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gina kucinski

gina kucinski

Description

School: Rensselaer Polytechnic Institute
Department: Engineering
Course: Calculus II
Professor: Gina kucinski
Term: Spring 2016
Tags: Math, Calculus, and Calc 2
Cost: Free
Name: Calc II Notes - Week of 2/21
Description: These are the notes for sections 11.1 and 11.2
Uploaded: 03/01/2016
4 Pages 226 Views 0 Unlocks
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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)

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