • BNo horizontal tangents, vertical tangents at (4,0) and 0,𝜋2
  • CHorizontal tangents at (4,0) and (22,𝜋4), vertical tangents at (0,𝜋2) and (22,𝜋4)
  • DHorizontal tangents at (4,0) and (22,𝜋4), vertical tangents at (0,𝜋2) and (22,𝜋4)
  • 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 dd𝑦𝑥 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 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
  • 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 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

Q18:

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

  • A 2 𝜋
  • B1
  • C 2 𝜋
  • DThe slope of the tangent line is undefined.
  • E 𝜋 2

Q19:

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

  • A 3
  • B 3
  • C 2 3
  • D0
  • E 3 2 3 3

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

Q23:

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

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

Q24:

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

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

Q25:

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

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

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