10.1

Parametric Functions

2

Use algebra or a trig identity to write an equation relating and .

1. 1 and 2 3

2. 3 and 54 3

3. sin and cos

4. sin cos and sin(2 )

5. tan and sec

6. sin and

x y

x t y t

x t y t

x t y t

x t t y t

x y

x y

cos(2 )

Quick Review

12 xy36 2 xy122 yxxy 2

22 1 xy 221 xy

What you’ll learn about Parametric Curves in the Plane Slope and Concavity Arc Length Cycloids

Essential QuestionsHow do we use parametric equations to definesome interesting and important curves that wouldbe difficult or impossible to define in the form y=f(x)?

Reviewing Some Parametric Curves1. Sketch the parametric curve and eliminate the parametric to find

an equation that relates x and y directly. 2 0, interval in the for sin and cos a. ttytx

1cossin 22 tt122 yx

Reviewing Some Parametric Curves

1. Sketch the parametric curve and eliminate the parametric to find an equation that relates x and y directly.

4 0, interval in the for sin2 and cos3 b. ttytx

1cossin 22 tt

3cos

xt

2sin

yt

123

22

yx

Reviewing Some Parametric Curves

1. Sketch the parametric curve and eliminate the parametric to find an equation that relates x and y directly.

4 0, interval in the for 2 and c. ttytx

2xt

22 xy

Parametric Differentiation Formulas

2

2

If and are both differentiable functions of and if / 0, then

/.

/If ' / is also a differentiable function of , then

'/' .

/

x y t dx dt

dy dy dt

dx dx dty dy dx t

d y d dy dty

dx dx dx dt

Analyzing a Parametric Curve2. Consider the curve defined parametrically by and 52 tx

.0for sin2 tty

a. Sketch a graph of the curve in the viewing window [-7, 7] by [-4, 4].

Analyzing a Parametric Curve2. Consider the curve defined parametrically by and 52 tx

.0for sin2 tty

b. Find the highest point on the curve. Justify your answer.

dtdxdtdy

dx

dy

tcos2

2

t

52

2

x

t2 t

tcos 0

2/

Max

533.2

2sin2

y 2

2 ,533.2

Analyzing a Parametric Curve2. Consider the curve defined parametrically by and 52 tx

.0for sin2 tty

c. Find all points of inflection on the curve. Justify your answer.

dtdxdtyd

dx

yd

2

2

sin tt

1cos t2tt2

32

cossin

t

ttt 0 Graph the function to solve.

7983.2t

57983.2 2 x 831.2 7983.2sin2y 673.0

.6730 ,831.2

Arc Length of a Parametrized Curve

2

1

1 2

2 2

Let be the length of a parametric curve that is traversed exactly once as

increases from to .

If / and / are continuous functions of , then

.t

t

L

t t t

dx dt dy dt t

dy dxL dt

dt dt

Example Measuring a Parametric Curve3. Find the length of the curve defined by ,cos ,sin tytx

.20for tThe curve is traced once as t goes from 0 to 2.

Because of the curve’s symmetry with respect to the

coordinate axis, its length is 4 times the length of the first

quadrant portion.

dtdt

dx

dt

dyS 4 2

0

22

dttt cossin4 2

0

22

2

0 14

dt 2

0 4

t

24

2 The length of the curve is 2, which is the circumference of a circle with radius 1.

CycloidsSuppose that a wheel of radius rolls along a horizontal line without

slipping. The path traced by a point on the wheel's edge is a

, where is originally at the origin.

a

P

Pcycloid

4. Find parametric equations for the path of point P in the figure above.

a opp.

hyp.

adj.

a

adjcos

cosadj a

cosaatx

a

oppsin

sinopp a

sinaay

Pg. 535, 10.1 #1-35 odd