Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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solve please an example will be in the second photo
### Determining the Zeros of \(f(x)\) We start by determining all places within the domain of \(f(x)\) where \(f(x) = 0\). This occurs whenever the numerator of the fraction is zero. We solve this by setting each individual factor equal to zero and solving those equations. #### Solving for Zeros: 1. \[(x + 3)\sqrt{4 - x} = 0\] For \(x + 3 = 0\): \[ x + 3 = 0 \implies x = -3 \] For \(\sqrt{4 - x} = 0\): \[ 4 - x = 0 \implies x = 4 \] Each solution, \(x = -3\) and \(x = 4\), lies within the domain of \(f(x)\). Therefore, \(f(x)\) has zeros at \(x = -3\) and \(x = 4\). ### Plotting Zeros on Number Line The zero \(x = 4\) is already plotted on the number line since it is the boundary of the domain. The other zero, \(x = -3\), is added to the number line. \[ \longleftarrow -3 \longrightarrow\quad 0 \quad \longrightarrow 4 \] ### Determining Sign in Each Interval We choose test values in each interval to determine the sign of \(f(x)\). Test values are indicated in parentheses as they are not uniquely chosen. #### Test Intervals and Values: \[ (-\infty, -3) : \text{Test value} = -4 \] \[ (-3, 0) : \text{Test value} = -1 \] \[ (0, 4) : \text{Test value} = 1 \] ### Sign Analysis of Each Interval 1. **Interval \((-∞, -3)\):** Test value \(x = -4\) - Factor \( \sqrt{4 - x} \) is positive. - Factor \( x^2 \) is positive. - \( x + 3 \) is negative. - \( x - 5 \) is negative. Product: Positive (since two negatives make a positive). 2. **Interval \((-3, 0
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Transcribed Image Text:### Determining the Zeros of \(f(x)\) We start by determining all places within the domain of \(f(x)\) where \(f(x) = 0\). This occurs whenever the numerator of the fraction is zero. We solve this by setting each individual factor equal to zero and solving those equations. #### Solving for Zeros: 1. \[(x + 3)\sqrt{4 - x} = 0\] For \(x + 3 = 0\): \[ x + 3 = 0 \implies x = -3 \] For \(\sqrt{4 - x} = 0\): \[ 4 - x = 0 \implies x = 4 \] Each solution, \(x = -3\) and \(x = 4\), lies within the domain of \(f(x)\). Therefore, \(f(x)\) has zeros at \(x = -3\) and \(x = 4\). ### Plotting Zeros on Number Line The zero \(x = 4\) is already plotted on the number line since it is the boundary of the domain. The other zero, \(x = -3\), is added to the number line. \[ \longleftarrow -3 \longrightarrow\quad 0 \quad \longrightarrow 4 \] ### Determining Sign in Each Interval We choose test values in each interval to determine the sign of \(f(x)\). Test values are indicated in parentheses as they are not uniquely chosen. #### Test Intervals and Values: \[ (-\infty, -3) : \text{Test value} = -4 \] \[ (-3, 0) : \text{Test value} = -1 \] \[ (0, 4) : \text{Test value} = 1 \] ### Sign Analysis of Each Interval 1. **Interval \((-∞, -3)\):** Test value \(x = -4\) - Factor \( \sqrt{4 - x} \) is positive. - Factor \( x^2 \) is positive. - \( x + 3 \) is negative. - \( x - 5 \) is negative. Product: Positive (since two negatives make a positive). 2. **Interval \((-3, 0
--- ### Show What You Know: Solving Inequalities with Various Function Types **MAT 190 - Precalculus** --- #### Objectives: The purpose of this assignment is for you to: 1. Demonstrate your ability to solve inequalities with various function types. 2. Improve your mathematical writing to include full solutions with justifications. 3. Integrate mathematical statements into grammatically correct expositions. --- #### Assignment: Find all intervals on which the function is positive and all intervals on which the function is negative: \[ f(x) = \frac{x - 3}{e^x (x - 5)^2 \sqrt{2x + 1}} \] Include a detailed explanation for each mathematical step written in grammatically correct complete sentences within a 2-column format. --- This transcription ensures clarity and precision for students tackling the assignment on solving inequalities with various function types in a structured manner.
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Transcribed Image Text:--- ### Show What You Know: Solving Inequalities with Various Function Types **MAT 190 - Precalculus** --- #### Objectives: The purpose of this assignment is for you to: 1. Demonstrate your ability to solve inequalities with various function types. 2. Improve your mathematical writing to include full solutions with justifications. 3. Integrate mathematical statements into grammatically correct expositions. --- #### Assignment: Find all intervals on which the function is positive and all intervals on which the function is negative: \[ f(x) = \frac{x - 3}{e^x (x - 5)^2 \sqrt{2x + 1}} \] Include a detailed explanation for each mathematical step written in grammatically correct complete sentences within a 2-column format. --- This transcription ensures clarity and precision for students tackling the assignment on solving inequalities with various function types in a structured manner.
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