Concept explainers
(a) By comparing areas, show that
(b) Use the Midpoint Rule with n = 10 to estimate In 1.5.
(a)
To show: The inequality
Explanation of Solution
Definition used:
Natural logarithmic function:
It is defined as
Geometrically the natural logarithmic function can be interpreted as the area under the curve
Calculation:
The given inequality is
Geometrically, the function
Use online graphing calculator and sketch the graph of the curve as shown below in Figure 1.
From Figure 1, the area under the curve
The area of the rectangle BCDE is computed below.
The area of the trapezoid ABCD is
Comparing these two areas with the function
Thus, it is clear that
(b)
To estimate: The function
Answer to Problem 1E
The estimated value of
Explanation of Solution
Definition used:
Midpoint rule:
It is defined as
Here,
Calculation:
The equation of the curve is
Given that
From part (a), the function
Then,
Use the definition of midpoint rule to estimate the value of
On further simplification,
Thus the estimated value of
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Chapter G Solutions
Single Variable Calculus: Early Transcendentals, Volume I
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