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
Concept explainers
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
Topic Video
Question
Can you help me fill out the charts as well
Transcribed Image Text:**Activity: Analyzing Functions for Intervals of Increase, Decrease, Concavity, and Inflection Points**
In this activity, students will explore how to determine where a function is increasing or decreasing and identify any relative extrema. They will also examine the concavity of the function and identify any inflection points.
---
### Task e: Determining Intervals of Increase and Decrease
**Objective:** Find the intervals on which the function is increasing and decreasing. Determine any relative extrema.
**Table for Analysis:**
| Interval | Test Value | Sign of \( f'(x) \) |
|-----------|------------|---------------------|
| | | |
| | | |
**Conclusion:**
- **Is \( f \) increasing or decreasing?**
- **Are there any relative extrema?**
---
### Task f: Identifying Concavity and Inflection Points
**Objective:** Determine the intervals where the function is concave up or concave down and find any inflection points.
**Table for Analysis:**
| Interval | Test Value | Sign of \( f''(x) \) |
|-----------|------------|----------------------|
| | | |
| | | |
**Conclusion:**
- **Is \( f \) concave up or concave down?**
- **Are there any inflection points?**
---
### Instructions:
1. Identify the critical points and evaluate the sign of the derivative (\( f'(x) \)) to ascertain where the function is increasing or decreasing.
2. Evaluate the second derivative (\( f''(x) \)) to determine the concavity of the function and locate any inflection points.
3. Fill out the tables with your analyses and conclusions.
This structured approach will aid students in thoroughly understanding the behavior of a given function.
![### Problem 5: \( f(x) = (x^2 - 9)^{2/3} \)
#### a. Find the domain.
- **Domain**: \( \mathbb{R} = (-\infty, \infty) \)
#### b. Find the x-intercept(s) and the y-intercept, if any.
- **To find x-intercepts**, set \( y = f(x) = 0 \Rightarrow (x^2 - 9)^{2/3} = 0 \)
- Solve \( x^2 - 9 = 0 \)
\[
x^2 = 9 \\
x = \pm 3
\]
- **x-intercepts**: \((3, 0)\) and \((-3, 0)\)
- **To find the y-intercept**, set \( x = 0 \)
- \( f(x) = y = (x^2 - 9)^{2/3} = ((-9)^{1/3})^{2/3} \)
- \( y = 8^{1/3} = 3^{2/3} = 3(3)^{1/3} \)
- **y-intercept**: \((0, 3(3)^{1/3} \approx (0, 4.3267)\)
#### c. Find any symmetry in the graph.
- To test for symmetry about the x-axis:
\[
y = -y \Rightarrow y = (-(x^2 - 9))^{2/3}
\]
- **No symmetry about x-axis because it’s not equivalent to original function.**
- To test for symmetry about the y-axis:
\[
x = -x \Rightarrow y = ((-x)^2 - 9)^{2/3}
\]
- **Symmetry about y-axis** because it’s equivalent to original function.
- **Symmetry**: About the y-axis
#### d. Find the end behavior of the graph and find all vertical, horizontal, or slant asymptotes.
- **Vertical Asymptote**:
- There are no such points in the function \( f(x) \) where the denominator of function equals zero or one.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae7a862e-0ee1-4a84-a4fd-63344cf59edd%2Fdd7847d7-5aff-453a-a6fc-5164b34d2c58%2F38ubvp4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 5: \( f(x) = (x^2 - 9)^{2/3} \)
#### a. Find the domain.
- **Domain**: \( \mathbb{R} = (-\infty, \infty) \)
#### b. Find the x-intercept(s) and the y-intercept, if any.
- **To find x-intercepts**, set \( y = f(x) = 0 \Rightarrow (x^2 - 9)^{2/3} = 0 \)
- Solve \( x^2 - 9 = 0 \)
\[
x^2 = 9 \\
x = \pm 3
\]
- **x-intercepts**: \((3, 0)\) and \((-3, 0)\)
- **To find the y-intercept**, set \( x = 0 \)
- \( f(x) = y = (x^2 - 9)^{2/3} = ((-9)^{1/3})^{2/3} \)
- \( y = 8^{1/3} = 3^{2/3} = 3(3)^{1/3} \)
- **y-intercept**: \((0, 3(3)^{1/3} \approx (0, 4.3267)\)
#### c. Find any symmetry in the graph.
- To test for symmetry about the x-axis:
\[
y = -y \Rightarrow y = (-(x^2 - 9))^{2/3}
\]
- **No symmetry about x-axis because it’s not equivalent to original function.**
- To test for symmetry about the y-axis:
\[
x = -x \Rightarrow y = ((-x)^2 - 9)^{2/3}
\]
- **Symmetry about y-axis** because it’s equivalent to original function.
- **Symmetry**: About the y-axis
#### d. Find the end behavior of the graph and find all vertical, horizontal, or slant asymptotes.
- **Vertical Asymptote**:
- There are no such points in the function \( f(x) \) where the denominator of function equals zero or one.
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 5 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.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