Составители:
Рубрика:
The tangent line is horizontal when
dy
dx
=0, which occurs when
cos t =0(and cos 2t =0), that is, when t =
π
2
or 3
π
2
. Thus, the
curve has horizontal tangents at the points (0, 2) and (0, −2).
The tangent is vertical when
dx
dt
=4cos2t =0(and cos t =0),
that is, when t =
π
4
, 3
π
4
, 5
π
4
, or 7
π
4
. The corresponding four points
on the curve are (2,
√
2), (−2,
√
2), (2, −
√
2), (−2, −
√
2).
4.12. Hyperbolic Functions
Certain combinations of the exponential functions e
x
and e
−x
arise
so frequently in mathematics and its applications that they deserve
to be given special names. Here we deal with hyperbolic functions:
sinh x =
e
x
− e
−x
2
, cosh x =
e
x
+ e
−x
2
,
tanh x =
sinh x
cosh x
, sechx =
1
cosh x
.
The reason for the names of these functions is that they are
related to the hyperbola in much the same way that the trigonometric
functions are related to the circle. Texts in Russian use notation
sh x, ch x, thx, cthx =
ch x
sh x
. Here are some identities similar to tri-
gonometric ones:
• ch
2
x − sh
2
x =1,
• ch(a + b)=cha ch b +sha sh b,
• sh(a + b)=sha ch b +cha sh b.
The proof is straightforward.
These functions can be used to parameterize a hyperbola: x =
ch t, y =sht. These equations are similar to parametric equations of
a unit circle (introduced earlier).
29
Страницы
- « первая
- ‹ предыдущая
- …
- 27
- 28
- 29
- 30
- 31
- …
- следующая ›
- последняя »
