calculus iii: section 11 - my webspace fileswebspace.ship.edu/deensley/m213/sec11_1.pdf ·...

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


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


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


Path of a particle

Trajectory of a bullet


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.



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



Circular motion


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



Eliminating the t


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.


Complex curves

Spirograph!


Parameterizing a Line


Insight

The line through (a, b) having slope m can be parameterized as

c(t) = (a + t, b + mt) −∞ < t <∞




c(t) = (a + t, b + mt) −∞ < t <∞


Translations and Circles

Translations and Circles


Parameterizing an Ellipse

Parameterizing an Ellipse


Paths vs. curves

Paths vs. curves

Compare each of the following to the parameterization c(t) = (t, t2) for−1 ≤ t ≤ 1.


Cycloid

Cycloid

c(t) = (t − sin t, 1− cos t)


Slope

Slope

Example. Let c(t) = (t2 + 1, t3 − 4t). Find the points where the tangentline is horizontal.


Horizontal tangent lines

Horizontal tangent lines


calculus iii: section 11 - my webspace fileswebspace.ship.edu/deensley/m213/sec11_1.pdf ·...

Documents