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Chapter 4 | Applications of Derivatives
4.44 Compare the growth rates of x 100 and 2 x .
Using the same ideas as in Example 4.45 a. it is not difficult to show that e x grows more rapidly than x p for any p >0. In Figure 4.75 and Table 4.11 , we compare e x with x 3 and x 4 as x →∞.
Figure 4.75 The exponential function e x grows faster than x p for any p >0. (a) A comparison of e x with x 3 . (b) A comparison of e x with x 4 .
5
10
15
20
x
x 3
125
1000
3375
8000
x 4
50,625
160,000
625 10,000
e x
148 22,026 3,269,017 485,165,195
Table 4.11 An exponential function grows at a faster rate than any power function Similarly, it is not difficult to show that x p grows more rapidly than ln x for any p >0. In Figure 4.76 and Table4.12 , we compare ln x with x 3 and x .
Figure 4.76 The function y = ln( x ) grows more slowly than x p for any p >0 as x →∞.
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