.lst-kix_ww5pi6gqxqho-3 > li{counter-increment:lst-ctn-kix_ww5pi6gqxqho-3}.lst-kix_ww5pi6gqxqho-6 > li:before{content:"" counter(lst-ctn-kix_ww5pi6gqxqho-6,lower-latin) ". "}ol.lst-kix_fun4t81ahut5-0.start{counter-reset:lst-ctn-kix_fun4t81ahut5-0 0}ol.lst-kix_ww5pi6gqxqho-6.start{counter-reset:lst-ctn-kix_ww5pi6gqxqho-6 0}.lst-kix_ww5pi6gqxqho-7 > li:before{content:"" counter(lst-ctn-kix_ww5pi6gqxqho-7,lower-roman) ". "}.lst-kix_ww5pi6gqxqho-8 > li:before{content:"" counter(lst-ctn-kix_ww5pi6gqxqho-8,decimal) ". "}.lst-kix_ww5pi6gqxqho-2 > li{counter-increment:lst-ctn-kix_ww5pi6gqxqho-2}ol.lst-kix_fun4t81ahut5-7.start{counter-reset:lst-ctn-kix_fun4t81ahut5-7 0}ol.lst-kix_ww5pi6gqxqho-3.start{counter-reset:lst-ctn-kix_ww5pi6gqxqho-3 0}.lst-kix_fun4t81ahut5-4 > li{counter-increment:lst-ctn-kix_fun4t81ahut5-4}.lst-kix_fun4t81ahut5-0 > li{counter-increment:lst-ctn-kix_fun4t81ahut5-0}ol.lst-kix_ww5pi6gqxqho-5.start{counter-reset:lst-ctn-kix_ww5pi6gqxqho-5 0}ol.lst-kix_fun4t81ahut5-4.start{counter-reset:lst-ctn-kix_fun4t81ahut5-4 0}.lst-kix_fun4t81ahut5-8 > li:before{content:"" counter(lst-ctn-kix_fun4t81ahut5-8,lower-roman) ". "}ol.lst-kix_fun4t81ahut5-8.start{counter-reset:lst-ctn-kix_fun4t81ahut5-8 0}.lst-kix_fun4t81ahut5-3 > li{counter-increment:lst-ctn-kix_fun4t81ahut5-3}ol.lst-kix_ww5pi6gqxqho-2.start{counter-reset:lst-ctn-kix_ww5pi6gqxqho-2 0}ol.lst-kix_fun4t81ahut5-1.start{counter-reset:lst-ctn-kix_fun4t81ahut5-1 0}.lst-kix_fun4t81ahut5-6 > li{counter-increment:lst-ctn-kix_fun4t81ahut5-6}.lst-kix_ww5pi6gqxqho-1 > li{counter-increment:lst-ctn-kix_ww5pi6gqxqho-1}.lst-kix_ww5pi6gqxqho-7 > li{counter-increment:lst-ctn-kix_ww5pi6gqxqho-7}.lst-kix_ww5pi6gqxqho-4 > li{counter-increment:lst-ctn-kix_ww5pi6gqxqho-4}ol.lst-kix_ww5pi6gqxqho-0{list-style-type:none}ol.lst-kix_ww5pi6gqxqho-4.start{counter-reset:lst-ctn-kix_ww5pi6gqxqho-4 0}.lst-kix_fun4t81ahut5-1 > li{counter-increment:lst-ctn-kix_fun4t81ahut5-1}ol.lst-kix_fun4t81ahut5-5.start{counter-reset:lst-ctn-kix_fun4t81ahut5-5 0}ol.lst-kix_ww5pi6gqxqho-4{list-style-type:none}.lst-kix_fun4t81ahut5-1 > li:before{content:"" counter(lst-ctn-kix_fun4t81ahut5-1,lower-latin) ". "}ol.lst-kix_fun4t81ahut5-8{list-style-type:none}ol.lst-kix_ww5pi6gqxqho-3{list-style-type:none}ol.lst-kix_ww5pi6gqxqho-2{list-style-type:none}.lst-kix_fun4t81ahut5-2 > li:before{content:"" counter(lst-ctn-kix_fun4t81ahut5-2,lower-roman) ". "}ol.lst-kix_ww5pi6gqxqho-1{list-style-type:none}.lst-kix_ww5pi6gqxqho-6 > li{counter-increment:lst-ctn-kix_ww5pi6gqxqho-6}ol.lst-kix_fun4t81ahut5-4{list-style-type:none}ol.lst-kix_fun4t81ahut5-5{list-style-type:none}ol.lst-kix_fun4t81ahut5-6{list-style-type:none}ol.lst-kix_fun4t81ahut5-7{list-style-type:none}ol.lst-kix_fun4t81ahut5-0{list-style-type:none}ol.lst-kix_fun4t81ahut5-1{list-style-type:none}.lst-kix_ww5pi6gqxqho-0 > li{counter-increment:lst-ctn-kix_ww5pi6gqxqho-0}.lst-kix_fun4t81ahut5-0 > li:before{content:"" counter(lst-ctn-kix_fun4t81ahut5-0,upper-latin) ". "}ol.lst-kix_fun4t81ahut5-2{list-style-type:none}ol.lst-kix_fun4t81ahut5-3{list-style-type:none}ol.lst-kix_ww5pi6gqxqho-1.start{counter-reset:lst-ctn-kix_ww5pi6gqxqho-1 0}.lst-kix_fun4t81ahut5-7 > li:before{content:"" counter(lst-ctn-kix_fun4t81ahut5-7,lower-latin) ". "}.lst-kix_ww5pi6gqxqho-5 > li{counter-increment:lst-ctn-kix_ww5pi6gqxqho-5}.lst-kix_fun4t81ahut5-5 > li:before{content:"" counter(lst-ctn-kix_fun4t81ahut5-5,lower-roman) ". "}ol.lst-kix_ww5pi6gqxqho-0.start{counter-reset:lst-ctn-kix_ww5pi6gqxqho-0 0}.lst-kix_fun4t81ahut5-6 > li:before{content:"" counter(lst-ctn-kix_fun4t81ahut5-6,decimal) ". "}ol.lst-kix_ww5pi6gqxqho-8{list-style-type:none}.lst-kix_fun4t81ahut5-3 > li:before{content:"" counter(lst-ctn-kix_fun4t81ahut5-3,decimal) ". "}ol.lst-kix_ww5pi6gqxqho-7{list-style-type:none}ol.lst-kix_ww5pi6gqxqho-6{list-style-type:none}ol.lst-kix_ww5pi6gqxqho-5{list-style-type:none}.lst-kix_fun4t81ahut5-7 > li{counter-increment:lst-ctn-kix_fun4t81ahut5-7}ol.lst-kix_ww5pi6gqxqho-8.start{counter-reset:lst-ctn-kix_ww5pi6gqxqho-8 0}ol.lst-kix_fun4t81ahut5-2.start{counter-reset:lst-ctn-kix_fun4t81ahut5-2 0}.lst-kix_fun4t81ahut5-4 > li:before{content:"" counter(lst-ctn-kix_fun4t81ahut5-4,lower-latin) ". "}.lst-kix_fun4t81ahut5-5 > li{counter-increment:lst-ctn-kix_fun4t81ahut5-5}.lst-kix_ww5pi6gqxqho-0 > li:before{content:"" counter(lst-ctn-kix_ww5pi6gqxqho-0,lower-latin) ". "}.lst-kix_fun4t81ahut5-8 > li{counter-increment:lst-ctn-kix_fun4t81ahut5-8}.lst-kix_ww5pi6gqxqho-2 > li:before{content:"" counter(lst-ctn-kix_ww5pi6gqxqho-2,decimal) ". "}.lst-kix_ww5pi6gqxqho-8 > li{counter-increment:lst-ctn-kix_ww5pi6gqxqho-8}.lst-kix_ww5pi6gqxqho-1 > li:before{content:"" counter(lst-ctn-kix_ww5pi6gqxqho-1,lower-roman) ". "}.lst-kix_ww5pi6gqxqho-3 > li:before{content:"" counter(lst-ctn-kix_ww5pi6gqxqho-3,lower-latin) ". "}ol.lst-kix_ww5pi6gqxqho-7.start{counter-reset:lst-ctn-kix_ww5pi6gqxqho-7 0}.lst-kix_fun4t81ahut5-2 > li{counter-increment:lst-ctn-kix_fun4t81ahut5-2}ol.lst-kix_fun4t81ahut5-3.start{counter-reset:lst-ctn-kix_fun4t81ahut5-3 0}.lst-kix_ww5pi6gqxqho-5 > li:before{content:"" counter(lst-ctn-kix_ww5pi6gqxqho-5,decimal) ". "}ol.lst-kix_fun4t81ahut5-6.start{counter-reset:lst-ctn-kix_fun4t81ahut5-6 0}.lst-kix_ww5pi6gqxqho-4 > li:before{content:"" counter(lst-ctn-kix_ww5pi6gqxqho-4,lower-roman) ". "}

