Calculus III: Section 11.1
Professor Ensley
Ship Math
August 29, 2011
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 1 / 17
Path of a particle
Path of a particle
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 2 / 17
Path of a particle
Path of a particle
It is not unusual that motion is easiest described using separate equationsfor the x- and y -coordinates. For example, an object might have initialhorizontal velocity of 200 feet per second and initial vertical veloctiy of 400feet per second. Ignoring wind resistance, simple physical laws tell us that
x(t) = 200t, y(t) = 400t − 16t2
for values of t with 0 ≤ t ≤ 25.
t 0 5 10 15 20 25
x
y
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 3 / 17
Path of a particle
Path of a particle
It is not unusual that motion is easiest described using separate equationsfor the x- and y -coordinates. For example, an object might have initialhorizontal velocity of 200 feet per second and initial vertical veloctiy of 400feet per second. Ignoring wind resistance, simple physical laws tell us that
x(t) = 200t, y(t) = 400t − 16t2
for values of t with 0 ≤ t ≤ 25.
t 0 5 10 15 20 25
x 0 1000 2000 3000 4000 5000
y 0 1600 2400 2400 1600 0
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 4 / 17
Path of a particle
Trajectory of a bullet
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 5 / 17
Functions/equations vs. parametric equations
Functions/equations vs. parametric equations
Insight
If you have a graph of the equation y = f (x), then the same curvecan be described by the parameterization (t, f (t)).
Given the parameterization c(t) = (x(t), y(t)), sometimes theparameter t can be eliminated, and we can combine to get anequation in x and y.
Example. Given x = 200t and y = 400t − 16t2, we can solve the firstequation for t (t = x/200) and substitute into the second equation to gety = 400(x/200)− 16(x/200)2, or y = 2x − x2/2500.
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 6 / 17
Functions/equations vs. parametric equations
Circular motion
We can imagine the motion of a particular car on a Ferris Wheel hascoordinates (70 sin (πt/50), 75− 70 cos (πt/50)) after t seconds, witht ≥ 0.
t 0 10 20 30 40 50 60 70 80 90 100
x
y
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 7 / 17
Functions/equations vs. parametric equations
Circular motion
We can imagine the motion of a particular car on a Ferris Wheel hascoordinates (70 sin (πt/50), 75− 70 cos (πt/50)) after t seconds, witht ≥ 0.
t 0 10 20 30 40 50 60 70 80 90 100
x 0 41.1 66.6 66.6 41.1 0 -41.1 -66.6 -66.6 -41.1 5y 5 18.4 53.4 96.6 131.6 145 131.6 96.6 53.4 18.4 5
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 8 / 17
Functions/equations vs. parametric equations
Eliminating the t
We can imagine the motion of a particular car on a Ferris Wheel hascoordinates (70 sin (πt/50), 75− 70 cos (πt/50)) after t seconds, witht ≥ 0.
To eliminate the variable t, we start with the identity
(sin (πt/50))2 + (cos (πt/50))2 = 1
Multiply through by 702:
(70 sin (πt/50))2 + (70 cos (πt/50))2 = 702
And substitute:x2 + (75− y)2 = 702
This is the equation of the circle described by original parametric equation.
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 9 / 17
Complex curves
Spirograph!
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 10 / 17
Parameterizing a Line
Parameterizing a Line
Insight
The line through (a, b) having slope m can be parameterized as
c(t) = (a + t, b + mt) −∞ < t <∞
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 11 / 17
Parameterizing a Line
Parameterizing a Line
c(t) = (a + t, b + mt) −∞ < t <∞
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 11 / 17
Translations and Circles
Translations and Circles
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 12 / 17
Parameterizing an Ellipse
Parameterizing an Ellipse
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 13 / 17
Paths vs. curves
Paths vs. curves
Compare each of the following to the parameterization c(t) = (t, t2) for−1 ≤ t ≤ 1.
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 14 / 17
Cycloid
Cycloid
c(t) = (t − sin t, 1− cos t)
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 15 / 17
Slope
Slope
Example. Let c(t) = (t2 + 1, t3 − 4t). Find the points where the tangentline is horizontal.
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 16 / 17
Horizontal tangent lines
Horizontal tangent lines
Professor Ensley (Ship Math) Calculus III: Section 11.1 August 29, 2011 17 / 17