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Lesson: Derivatives of Polar Functions

Worksheet • 2 Questions

Q1:

Consider the polar equation π‘Ÿ = 4 πœƒ s i n 2 .

Calculate d d 𝑦 π‘₯ for π‘Ÿ = 4 πœƒ s i n 2 .

  • A d d s i n c o s c o s s i n 𝑦 π‘₯ = 3 πœƒ πœƒ 2 πœƒ βˆ’ πœƒ 2 2
  • B 8 πœƒ πœƒ s i n c o s
  • C d d c o s s i n s i n 𝑦 π‘₯ = 2 πœƒ βˆ’ πœƒ 3 πœƒ 2
  • D d d c o s s i n s i n c o s 𝑦 π‘₯ = 2 πœƒ βˆ’ πœƒ 3 πœƒ πœƒ 2 2
  • E d d s i n c o s s i n 𝑦 π‘₯ = 3 πœƒ 2 πœƒ βˆ’ πœƒ 2

Find the slope of the tangent to π‘Ÿ = 4 πœƒ s i n 2 when πœƒ = πœ‹ 8 . Give your answer accurate to three significant figures.

Q2:

Consider the polar equation π‘Ÿ = 2 πœƒ s i n . We can calculate the derivative d d 𝑦 π‘₯ by dividing the derivative d d 𝑦 πœƒ by the derivative d d π‘₯ πœƒ .

To calculate the derivative d d 𝑦 πœƒ , we first need to introduce the variable 𝑦 by multiplying both sides of the equation by s i n πœƒ and then substituting. Write this equation 𝑦 in terms of πœƒ .

  • A 𝑦 = 2 πœƒ s i n 
  • B 𝑦 = 2 2 πœƒ s i n
  • C 𝑦 = 2 πœƒ s i n
  • D 𝑦 = 4 πœƒ s i n 
  • E 𝑦 = 2 πœƒ s i n 

Calculate the derivative d d 𝑦 πœƒ .

  • A d d s i n c o s 𝑦 πœƒ = 4 πœƒ πœƒ
  • B d d c o s 𝑦 πœƒ = 4 2 πœƒ
  • C d d s i n c o s 𝑦 πœƒ = βˆ’ 4 πœƒ πœƒ
  • D d d s i n c o s 𝑦 πœƒ = 8 πœƒ πœƒ
  • E d d s i n 𝑦 πœƒ = 4 πœƒ

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

  • A π‘₯ = 2 πœƒ πœƒ s i n c o s
  • B π‘₯ = βˆ’ 𝑦 πœƒ c o t
  • C π‘₯ = 2 πœƒ c o s
  • D π‘₯ = 𝑦 πœƒ c o s
  • E π‘₯ = 2 πœƒ s i n

Calculate the derivative d d π‘₯ πœƒ .

  • A d d c o s π‘₯ πœƒ = 2 2 πœƒ
  • B π‘₯ = 2 πœƒ c o s
  • C d d c o s π‘₯ πœƒ = 2 πœƒ
  • D d d c o s s i n π‘₯ πœƒ = ο€Ί πœƒ + πœƒ   
  • E d d c o s s i n π‘₯ πœƒ = 2 ο€Ί πœƒ + πœƒ   

The derivative d d 𝑦 π‘₯ is equal to d d d d     . Calculate d d 𝑦 π‘₯ .

  • A d d s i n c o s c o s 𝑦 π‘₯ = 4 πœƒ πœƒ 2 2 πœƒ
  • B d d s i n c o s c o s 𝑦 π‘₯ = βˆ’ 4 πœƒ πœƒ 2 πœƒ
  • C d d s i n c o s c o s 𝑦 π‘₯ = 4 πœƒ πœƒ 2 πœƒ
  • D d d s i n c o s c o s s i n 𝑦 π‘₯ = 4 πœƒ πœƒ 2 ο€Ί πœƒ + πœƒ   
  • E d d s i n c o s c o s 𝑦 π‘₯ = βˆ’ 4 πœƒ πœƒ 2 2 πœƒ

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

  • A √ 3
  • B βˆ’ 2 √ 3
  • C 2 √ 3
  • D √ 3 3
  • E βˆ’ √ 3
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