College Algebra & Trigonometry, 2018a

144 CHAPTER 3. EXPONENTS AND LOGARITHMS
As mathematicians examined these graphs during the 17th and 18th centuries,
they began to question what the value of the base ”b” should be in the equation
y = b x so that the slope of the tangent line at the point (0, 1) would be equal to
exactly 1. The answer was e ≈ 2.71828.
Another way to derive the value of e uses Integral Calculus. Integral Calculus is
often concerned with finding the area under a curve. This process can then be
generalized and used to make many other types of calculations that are similar
to finding area.
Consider the graph of the curve y = 1 x :
2
1
y = 1 x
−1 1 2 3 4 5
−1
We can delineate borders on the x values and determine the area of the resulting
region using the techniques of calculus:
0.8
0.6
1 y = 1 x
1
0.8
0.6
y = 1 x
0.4
0.2
A ≈ 0.693
0.4
0.2
A ≈ 1.0986
−0.2
0.6 0.8 1 1.2 1.4 1.6 1.8 2
−0.2
1 1.5 2 2.5 3