11.1 Parametric EquationsWe also discuss several other topics like What is the only nuclear meltdown to have caused illness and death?

Let C be a curve in the plane. We can describe a particle’s motion along C by giving its coordinates as functions of time tIf you want to learn more check out How do you determine a more stable resonance?

        X = f(t), y = g(t)

These equations are called parametric equations and t is called a parameter

As t varies, the point (x,y) = (f(t), g(+)) varies and traces the parametric curve C

We also discuss several other topics like Give an example of a cdf.

Example: Sketch and identify the curve given by x = t + 1, y = t2 - 2t, - < t< We also discuss several other topics like Define the use of augmented matrix.

        X = t + 1 → t = x - 1Don't forget about the age old question of What is the persistence of learning over time through the storage and retrieval of information?

        Y = (x - 1)2 - 2(x - 1) = x2 - 2x + 1 - 2x + 2 = x2 - 4x + 3If you want to learn more check out How should human rights be viewed?

T

X

Y

-1

0

3

0

1

0

1

2

-1

2

3

0

Example: Sketch curve given by x = cost, y = sin t, 0 < + < 2yr

        Eliminate parameter: cos2t + sin2t = 1 → x2 + y2 = 1 (UC)

T

X

Y

0

1

0

/2

0

1

-1

0

3/2

0

-1

2

1

0

Example: Sketch curve given by x = sin(2t), y = cos(2t), 0 < t < 2

        sin2(2t) + cos2(2t) = 1 → x2 + y2 = 1 (UC)

T

X

Y

0

0

1

/1

1

0

/2

0

-1

3/4

-1

0

0

1

If form(x - h)2 + (y - k)2 = R2 → (x - h)2 = R2 cos2t, (y - k)2 = R2 sin2t

        → x = h + Rcost, y = k + Rsint, 0 < t < 2

If form (a,b) through sope m → x = a + t, y = btmt, t = x -a, -s + s

If form P(a,b) and Q(c, d) → x = a + t(c -a), y = b + t(d - b), 0 < t < 1

If form segment joining from P(a,b) to Q(c,d) → x = a + t(c - a), y = b + t(d - b)

=: this is the slope of the target line to a parametric curve

Example: Let C be given by x = t2, y = t3 - 3t

1. Find an equation of tangent line to C when t = 2
2. Find (x,y) where tangent line is horizontal

==== slope of tangent line

When t = 2, x = 22 = 4, y = 23 - 3(2) = 8 - 6 = 2

                        (4,2)

Y -2 = (x - 4)

Tangent line is horizontal when  = 0, therefore = 0 but 0

3t2 - 3 = 0 → 3(t2 - 1) = 0 → 3(t + 1)(t - 1) = 0 → t = -1, 1

When t = -1 → x = -12 = 1, y = -13 + 3 = 2 → (1,2)

When t = 1, → x = 12 = 1, y = 13 - 3(1) = -2 → (1,-2)

11.2 Arc Length and Speed

→ S =

Example: x = t2, y = t3, 0 < t < 2

S = dt = dt

= tdt = u1/2du = u3/2 = (403/2) - (y3/2)

References: