Concept explainers
(a)
To sketch: The graph of the function and obtain the point when the function crosses the y-axis.
(a)
Answer to Problem 2E
The graph of the function
The function
Explanation of Solution
Given:
The function is
Formula used: Derivative of Exponential Function
Calculation:
Obtain the derivative of
Use the derivative of exponential function in equation (1).
Since the given function and its derivative are same,
The graph of the function
From the graph, it is observed that the function
Moreover, the function is closer to zero as x approaches minus infinity and it is closer to infinity as x approaches plus infinity.
That is,
Substitute 0 for x in
Therefore, the function is crosses the y axis at
(b)
To describe: The type of functions
(b)
Answer to Problem 2E
Both the functions
The differentiation formulas for
Explanation of Solution
Given:
The function are
Formula used: Power Rule
If n is a real number, then
Calculation:
The graph of the function
From the graph, it is observed that the function
Thus,
From part (a),
The derivatives of both the functions
Therefore, both the functions
Obtain the derivatives of
From part (a), the derivative of
Since the derivative of
Apply the Power rule (2),
Thus, the derivative of
Therefore, the differentiation formulas for
(c)
To identify: The function which grows more rapidly when x is large.
(c)
Answer to Problem 2E
The function
Explanation of Solution
Given:
The function are
The graph of the functions
From the graph, it is observed that the value of
That is,
Therefore, the function
Chapter 3 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition