Concept explainers
(a)
To show: The inequality
(a)
Explanation of Solution
Definition used:
The natural logarithmic function is the function defined by
If
Calculation:
The given inequality is
Here,
By using the above mentioned definition,
Use online graphing calculator, graph of the curve
From Figure 1, it is observed that the area is larger than the area of rectangle BCDE and smaller than the area of trapezoid ABCD.
The area of the rectangle BCDE is calculated as follows.
The area of the trapezoid ABCD is calculated as follows.
It is known that the value of
The decimal value of the fraction
By comparing the above values with
Hence proved.
(b)
To show: The function
(b)
Answer to Problem 1E
The estimated value of
Explanation of Solution
Definition used:
Midpoint Rule:
Midpoint rule can be defined as
Where,
Natural logarithmic:
The natural logarithmic function is the function defined by
Calculation:
It is known that the equation of the curve is
Here,
From definition of natural logarithmic, the function
Here,
Find
Estimate the value of
Further simplification,
Therefore, the estimated value of
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Chapter C Solutions
Essential Calculus: Early Transcendentals
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