John Zhu

Parametric Curves

Slide Duration:

Table of Contents

Section 1: Prerequisites

Parametric Curves

10m 10s

Intro

0:00

Parametric Equations

0:23

Familiar Functions

0:32

Parametric Equation/ Function

0:56

Example 1: Graph Parametric Equation

1:48

Example 2

4:30

Example 3

6:01

Example 4

7:12

Example 5

8:10

Polar Coordinates

14m 54s

Intro

0:00

Polar Coordinates

0:09

Definition

0:11

Example

0:23

Converting Polar Coordinates

1:49

Example: Convert Polar Equation to Cartesian Coordinates

3:06

Example: Convert Polar Equation to Cartesian Coordinates

5:21

Example: Find the Polar Representation

6:51

Example: Find the Polar Representation

8:39

Example: Sketch the Graph

10:02

Vector Functions

12m 5s

Intro

0:00

Vector Functions

0:11

2 Parts

0:18

Example 1

1:44

Example 2

3:52

Example 3

4:59

Example 4

6:03

Example 5

8:00

Example 6

9:23

Example 7

10:16

Section 2: Differentiation

Parametric Differentiation

13m 15s

Intro

0:00

Parametric Differentiation

0:15

Example 1

1:16

Example 2

1:54

Example 3

3:15

Example 4

4:59

Parametric Differentiation: Position, Speed & Acceleration

7:45

Example 5

8:32

Example 6

10:37

Polar Differentiation

10m 14s

Intro

0:00

Polar Differentiation

0:11

Goal

0:13

Method

0:24

Example 1

0:52

Example 2

4:27

Example 3

7:03

L'Hopital's Rule

6m 43s

Intro

0:00

L'Hopitals Rule

0:09

Conditions

0:14

Limit

0:36

Example 1

1:21

Example 2

2:38

Example 3

4:09

Example 4

5:01

Section 3: Integration

Integration By Parts

19m 4s

Intro

0:00

Integration by Parts

0:12

Formula

0:31

How It's Derived From Product Rule

1:09

Integration by Parts: Rules to Follow

3:24

Recap of the Rules

4:13

Example 1

4:29

Example 2

7:04

Example 3

8:40

Example 4

13:48

Example 5

15:36

Integration By Partial Fractions

22m 4s

Intro

0:00

Integration by Partial Fractions: Goal

0:08

Example 1

2:10

Example 2

7:10

Example 3

10:06

Example 4

14:23

Example 5

16:26

Improper Integrals

19m 1s

Intro

0:00

Improper Integrals

0:06

3 Steps

1:11

Example 1

1:46

Example 2

3:20

Example 3

6:05

Example 4

9:02

Example 5

11:21

Example 6

15:54

Section 4: Applications of Integration

Logistic Growth

19m 16s

Intro

0:00

Logistic Growth Function

0:07

Defining Variables

0:40

Equation Parts

1:51

Logistic Growth

2:04

Example 1

2:59

Example 2

7:13

Example 3

11:29

Example 4

15:21

Arc Length for Parametric & Polar Curves

17m 40s

Intro

0:00

Arc Length

0:13

Arc Length of a Normal Function

0:24

Example

1:27

Example 2: Arc Length for Parametric Curves

3:31

Example 3: Arc Length for Parametric Curves

4:23

Example 4: Arc Length for Parametric Curves

8:05

Example 5: Arc Length for Parametric Curves

12:22

Example 6: Arc Length for Polar Curves

15:36

Example 7: Arc Length for Polar Curves

16:03

Area for Parametric & Polar Curves

4m 33s

Intro

0:00

Area for Parametric Curves: Parametric Function

0:10

Example 1: Area for Parametric Curves

0:35

Area for Parametric Curves: Polar Function

2:50

Example 2: Area for Polar Curves

3:18

Section 5: Sequences, Series, & Approximations

Definition & Convergence

7m 10s

Intro

0:00

Sequences: Definition

0:09

Definition

0:31

Example 1

1:07

Sequences: Convergence

2:02

Example 1

2:52

Example 2

3:36

Example 3

4:47

Example 4

6:16

Geometric Series

10m 59s

Intro

0:00

Geometric Series

0:07

Definition

0:24

Expanded Form

0:39

Convergence Rules

1:00

Example 1: Convergence

1:22

Example 2: Convergence

2:36

Example 3: Convergence

