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this latter equation implicitly with respect to x, we have sec
2
y
dy
dx
=1
dy
dx
=
1
sec
2
y
=
1
1+tan
2
y
=
1
1+x
2
,
d
dx
(tan
−1
x)=
1
1+x
2
.
Here are the remaining differentiation formulas
d
dx
(cos
−1
x)=
−1
√
1 − x
2
,
d
dx
(arcctg x)=
−1
1+x
2
.
4.10. Parametric Curves
Reminder
The Vertical Line Test (or VLT): a curve in the xy− plane is the
graph of a function of x if and only if no vertical line intersects the
curve more than once.
Imagine that a particle moves along a curve C in the xy−plane.
It is, sometimes, impossible to describe C by an equation of the
form y = f(x) because C fails the Vertical Line Test. But the x−
and y−coordinates of the particle are functions of time and so we
can write x = x(t) and y = y(t). Such a pair of functions is often a
convenient way of describing a curve and gives rise to the following
definition.
Suppose that x and y are both given as continuous functions of
a third variable t (called a parameter ) by the equations
x = x(t),y= y(t)
(called parametric equations). Each value of t determines a point
(x, y), which we can plot in a coordinate plane. As t varies, the
point (x(t),y(t)) varies and traces out a curve C, which we call a
parametric curve. The parameter t does not necessarily represent
time. Other letter can be used for the parameter. However, in many
applications of parametric curves, t does denote time and we can
interpret (x, y)=(x(t),y(t)) as the position of a particle at time t.
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