To find: The first and second derivative of the function
Answer to Problem 45E
The first and second derivatives of the function are,
Explanation of Solution
Formula used:
The derivative of a function f , denoted by
Calculation:
Obtain the derivative of the function
Compute
Simplify the numerator and to obtain the value of
Since the limit h approaches zero but not equal to zero, cancel the common term h from both the numerator and the denominator,
Thus, the first derivative of the function is,
Obtain the second derivative of the function.
Compute
Simplify the numerator and to obtain the derivatives
Since the limit h approaches zero but not equal to zero, cancel the common term h from both the numerator and the denominator,
Thus, the second derivative of the function is,
To check: The derivatives
The derivative
Graph:
Use the online graphing calculator to draw the graph of the functions
From Figure 1, it is observed that the graph of
Take several points on the domain and estimate the slope of the tangent to the function
Thus, the derivative
The
The straight line has same slope at all points. That is,
From the Figure 1, the estimated slope of the function
Therefore, the function
Chapter 2 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning