Lesson Worksheet: Slope of a Polar Curve Mathematics • Higher Education

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 polar curve π‘Ÿ=2πœƒcos at the point πœƒ=πœ‹6.

  • A7√33
  • B0
  • C7√316
  • D√37

Q2:

Find the slope of the tangent line to the polar curve π‘Ÿ=1+πœƒcos at the point πœƒ=πœ‹4.

  • Aβˆ’βˆš2+1
  • Bβˆ’βˆš2βˆ’1
  • Cβˆ’1βˆ’βˆš22
  • Dβˆ’2+√2
  • Eβˆ’βˆš22βˆ’12

Q3:

Consider the polar equation π‘Ÿ=4πœƒsin.

Calculate dd𝑦π‘₯ for π‘Ÿ=4πœƒsin.

  • Addsincoscossin𝑦π‘₯=3πœƒπœƒ2πœƒβˆ’πœƒοŠ¨οŠ¨
  • B8πœƒπœƒsincos
  • Cddcossinsin𝑦π‘₯=2πœƒβˆ’πœƒ3πœƒοŠ¨
  • Dddcossinsincos𝑦π‘₯=2πœƒβˆ’πœƒ3πœƒπœƒοŠ¨οŠ¨
  • Eddsincossin𝑦π‘₯=3πœƒ2πœƒβˆ’πœƒοŠ¨

Find the slope of the tangent to π‘Ÿ=4πœƒsin when πœƒ=πœ‹8. Give your answer accurate to two decimal places.

Q4:

Consider the polar equation π‘Ÿ=2πœƒsin. We can calculate the derivative dd𝑦π‘₯ by dividing the derivative ddπ‘¦πœƒ by the derivative ddπ‘₯πœƒ.

To calculate the derivative ddπ‘¦πœƒ, we first need to introduce the variable 𝑦 by multiplying both sides of the equation by sinπœƒ and then substituting. Write this equation 𝑦 in terms of πœƒ.

  • A𝑦=22πœƒsin
  • B𝑦=2πœƒsin
  • C𝑦=4πœƒsin
  • D𝑦=2πœƒsin
  • E𝑦=2πœƒsin

Calculate the derivative ddπ‘¦πœƒ.

  • Addsincosπ‘¦πœƒ=4πœƒπœƒ
  • Bddsinπ‘¦πœƒ=4πœƒ
  • Cddsincosπ‘¦πœƒ=8πœƒπœƒ
  • Dddcosπ‘¦πœƒ=42πœƒ
  • Eddsincosπ‘¦πœƒ=βˆ’4πœƒπœƒ

Similarly, to calculate the derivative ddπ‘₯πœƒ, we first need to introduce the variable π‘₯ by multiplying both sides of the original equation by cosπœƒ and then substituting. Write this equation π‘₯ in terms of πœƒ.

  • Aπ‘₯=π‘¦πœƒcos
  • Bπ‘₯=2πœƒcos
  • Cπ‘₯=2πœƒsin
  • Dπ‘₯=2πœƒπœƒsincos
  • Eπ‘₯=βˆ’π‘¦πœƒcot

Calculate the derivative ddπ‘₯πœƒ.

  • Addcossinπ‘₯πœƒ=2ο€Ίπœƒ+πœƒο†οŠ¨οŠ¨
  • Bddcossinπ‘₯πœƒ=ο€Ίπœƒ+πœƒο†οŠ¨οŠ¨
  • Cπ‘₯=2πœƒcos
  • Dddcosπ‘₯πœƒ=2πœƒ
  • Eddcosπ‘₯πœƒ=22πœƒ

The derivative dd𝑦π‘₯ is equal to ddddο˜οΌο—οΌ. Calculate dd𝑦π‘₯.

  • Addsincoscossin𝑦π‘₯=4πœƒπœƒ2ο€Ίπœƒ+πœƒο†οŠ¨οŠ¨
  • Bddsincoscos𝑦π‘₯=4πœƒπœƒ2πœƒ
  • Cddsincoscos𝑦π‘₯=4πœƒπœƒ22πœƒ
  • Dddsincoscos𝑦π‘₯=βˆ’4πœƒπœƒ22πœƒ
  • Eddsincoscos𝑦π‘₯=βˆ’4πœƒπœƒ2πœƒ

Use the derivative function to calculate the slope of the tangent to π‘Ÿ=2πœƒsin at πœƒ=πœ‹6.

  • A√33
  • Bβˆ’βˆš3
  • C√3
  • D2√3
  • Eβˆ’2√3

Q5:

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

  • Aβˆ’1πœ‹
  • Bβˆ’πœ‹
  • C0
  • D1πœ‹
  • Eπœ‹

Q6:

Find the slope of the tangent line to the curve π‘Ÿ=1+πœƒsin at πœƒ=πœ‹4.

  • Aβˆ’1+√2
  • Bβˆ’βˆš2+1
  • C1+√2
  • Dβˆ’βˆš22βˆ’12
  • Eβˆ’βˆš2βˆ’1

Q7:

Find the slope of the tangent line to the curve π‘Ÿ=2βˆ’3πœƒsin at πœƒ=5πœ‹4.

  • Aβˆ’βˆš2
  • Bβˆ’βˆš2+2√2βˆ’1
  • Cβˆ’2βˆ’βˆš2
  • Dβˆ’βˆš2βˆ’1√2+2
  • Eβˆ’βˆš21+√2

Q8:

Find the slope of the tangent line to the curve π‘Ÿ=2πœƒsin at πœƒ=πœ‹6.

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

Q9:

Find the slope of the tangent line to the curve π‘Ÿ=πœƒcos at πœƒ=πœ‹6.

  • Aβˆ’βˆš33
  • Bβˆ’βˆš34
  • Cβˆ’βˆš3
  • D√33
  • E√3

Q10:

Find the slope of the tangent line to the curve π‘Ÿ=ο€½πœƒ3cos at πœƒ=πœ‹2.

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

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