Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Concept explainers

Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
Numbers
Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
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**Concavity and Points of Inflection Analysis** Use the given graph of the function \( f \) to answer the following questions. ![Graph of function \( f \)](image-link) The graph provided shows a blue curve representing the function \( f \). The x-axis ranges approximately from -1 to 8 and the y-axis ranges from -1 to 8. The curve exhibits changes in concavity and includes points which may be identified as points of inflection. 1. **Find the open interval(s) on which \( f \) is concave upward.** Answer (in [interval notation](https://www.mathsisfun.com/sets/interval-notation.html)): 2. **Find the open interval(s) on which \( f \) is concave downward.** Answer (in [interval notation](https://www.mathsisfun.com/sets/interval-notation.html)): 3. **Find the coordinates of the points of inflection. List your answers as points in the form \( (a, b) \).** Answer (separate by commas): In the context of the graph: - **Concave Upward**: This is where the graph of the function is curving upwards, resembling the shape of a cup. The intervals where the function changes from decreasing to increasing can be noted. - **Concave Downward**: This is where the graph of the function is curving downwards, resembling the shape of a cap. The intervals where the function changes from increasing to decreasing are noted. - **Points of Inflection**: These are points where the graph changes concavity from upward to downward or vice versa. They represent transitions in the curvature of the function. To complete the answers, carefully observe the graph and identify the specified intervals and points based on changes in concavity.
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Transcribed Image Text:**Concavity and Points of Inflection Analysis** Use the given graph of the function \( f \) to answer the following questions. ![Graph of function \( f \)](image-link) The graph provided shows a blue curve representing the function \( f \). The x-axis ranges approximately from -1 to 8 and the y-axis ranges from -1 to 8. The curve exhibits changes in concavity and includes points which may be identified as points of inflection. 1. **Find the open interval(s) on which \( f \) is concave upward.** Answer (in [interval notation](https://www.mathsisfun.com/sets/interval-notation.html)): 2. **Find the open interval(s) on which \( f \) is concave downward.** Answer (in [interval notation](https://www.mathsisfun.com/sets/interval-notation.html)): 3. **Find the coordinates of the points of inflection. List your answers as points in the form \( (a, b) \).** Answer (separate by commas): In the context of the graph: - **Concave Upward**: This is where the graph of the function is curving upwards, resembling the shape of a cup. The intervals where the function changes from decreasing to increasing can be noted. - **Concave Downward**: This is where the graph of the function is curving downwards, resembling the shape of a cap. The intervals where the function changes from increasing to decreasing are noted. - **Points of Inflection**: These are points where the graph changes concavity from upward to downward or vice versa. They represent transitions in the curvature of the function. To complete the answers, carefully observe the graph and identify the specified intervals and points based on changes in concavity.
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