Concept explainers
a.
Use a graphing utility to graph the sine function and its polynomial approximation in the same viewing window
a.
Answer to Problem 94E
Explanation of Solution
Given information:
Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials
And,
where
Use a graphing utility to graph the sine function and its polynomial approximation in the same viewing window. How do the graphs compare?
Calculation:
The sine function
From the graph above, we see that
Hence, the polynomial function is
b.
Use the graphing utility to graph the cosine function and its polynomial approximation in the same viewing window.
b.
Answer to Problem 94E
Explanation of Solution
Given information:
Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials
And,
Where
Use the graphing utility to graph the cosine function and its polynomial approximation in the same viewing window. How do the graphs compare?
Calculation:
The cosine function
From the graph above, we see that
Hence, the polynomial function is
c.
How does the accuracy of the approximations change when an additional term is added?
c.
Answer to Problem 94E
Accuracy is increased as the additional term is added to the approximation.
Explanation of Solution
Given information:
Using calculus, it can be shown that the sine and cosine functions can be approximated by the polynomials
And,
Where
Study the patterns in the polynomial approximations of the sine and cosine functions and predict the next term in each. Then repeat parts (a) and (b). How does the accuracy of the approximations change when an additional term is added?
Calculation
Studying the pattern of polynomial function from part(a), it can be written including the next term as follows,
Above polynomial function is graphed along with the sine function as follows,
From the graph above, we see that
Studying the pattern of polynomial function from part (b), it can be written including the next term as follows,
Above polynomial function is graphed along with the sine function as follows,
From the graph above, we see that
Hence, Accuracy is increased as the additional term is added to the approximation.
Chapter 4 Solutions
EBK PRECALCULUS W/LIMITS
- 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