3:45

Sum of Series

5:04

Sum of First n Terms

5:14

Sum of Series

5:24

Example 1: Sum of Series

5:39

Example 2: Sum of Series

7:15

Example 3: Sum of Series

8:24

Example 4: Sum of Series

9:36

Harmonic & P Series

3m 58s

Intro

0:00

Harmonic Series

0:08

P-Series

1:17

Example 1: P-Series Test

2:22

Example 2: P-Series Test

3:06

Integral Test

11m 31s

Intro

0:00

Integral Test

0:09

Example 1

0:54

Example 2

4:00

Example 3

7:59

Example 4

9:47

Comparison Test

7m 47s

Intro

0:00

Comparison Test

0:10

Example 1

1:07

Example 2

2:33

Example 3

4:20

Example 4

6:29

Limit Comparison Test

9m 40s

Intro

0:00

Limit Comparison Test

0:09

Conditions

0:31

Example 1

1:01

Example 2

2:53

Example 3

4:15

Example 4

6:19

Ratio Test

11m 20s

Intro

0:00

Ratio Test

0:09

Example 1

0:57

Example 2

2:55

Example 3

6:27

Example 4

8:36

Alternating Series

9m 11s

Intro

0:00

Alternating Series

0:08

Convergence Test

0:59

Example 1

1:27

Example 2

2:42

Example 3

4:57

Example 4

6:37

Absolute Convergence

7m 13s

Intro

0:00

Absolute Convergence

0:12

Example 1

0:52

Example 2

3:42

Example 3

5:21

Power Series Convergence

12m 52s

Intro

0:00

Power Series

0:09

Definition

0:19

Radius & Interval of Convergence

2:07

Example 1

2:28

Example 2

4:18

Example 3

6:20

Example 4

9:11

Taylor Series

15m 11s

Intro

0:00

Taylor Series

0:06

MacLaurin Series

0:45

Example 1

1:02

Example 2

2:45

Example 3

6:04

Example 4

8:23

Example 5

11:49

Power Series Operations

16m 40s

Intro

0:00

Operations

0:07

Example 1: Substitution

1:05

Example 2: Substitution

3:41

Example 3: Differentiation/ Integration

5:39

Example 4: Differentiation/ Integration

9:55

Example 5: Multiplying/ Dividing

12:32

Example 6: Multiplying/ Dividing

14:50

Lagrange Error

7m 9s

Intro

0:00

Power Series: Lagrange Error

0:06

Lagrange Remainder

0:21

Lagrange Error Bound

0:50

Example 1

1:06

Example 2

3:27

Section 6: Practice Test

AP Calc BC Practice Test Section 1: Multi Choice Part 1

15m 45s

Intro

0:00

Practice Problem 1

0:10

Practice Problem 2

1:19

Practice Problem 3

2:33

Practice Problem 4

5:25

Practice Problem 5

6:32

Practice Problem 6

9:11

Practice Problem 7

11:31

Practice Problem 8

13:08

Practice Problem 9

14:16

AP Calc BC Practice Test Section 1: Multi Choice Part 2

21m 38s

Intro

0:00

Practice Problem 10

0:12

Practice Problem 11

3:03

Practice Problem 12

4:53

Practice Problem 13

6:56

Practice Problem 14

9:10

Practice Problem 15

12:17

Practice Problem 16

14:43

Practice Problem 17

16:18

Practice Problem 18

18:34

AP Calc BC Practice Test Section 1: Multi Choice Part 3

17m 31s

Intro

0:00

Practice Problem 19

0:09

Practice Problem 20

3:03

Practice Problem 21

4:07

Practice Problem 22

5:41

Practice Problem 23

8:48

Practice Problem 24

11:16

Practice Problem 25

12:53

Practice Problem 26

14:12

Practice Problem 27

15:21

AP Calc BC Practice Test Section 1: Multi Choice Part 4

14m 17s

Intro

0:00

Practice Problem 28

0:08

Practice Problem 29

1:19

Practice Problem 30

2:15

Practice Problem 31

4:19

Practice Problem 32

6:54

Practice Problem 33

8:13

Practice Problem 34

9:03

Practice Problem 35

10:21

Practice Problem 36

11:42

AP Calc BC Practice Test Section 1: Multi Choice Part 5

22m 45s

Intro

0:00

Practice Problem 37

0:10

Practice Problem 38

4:09

Practice Problem 39

7:21

Practice Problem 40

9:42

Practice Problem 41

11:35

Practice Problem 42

14:01

Practice Problem 43

15:50

Practice Problem 44

16:43

