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CHAPTER 6
Inverse Functions
EXAMPLE 10 Find the area under the curve y − e23x from 0 to l.
SOLUTION The area is
g
A − y e23x dx − 213 e23x 0 − 13 s1 2 e23 d
1
1
0
■ Exponential Graphs
y
The exponential function f sxd − e x is one of the most frequently occurring functions in
calculus and its applications, so it is important to be familiar with its graph (Figure 14)fiand
properties. We summarize these properties as follows, using the fact that this function
is just a special case of the exponential functions considered in Theorem 2 but with base
b − e . 1.
y=´
1
0
n
1
FIGURE 14
x
11 Properties of the Natural Exponential Function The exponential
function f sxd − e x is an increasing continuous function with domain R and
rangefis0, `d. Thus e x . 0 for all x. Also
The natural exponential function
lim e x − 0
lim e x − `
x l2`
xl`
So the x-axis is a horizontal asymptote of f sxd − e x.
e 2x
.
x l ` e 2x 1 1
EXAMPLE 11 Find lim
SOLUTION We divide numerator and denominator by e 2x:
e 2x
1
1
− lim
−
xl` e
x l ` 1 1 e22x
11
1 1 lim e22x
lim
2x
xl`
−
1
−1
110
We have used the fact that t − 22x l 2` as x l ` and so
lim e22x − lim e t − 0
xl`
t l2`
n
EXAMPLE 12 Use the rst and second derivatives of f sxd − e 1yx, together with
asymptotes, to sketch its graph.
|
SOLUTION Notice that the domain of f is h x x ± 0j, so we check for vertical
asymptotes by computing the left and right limits as x l 0. As x l 01, we know that
t − 1yx l `, so
lim e 1yx − lim e t − `
x l 01
tl`
and this shows that x − 0 is a vertical asymptote. As x l 02, we have
t − 1yx l 2`, so
lim e 1yx − lim e t − 0
x l 02
t l2`
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