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

262 Chapter 3 Differentiation Rules
We can sketch the graphs of sinh 21 , cosh 21 , and tanh 21 in Figures 8, 9, and 10 by using
Figures 1, 2, and 3.
y
y
y
0
x
_1
0
1
x
0
1
x
FIGURE 8 y − sinh 21 x
domain − R range − R
FIGURE 9 y − cosh 21 x
domain − f1, `d range − f0, `d
FIGURE 10 y − tanh 21 x
domain − s21, 1d range − R
Since the hyperbolic functions are defined in terms of exponential functions, it’s not
surprising to learn that the inverse hyperbolic functions can be expressed in terms of
logarithms. In particular, we have:
3 sinh 21 x − lnsx 1 sx 2 1 1d x [ R
Formula 3 is proved in Example 3.
The proofs of Formulas 4 and 5 are
requested in Exercises 26 and 27.
4 cosh 21 x − lnsx 1 sx 2 2 1d x > 1
5 tanh 21 x − 1 2 ln S 1 1 x
1 2 xD 21 , x , 1
ExamplE 3 Show that sinh 21 x − lnsx 1 sx 2 1 1d.
SOLUTION Let y − sinh 21 x. Then
x − sinh y − e y 2 e 2y
2
so e y 2 2x 2 e 2y − 0
or, multiplying by e y ,
e 2y 2 2xe y 2 1 − 0
This is really a quadratic equation in e y :
se y d 2 2 2xse y d 2 1 − 0
Solving by the quadratic formula, we get
e y − 2x 6 s4x 2 1 4
2
− x 6 sx 2 1 1
Note that e y . 0, but x 2 sx 2 1 1 , 0 (because x , sx 2 1 1). Thus the minus sign
is inadmissible and we have
e y − x 1 sx 2 1 1
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