Practice Problem 45

19:35

AP Calc BC Practice Test Section 1: Free Response Part 1

9m 55s

Intro

0:00

Practice Problem 1a

0:08

Practice Problem 1b

1:30

Practice Problem 1c

3:08

Practice Problem 2a

3:50

Practice Problem 2b

5:40

Practice Problem 2c

7:24

AP Calc BC Practice Test Section 1: Free Response Part 2

10m 8s

Intro

0:00

Practice Problem 3a

0:09

Practice Problem 3b

3:01

Practice Problem 4a

3:55

Practice Problem 4b

5:43

AP Calc BC Practice Test Section 1: Free Response Part 3

13m 58s

Intro

0:00

Practice Problem 5a

0:07

Practice Problem 5b

2:21

Practice Problem 5c

4:21

Practice Problem 6a

6:16

Practice Problem 6b

8:57

Practice Problem 6c

11:03

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For more information, please see full course syllabus of Calculus BC

Calculus BC Parametric Curves

Section 1: Prerequisites: Lecture 1 | 10:10 min

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Lecture Comments (11)

	0 answers Post by Mohamed E Sowaileh on July 10, 2017 Hello Dr. John Zhu, I hope you are very well. I am a student who is extremely weak in math. In order to be very strong in math, specially for engineering field, could you provide me with sequential order of mathematical topics and textbooks. With what should I begin so that I can master big topics like calculus, statistics, probability ... etc. Your guidance is precious to me. Thank you so much.
	0 answers Post by Firebird wang on November 2, 2016 Are there any way to watch the two video which called AP Statistics Practice Test 2013 an AP Statistics Practice Test 2014?
	3 answers Last reply by: Firebird wang Wed Oct 12, 2016 10:47 PM Post by Jimmy Jones on September 8, 2015 Hello Professor, I am contacting to know if I can go straight to learning this course instead of learning Calculus AB. Is that made possible through these lessons Thanks
	0 answers Post by Brady Dill on July 13, 2014 Great video! But I was thrown by your use of 'comprise', which means something entirely different. https://www.google.com/search?q=comprise+definition&ie=utf-8&oe=utf-8&aq=t&rls=org.mozilla:en-US:official&client=firefox-a&channel=sb Otherwise, well-explained! I understand now.
	0 answers Post by John Zhu on August 12, 2013 Yes folks, Ln(0) is never defined! That is a drawing error that SHOULD include a vertical asymptote as Sumant has suggested. Thanks for the feedback and notes guys. Cheers! =)
	0 answers Post by Him Tam on July 30, 2013 Yeah, the graph shouldn't intersect the y-axis
	0 answers Post by Sumant Nigam on March 8, 2013 I thought the video was great, but for the last problem: f(x)=ln(x)-1 the graph should not intersect the y-axis, right? There should be a vertical asymptote there.
	0 answers Post by Maimouna Louche on June 15, 2012 Yey I am first! I loved it all!

Answer EngineGet answers to any question!Ask any question related to Calculus BC

Working on the solution...

Navigating the High School Math Maze

Parametric Curves

Parametric equation: takes in 2 dependent variables
Graphing: treat each dependent input variable separately
Converting to Cartesian equation

Solve for t in terms of x
Substitute solved tterm into yterm

Parametric Curves

Graph f(t) = (3t + 6, cos(t)), for − 2 ≤ t ≤ 2

Make a table
t
− 2
− 1
0
1
2
x(t)
− 2
1
4
7
10
y(t)
− 0.42
0.54
1
0.54
− 0.42

Plot the points and graph

Graph f(t) = (sin(x), √x) for 0 ≤ t ≤ 2π

Make a table
t
0
2
π
[(3π)/2]
2π
x(t)
0
1
0
− 1
0
y(t)
0
1.25
1.77
2.17
2.51

Plot the points and graph

A parametric curve is defined by
y(t) = t^²
x(t) = t + 2
What is its Cartesian equation?

