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Calculus BC Polar Differentiation

Section 2: Differentiation: Lecture 2 | 10:14 min

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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{??

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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 = e^θ cos(θ), what is [dy/dx]?

Find f′(θ)
f(θ) = e^θ cos(θ)
f′(θ) = − e^θ sinθ+ e^θ cosθ
= e^θ (cosθ− sinθ)
Apply definitions
x = rcosθ
= e^2θcosθcosθ
= e^2θcos²θ
y = rsinθ
= e^2θcosθsinθ
Find [dy/(dθ)]
[dy/(dθ)] = f(θ)( − sinθ) + f′(θ)(cosθ)
= − e^θ cosθsinθ+ ( − e^θ sinθ+ e^θ cosθ)(cosθ)
= − e^θ cosθsinθ− e^θ sinθcosθ+ e^θ cos²θ
Find [dx/(dθ)]
[dx/(dθ)] = f(θ)(cosθ) + f′(θ)( − sinθ)
= e^θ cosθsinθ(cosθ) − ( − e^θ sinθ+ e^θ cosθ)(sinθ)
= e^θ cos²θsinθ+ e^θ sin²θ+ e^θ cosθsinθ
Divide [dy/(dθ)] by [dx/(dθ)] to find [dy/dx]
[dy/dx] = [( − e^θ cosθsinθ− e^θ sinθcosθ+ e^θ cos²θ)/(e^θ cos²θsinθ+ e^θ sin²θ+ e^θ cosθsinθ)]

= [( − cosθsinθ− sinθcosθ+ cos²θ)/(cos²θsinθ+ sin²θ+ cosθsinθ)]

If r = e^θ 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

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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

Polar Differentiation

Example 1

Example 2

Example 3

Intro 0:00
Polar Differentiation 0:11

Goal
Method

Example 1 0:52
Example 2 4:27
Example 3 7:03

Mathematics: AP Calculus BC

Section 1: Prerequisites
	Parametric Curves	10:10
	Polar Coordinates	14:54
	Vector Functions	12:05
Section 2: Differentiation
	Parametric Differentiation	13:15
	Polar Differentiation	10:14
	L'Hopital's Rule	6:43
Section 3: Integration
	Integration By Parts	19:04
	Integration By Partial Fractions	22:04
	Improper Integrals	19:01
Section 4: Applications of Integration
	Logistic Growth	19:16
	Arc Length for Parametric & Polar Curves	17:40
	Area for Parametric & Polar Curves	4:33
Section 5: Sequences, Series, & Approximations
	Definition & Convergence	7:10
	Geometric Series	10:59
	Harmonic & P Series	3:58
	Integral Test	11:31
	Comparison Test	7:47
	Limit Comparison Test	9:40
	Ratio Test	11:20
	Alternating Series	9:11
	Absolute Convergence	7:13
	Power Series Convergence	12:52
	Taylor Series	15:11
	Power Series Operations	16:40
	Lagrange Error	7:09
Section 6: Practice Test
	AP Calc BC Practice Test Section 1: Multi Choice Part 1	15:45
	AP Calc BC Practice Test Section 1: Multi Choice Part 2	21:38
	AP Calc BC Practice Test Section 1: Multi Choice Part 3	17:31
	AP Calc BC Practice Test Section 1: Multi Choice Part 4	14:17
	AP Calc BC Practice Test Section 1: Multi Choice Part 5	22:45
	AP Calc BC Practice Test Section 1: Free Response Part 1	9:55
	AP Calc BC Practice Test Section 1: Free Response Part 2	10:08
	AP Calc BC Practice Test Section 1: Free Response Part 3	13:58

Related Books

	Cracking the AP Calculus BC Exam, 2015 Edition Author: Princeton Review ISBN: 804124825 Publisher: Princeton Review Year: 2014 The book includes 3 full length practice tests with detailed explanations, a review of all the key concepts, and targeted strateeies to ace th exam for your highest score.
	Cliffs AP: Calculus AB & BC, 3rd Edition Authors: Dale W Johnson, Kerry J King ISBN: 764586831 Publisher: Cliffs Notes Year: 2001 The book has a topic-by topic breakdown and lots of problem approach suggestions for both free response and multiple choice calculus questions. The book also includes two full length tests.

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John Zhu

Polar Differentiation

Slide Duration:

Table of Contents

Section 1: Prerequisites

Parametric Curves

10m 10s

Intro