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Lecture Comments (10)

0 answers

Post by Firebird wang on November 2 at 09:52:51 PM

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!

Related Articles:

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 t term into y term

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]
    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) = t2
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.

*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

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 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