๐Ÿ“š

All Subjects

ย >ย 

โ™พ๏ธย 

AP Calc

ย >ย 

๐Ÿฆ–

Unit 9

9.7 Defining Polar Coordinates and Differentiating in Polar Form

1 min readโ€ขjune 8, 2020

sumi-vora

Sumi Vora


https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(565).png?alt=media&token=ed3a3ae2-b1e7-4c15-b409-8f8cfcefc696
๐ŸŽฅWatch: AP Calculus BC - Polar Coordinates and Calculus (for teachers)
Polar functions are functions that are graphed around a pole in a circular system rather than the Cartesian rectangular system. Polar functions are graphed with the points (r, ฮธ) rather than (x, y).
When we are working with polar graphs, we canโ€™t differentiate them right away. We have to convert them to Cartesian graphs. Converting polar equations to Cartesian also helps us visualize them.ย 
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1183).png?alt=media&token=c9db3c65-7ad8-4c6d-baaa-96529a43e114
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1184).png?alt=media&token=ce7ef20a-e693-4935-bbc9-96f345d8af1e
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1185).png?alt=media&token=3ffc2c3a-5eab-4d11-9ded-7e363db131cb
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1186).png?alt=media&token=e01aa053-ebf1-4b21-88fb-99992e20258c
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1187).png?alt=media&token=30f5d5e5-ec17-414d-b308-0bc712a6935c

Resources:

Derivatives of Polar Functions

When we take derivatives of polar functions, we can take them as dr/dฮธ, which would give us the points that are furthest away from the origin on the polar coordinate system. We find dr/dฮธ in the same way we would find any normal derivative: by taking the derivative of the polar function:ย 
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1189).png?alt=media&token=196b8ff6-17ca-4429-942f-cbd47d4bbbe7
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1193).png?alt=media&token=dae0edea-a087-4073-8d09-3e7d9f0c1b87
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1194).png?alt=media&token=00484d2a-df15-4c83-93ca-c7ad75201950
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1195).png?alt=media&token=98c8b289-3a67-4f06-a69e-c9b05d2eb101
While dr/dฮธ can tell us relative maximum and minimum values, it doesnโ€™t tell us the slope of the tangent line, since we canโ€™t have linear graphs on the polar coordinate system. In order to find the slope of the tangent line, we need to find the derivative on the Cartesian system, which requires us to calculate dy/dx.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1197).png?alt=media&token=6d3a14e1-d1c0-447e-8518-d8c5ec03b431
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1199).png?alt=media&token=11a80644-00e0-4f67-881f-e9f62f53e624
Of course, you can memorize this formula, but most students find it much easier to simply derive it using the chain rule.ย 
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1203).png?alt=media&token=9ab5d5a1-8752-4c62-a4f5-64be489ad380
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(1201).png?alt=media&token=838e9f3c-a7ca-4402-8694-7c89494cfd82

Was this guide helpful?

๐Ÿ” Are you ready for college apps?
Take this quiz and find out!
Start Quiz
FREE AP calc Survival Pack + Cram Chart PDF
Sign up now for instant access to 2 amazing downloads to help you get a 5
Browse Study Guides By Unit
๐Ÿ“†
Big Reviews: Finals & Exam Prep
โœ๏ธ
Free Response Questions (FRQ)
๐Ÿง
Multiple Choice Questions (MCQ)
โ™พ
Unit 10: Infinite Sequences and Series (BC Only)
๐Ÿ‘‘
Unit 1: Limits & Continuity
๐Ÿค“
Unit 2: Differentiation: Definition & Fundamental Properties
๐Ÿค™๐Ÿฝ
Unit 3: Differentiation: Composite, Implicit & Inverse Functions
๐Ÿ‘€
Unit 4: Contextual Applications of the Differentiation
โœจ
Unit 5: Analytical Applications of Differentiation
๐Ÿ”ฅ
Unit 6: Integration and Accumulation of Change
๐Ÿ’Ž
Unit 7: Differential Equations
๐Ÿถ
Unit 8: Applications of Integration
Join us on Discord
Thousands of students are studying with us for the AP Calculus AB/BC exam.
join now
๐Ÿ’ช๐Ÿฝ Are you ready for the Calc AB exam?
Take this quiz for a progress check on what youโ€™ve learned this year and get a personalized study plan to grab that 5!
START QUIZ
๐Ÿ’ช๐Ÿฝ Are you ready for the Calc BC exam?
Take this quiz for a progress check on what youโ€™ve learned this year and get a personalized study plan to grab that 5!
START QUIZ
Hours Logo
Studying with Hours = the ultimate focus mode
Start a free study session
๐Ÿ“ฑ Stressed or struggling and need to talk to someone?
Talk to a trained counselor for free. It's 100% anonymous.
Text FIVEABLE to 741741 to get started.
ยฉ 2021 Fiveable, Inc.