320
Chapter 3 | Derivatives
4 x
4 x
x
x
4 3
4 3.141593
64
77.8802710486
4 3.1
4 3.1416
73.5166947198
77.8810268071
4 3.14
4 3.142
77.7084726013
77.9242251944
4 3.141
4 3.15
77.8162741237
78.7932424541
4 3.2
4 3.1415
77.8702309526
84.4485062895
4 4
4 3.14159
77.8799471543
256
Table 3.6 Approximating a Value of 4 π We also assume that for B ( x ) = b x , b >0, the value B ′(0) of the derivative exists. In this section, we show that by making this one additional assumption, it is possible to prove that the function B ( x ) is differentiable everywhere. We make one final assumption: that there is a unique value of b >0 for which B ′(0) =1. We define e to be this unique value, as we did in Introduction to Functions and Graphs . Figure 3.33 provides graphs of the functions y =2 x , y =3 x , y =2.7 x , and y =2.8 x . A visual estimate of the slopes of the tangent lines to these functions at 0 provides evidence that the value of e lies somewhere between 2.7 and 2.8. The function E ( x ) = e x is called the natural exponential function . Its inverse, L ( x ) = log e x = ln x is called the natural logarithmic function .
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