322
Chapter 3 | Derivatives
b −0.00001 −1
0.00001 −1 0.00001
b −0.00001 −1
0.00001 −1 0.00001
b
b
( 0 ) < b
( 0 ) < b
−0.00001 < B ′
−0.00001 < B ′
2
0.693145< B ′ (0) < 0.69315
2.7183 1.000002 < B ′ (0) < 1.000012
2.7
0.993247< B ′ (0) < 0.993257
2.719
1.000259< B ′ (0) < 1.000269
2.71
0.996944< B ′ (0) < 0.996954
2.72
1.000627< B ′ (0) < 1.000637
2.718
0.999891< B ′ (0) < 0.999901
2.8
1.029614< B ′ (0) < 1.029625
2.7182 0.999965 < B ′ (0) < 0.999975
3
1.098606< B ′ (0) < 1.098618
Table 3.7 Estimating a Value of e
The evidence from the table suggests that 2.7182< e <2.7183. The graph of E ( x ) = e x together with the line y = x +1 are shown in Figure 3.34 . This line is tangent to the graph of E ( x ) = e x at x =0.
Figure 3.34 The tangent line to E ( x ) = e x at x =0 has slope 1.
Now that we have laid out our basic assumptions, we begin our investigation by exploring the derivative of B ( x ) = b x , b >0. Recall that we have assumed that B ′(0) exists. By applying the limit definition to the derivative we conclude that (3.28) B ′(0) = lim h →0 b 0+ h − b 0 h = lim h →0 b h −1 h . Turning to B ′( x ), we obtain the following.
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