Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
100%
![**Transcription for Educational Website**
---
**Title: Quadratic Approximation of Trigonometric Functions**
**Introduction:**
In this exercise, we will explore how to use a local quadratic approximation to estimate the value of the tangent function at a specific angle.
**Task:**
Use a local quadratic approximation to approximate \(\tan 61^\circ\).
**Explanation:**
To perform a quadratic approximation, we typically use a Taylor series expansion up to the second degree (quadratic term). The general form of a quadratic approximation for a function \( f(x) \) around a point \( x = a \) is:
\[ f(x) \approx f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2 \]
Here, \( f(a) \), \( f'(a) \), and \( f''(a) \) are the function, the first derivative, and the second derivative evaluated at the point \( a \) respectively. For this particular task, you would:
1. Choose a point \( a \) close to \( 61^\circ \) where the trigonometric values are known or easier to calculate.
2. Compute the value of \(\tan a\).
3. Determine the first and second derivatives of \(\tan x\) with respect to \( x \).
4. Plug these values into the quadratic approximation formula.
Note: This method gives a good approximation for small deviations around the point \( a \).
**Conclusion:**
Through quadratic approximation, you can derive close estimations of trigonometric functions without a calculator, relying on calculus concepts.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80f7a2c1-bbd9-492a-a9a7-34302e2af2a3%2F8750bf14-5667-4e55-9f6b-77a406a525c4%2F1rwb0wp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website**
---
**Title: Quadratic Approximation of Trigonometric Functions**
**Introduction:**
In this exercise, we will explore how to use a local quadratic approximation to estimate the value of the tangent function at a specific angle.
**Task:**
Use a local quadratic approximation to approximate \(\tan 61^\circ\).
**Explanation:**
To perform a quadratic approximation, we typically use a Taylor series expansion up to the second degree (quadratic term). The general form of a quadratic approximation for a function \( f(x) \) around a point \( x = a \) is:
\[ f(x) \approx f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2 \]
Here, \( f(a) \), \( f'(a) \), and \( f''(a) \) are the function, the first derivative, and the second derivative evaluated at the point \( a \) respectively. For this particular task, you would:
1. Choose a point \( a \) close to \( 61^\circ \) where the trigonometric values are known or easier to calculate.
2. Compute the value of \(\tan a\).
3. Determine the first and second derivatives of \(\tan x\) with respect to \( x \).
4. Plug these values into the quadratic approximation formula.
Note: This method gives a good approximation for small deviations around the point \( a \).
**Conclusion:**
Through quadratic approximation, you can derive close estimations of trigonometric functions without a calculator, relying on calculus concepts.
---
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning