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