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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!

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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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John Zhu

Parametric Curves

Duration: 10:11 min.

Table of Contents

Section 1: Prerequisites

Parametric Curves

10m 10s

Intro