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Lecture Comments (6)

5 answers

Last reply by: satwinder kaur
Mon Jan 4, 2016 5:19 PM

Post by adam matthews on December 10, 2012

i got dy/dx is -4/sqroot 2 or -2.414, also that is what my graphing calc says{??

Polar Differentiation

  • Solving for slope:
    • Convert polar equation to parametric equations
    • Differentiate the 2 parametric parts separately
    • Divide

Polar Differentiation

Given r = 2sinθ, what is it's parametric form?
  • Apply definitions
  • x = rcosθ
  • = 2sinθcosθ
  • y = rsinθ
  • = 2sin2θ
(2sinθcosθ,2sin2θ)
Given (3cosθ+ cos2θ, − sinθcosθ), what is the slope at θ = [(7π)/8]
  • Find [dy/dx] by finding [dx/(dθ)] and [dy/(dθ)]
  • x = 3cosθ+ cos2θ
  • [dx/(dθ)] = − 3sinθ− 2cosθsinθ
  • y = − sinθcosθ
  • [dy/(dθ)] = sin2θ− cos2θ
  • Divide [dy/(dθ)] by [dx/(dθ)]
  • [dy/dx] = [(sin2θ− cos2θ)/( − 3sinθ− 2cosθsinθ)]
  • Solve for θ = [(7π)/8]
  • [(sin2θ− cos2θ)/( − 3sinθ− 2cosθsinθ)] = [(sin2[(7π)/8] − cos2[(7π)/8])/( − 3sin[(7π)/8] − 2cos[(7π)/8]sin[(7π)/8])]
= 0.81
If r = sinθ, what is the y parametric form, and [dy/(dθ)]?
  • Apply definition
  • y = rsinθ
  • = sin2θ
Find the derivative with Power - Reducing identity
sin2θ = [(1 − cos(2θ))/2] = [1/2] − [(cos(2θ))/2][dy/(dθ)] = sin2θ
If r = sinθ, what is the x parametric form, and [dx/(dθ)]?
  • Apply definition
  • x = rcosθ
  • = sinθcosθ
Find the derivative with Product Rule
[dx/(dθ)] = sinθ( − sinθ) + cosθ(cosθ)
If r = sinθ, what is its slope at θ = [( − 5π)/3]
  • Divide [dy/(dθ)] by [dx/(dθ)] to find [dy/dx]
  • [dy/dx] = [(sin2θ)/( − sin2θ+ cos2θ)]
  • Solve for θ = [( − 5π)/3]
  • [(sin2θ)/( − sin2θ+ cos2θ)] = [(sin2([( − 5π)/3]))/( − sin2([( − 5π)/3]) + cos2([( − 5π)/3]))]
= − √3
If r = √3 (1 − cos θ), what is the y parametric form, and [dx/(dθ)]?
  • Apply definition
  • x = rcosθ
  • = √3 (1 − cosθ)(cosθ)
  • = √3 cosθ− cos2θ
Find the derivative with Product Rule
√3 cosθ− cos2θ = √3 cosθ− [(1 + cos(2θ))/2] = √3 cosθ− [1/2] − [(cos(2θ))/2][dx/(dθ)] = − √3 sinθ+ sin2θ
If r = √3 (1 − cosθ), what is the y parametric form, and [dy/(dθ)]?
  • Apply definition
  • y = rsinθ
  • = √3 (1 − cosθ)(sinθ)
  • = √3 sinθ− sinθcosθ
Find the derivative
[dy/(dθ)] = √3 cosθ− cos(2θ)
If r = sinθ, what is its slope at θ = [7p/4]
  • Divide [dy/(dθ)] by [dx/(dθ)] to find [dy/dx]
  • [dy/dx] = [(√3 cosθ− cos(2θ))/( − √3 sinθ+ sin2θ)]
  • Solve for θ = [7p/4]
  • [(√3 cosθ− cos(2θ))/( − √3 sinθ+ sin2θ)] = [(√3 cos( [7p/4] ) − cos((2)( [7p/4] )))/( − √3 sin( [7p/4] ) + sin((2)( [7p/4] )))]
  • = [(√6 )/(√6 − 2)]
= 5.45
If r = eθ cos(θ), what is [dy/dx]?
  • Find f′(θ)
  • f(θ) = eθ cos(θ)
  • f′(θ) = − eθ sinθ+ eθ cosθ
  • = eθ (cosθ− sinθ)
  • Apply definitions
  • x = rcosθ
  • = ecosθcosθ
  • = ecos2θ
  • y = rsinθ
  • = ecosθsinθ
  • Find [dy/(dθ)]
  • [dy/(dθ)] = f(θ)( − sinθ) + f′(θ)(cosθ)
  • = − eθ cosθsinθ+ ( − eθ sinθ+ eθ cosθ)(cosθ)
  • = − eθ cosθsinθ− eθ sinθcosθ+ eθ cos2θ
  • Find [dx/(dθ)]
  • [dx/(dθ)] = f(θ)(cosθ) + f′(θ)( − sinθ)
  • = eθ cosθsinθ(cosθ) − ( − eθ sinθ+ eθ cosθ)(sinθ)
  • = eθ cos2θsinθ+ eθ sin2θ+ eθ cosθsinθ
  • Divide [dy/(dθ)] by [dx/(dθ)] to find [dy/dx]
  • [dy/dx] = [( − eθ cosθsinθ− eθ sinθcosθ+ eθ cos2θ)/(eθ cos2θsinθ+ eθ sin2θ+ eθ cosθsinθ)]
= [( − cosθsinθ− sinθcosθ+ cos2θ)/(cos2θsinθ+ sin2θ+ cosθsinθ)]
If r = eθ cos(θ), what is its slope at θ = [(π)/2]?
  • Divide [dy/(dθ)] by [dx/(dθ)] to find [dy/dx]
  • [( − cosθsinθ− sinθcosθ+ cos2θ)/(cos2θsinθ+ sin2θ+ cosθsinθ)] = [( − cos[(π)/2]sin[(π)/2] − sin[(π)/2]cos[(π)/2] + cos2[(π)/2])/(cos2[(π)/2]sin[(π)/2] + sin2[(π)/2] + cos[(π)/2]sin[(π)/2])]
  • = [( − 0 − 0 + 0)/(0 + 1 + 0)]
= 0

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Polar Differentiation

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Polar Differentiation 0:11
    • Goal
    • Method
  • Example 1 0:52
  • Example 2 4:27
  • Example 3 7:03