Solve for t with x
x = t + 2
t = x − 2

Substitute into y(t)
y(t) = t²
y = (x − 2)²

A parametric curve is defined by:
y(t) = t − 9
x(t) = √t
for t ≥ 0
What is its Cartesian equation?

Solve for t, we start with x
x = √t
t = x²

Substitute into y(t)
y(t) = t − 9
y = x² − 9

Find the Cartesian equation of the following parametric equations:
x(t) = [cos(t)/3]
y(t) = [sin(t)/4]
for t ≥ 0

Isolate x and y
x = [cos(t)/3]
3x = cos(t)
y = [sin(t)/4]
4y = sin(t)

Apply Pythagorean identity
cos²(t) + sin²(t) = 1
(3x)² + (4y)² = 1
9x² + 16y² = 1

Find the Cartesian equation of the following parametric equations:
x(t) = √2 sin(t)
y(t) = 2cos(t) + 3
for t ≥ 0

Isolate x
x = √2 sin(t)
[x/(√2 )] = sin(t)
Isolate y
y = 2cos(t) + 3
y − 3 = 2cos(t)
[(y − 3)/2] = cos(t)

sin²(t) + cos²(t) = 1
([x/(√2 )])² + ([(y − 3)/2])² = 1
[(x²)/2] + [((y − 3)²)/4] = 1

Find the Cartesian equation of the following parametric equations:
x(t) = t + 5
y(t) = t³ + 5

Isolate t with x
x = t + 5
x − 5 = t

y = (x − 5)3 + 125
y = x³ − 15x² + 75x − 125 + 125
y = x³ − 15x² + 75x

Find the Cartesian equation of the following parametric equations:
x(t) = 1 + t
y(t) = t² − 9

Isolate t with x
x = 1 + t
− t = 1 − x
t = x − 1

Substitute into y(t)
y = (x − 1)² − 9
y = x² − 2x + 1 − 9
y = x² − 2x − 8

Find the Cartesian equation of the following parametric equations, and graph it:
x(t) = t² − 3
y(t) = 1 − t
for t − 3

Isolate t with x
x = t² − 3
x + 3 = t²
√{x + 3} = t
Substitute into y(t)
y = 1 − √{x + 3}
Graph the equation using shifts
y = 1 − √{x + 3}

The graph √x has been shifted up by 1 unit to y = 1, and left by 3 units. It also has been reflected.

Find the Cartesian equation of the following parametric equations, and graph it:
x(t) = arccos([t/4]) − [(π)/2]
y(t) = [t/8] − 3 for − 2π ≤ t ≤ 2π

Isolate t with x
x = arccos([t/4]) − [(π)/2]
x + [(π)/2] = arccos([t/4])
cos(x + [(π)/2]) = [t/4]
4cos(x + [(π)/2]) = t
Observe half - angle trig identity
cos(u + [(π)/2]) = − sinu
4cos(x + [(π)/2]) = t
− 4sinx = t
Substitute into y(t)
y = [( − 4sin(x))/8] − 3
y = − [sinx/2] − 3
Graph using shifts, reflections, and stretching
y = − [sinx/2] − 3
The graph y = sinx has been shifted 3 units down to y = − 3, and reflected. It also has been compressed by a factor of 2.
Remember to graph within the bounds − 2π ≤ t ≤ 2π.

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Answer

Parametric Curves

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

Parametric Equations

Example 1: Graph Parametric Equation

Example 2

Example 3

Example 4

Example 5

Intro 0:00
Parametric Equations 0:23

Familiar Functions
Parametric Equation/ Function

Example 1: Graph Parametric Equation 1:48
Example 2 4:30
Example 3 6:01
Example 4 7:12
Example 5 8:10

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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Calculus BC Parametric Curves

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Calculus BC Parametric Curves

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