James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 10.2 Calculus with Parametric Curves 649
TEC Look at Module 10.1B to see
how hypocycloids and epicycloids
are formed by the motion of rolling
circles.
graph? Show that if we take a − 4, then the parametric equations of the hypocycloid
reduce to
x − 4 cos 3 y − 4 sin 3
This curve is called a hypocycloid of four cusps, or an astroid.
3. Now try b − 1 and a − nyd, a fraction where n and d have no common factor. First let
n − 1 and try to determine graphically the effect of the denominator d on the shape of the
graph. Then let n vary while keeping d constant. What happens when n − d 1 1?
4. What happens if b − 1 and a is irrational? Experiment with an irrational number like s2
or e 2 2. Take larger and larger values for and speculate on what would happen if we
were to graph the hypocycloid for all real values of .
5. If the circle C rolls on the outside of the fixed circle, the curve traced out by P is called an
epicycloid. Find parametric equations for the epicycloid.
6. Investigate the possible shapes for epicycloids. Use methods similar to Problems 2–4.
Having seen how to represent curves by parametric equations, we now apply the methods
of calculus to these parametric curves. In particular, we solve problems involving tangents,
areas, arc length, and surface area.
Tangents
Suppose f and t are differentiable functions and we want to find the tangent line at a
point on the parametric curve x − f std, y − tstd, where y is also a differentiable function
of x. Then the Chain Rule gives
dy
dt − dy
dx ? dx
dt
If dxydt ± 0, we can solve for dyydx:
If we think of the curve as being
traced out by a moving particle, then
dyydt and dxydt are the vertical and
horizontal velocities of the particle and
Formula 1 says that the slope of the
tangent is the ratio of these velocities.
1
dy
dy
dx − dt
dx
dt
if
dx
dt ± 0
d 2 y
Note that d 2 y
dx Þ dt 2
2 d 2 x
dt 2
Equation 1 (which you can remember by thinking of canceling the dt’s) enables us
to find the slope dyydx of the tangent to a parametric curve without having to eliminate
the parameter t. We see from (1) that the curve has a horizontal tangent when dyydt − 0
(provided that dxydt ± 0) and it has a vertical tangent when dxydt − 0 (provided that
dyydt ± 0). This information is useful for sketching parametric curves.
As we know from Chapter 4, it is also useful to consider d 2 yydx 2 . This can be found
by replacing y by dyydx in Equation 1:
d 2 y
dx 2
S − d dy −
dx dxD
d
S dy
dt dxD
dx
dt
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