Assignment 3, Spring 2008
This assignment is due Monday, February 25, 2008, at the beginning of class.
1. In each problem below,
y
is deﬁned implicitly by a certain equation. Use implicit di±erentiation to
ﬁnd
dy/dx.
If possible
, put your ﬁnal answer in terms of just the variable
x
.
(a) sin
y
=
x.
(b) ln
y
=
x
ln
x.
(c)
x
2
+
y
2
= 1
.
(d) (
x
+
y
)
3
=
xy.
2. We deﬁne a certain function cosh(
t
) in terms of the variable
t
as follows:
cosh(
t
) =
1
2
(
e
t
+
e

t
)
.
This function is sometimes called the
hyperbolic
cosine of
t
. The
hyperbolic
sine function is similarly
deﬁned by
sinh(
t
) =
1
2
(
e
t

e

t
)
.
Use these deﬁnitions in the following problems.
(a) By analogy to the ordinary trig functions, develop formulae for the hyperbolic tangent, hyper
bolic cotangent, hyperbolic secant and hyperbolic cosecant functions.
(b) Show that the derivatives of the hyperbolic sine function is the hyperbolic cosine function, and
conversely, that the derivative of the hyperbolic cosine is the hyperbolic sine.
(c) Compute the derivative of the remaining four hyperbolic trig functions.
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 Spring '08
 STAFF
 Calculus, Exponential Function, Inverse function, Hyperbolic function, Elementary special functions, Bernoulli number

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