10.1 parametric functions quick review what you’ll learn about parametric curves in the plane...
TRANSCRIPT
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10.1
Parametric Functions
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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
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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)?
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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
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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
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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
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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
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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].
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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
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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
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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
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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.
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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
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Pg. 535, 10.1 #1-35 odd