Find the interval(s) on which f(x) = |x² - 9| is differentiable. Enter your answer using interval notation, for example [2,5). Use U for union and for infinity use oo. Example: (-00,2]U (2,5)U[10,00). Use DNE for the empty set. Domain=
Find the interval(s) on which f(x) = |x² - 9| is differentiable. Enter your answer using interval notation, for example [2,5). Use U for union and for infinity use oo. Example: (-00,2]U (2,5)U[10,00). Use DNE for the empty set. Domain=
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...
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![**Differentiability of the Function \( f(x) = |x^2 - 9| **
**Instructions:**
Find the interval(s) on which the function \( f(x) = |x^2 - 9| \) is differentiable.
**Guidelines for Answering:**
- Enter your answer using interval notation. For example, use [2,5] to denote a closed interval.
- Use 'U' for union and 'oo' for infinity.
- Example of interval notation: \((-oo,2] \cup (2,5) \cup [10,oo)\).
- Use ‘DNE’ for the empty set if applicable.
**Input Box:**
- Domain = [Input box here]
**Notes:**
- To determine differentiability, consider where the absolute value function \(x^2 - 9 = 0\) changes behavior. This occurs where \(x^2 - 9 = 0\) or \(x = 3\) and \(x = -3\).
- Analyze differentiability on intervals \((-oo, -3)\), \((-3, 3)\), \((3, oo)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F54e8c682-be67-4ef0-bd1d-cf8b2e543b76%2F16f45814-4c9f-4026-96ab-438bdd24991d%2F4qyo4ah_processed.png&w=3840&q=75)
Transcribed Image Text:**Differentiability of the Function \( f(x) = |x^2 - 9| **
**Instructions:**
Find the interval(s) on which the function \( f(x) = |x^2 - 9| \) is differentiable.
**Guidelines for Answering:**
- Enter your answer using interval notation. For example, use [2,5] to denote a closed interval.
- Use 'U' for union and 'oo' for infinity.
- Example of interval notation: \((-oo,2] \cup (2,5) \cup [10,oo)\).
- Use ‘DNE’ for the empty set if applicable.
**Input Box:**
- Domain = [Input box here]
**Notes:**
- To determine differentiability, consider where the absolute value function \(x^2 - 9 = 0\) changes behavior. This occurs where \(x^2 - 9 = 0\) or \(x = 3\) and \(x = -3\).
- Analyze differentiability on intervals \((-oo, -3)\), \((-3, 3)\), \((3, oo)\).
![The image displays two graphs and an equation involving functions and derivatives. Here's a detailed transcription and explanation:
### Graph Descriptions:
#### Left Graph (`g(x)`):
- **Axes Labeling**:
- Horizontal axis is labeled as "x-values."
- Vertical axis is labeled as "y-values."
- **Graph Details**:
- This graph plots the function \(g(x)\) with a decreasing linear trend.
- The graph passes through the points approximately \((1, 9)\) and \((5, 5)\).
#### Right Graph (`h(x)`):
- **Axes Labeling**:
- Horizontal axis is labeled as "x-values."
- Vertical axis is labeled as "y-values."
- **Graph Details**:
- This graph plots the function \(h(x)\) with an initial increasing trend, peaking, and then decreasing.
- The graph approximately starts at point \((1, 1)\), peaks at \((3, 5)\), and then descends to \((5, 1)\).
### Equation and Derivative:
- The function \(f(x)\) is defined as the product of two functions: \(f(x) = g(x) \cdot h(x)\).
- Below the graphs, there is an expression for the derivative of \(f(x)\), specifically asking for the value \(f'(2)\).
### Task:
- Calculate the derivative of the product function \(f(x)\) and evaluate it at \(x = 2\), using the given graphs of \(g(x)\) and \(h(x)\).
- **Given Expression for Derivative**:
\[
f'(2) =
\]
**Note**: To find \(f'(2)\), you would typically use the product rule for derivatives: \((g \cdot h)' = g' \cdot h + g \cdot h'\). You'll need to determine the slopes of the tangent lines (derivatives) for \(g(x)\) and \(h(x)\) at \(x = 2\) from the graphs.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F54e8c682-be67-4ef0-bd1d-cf8b2e543b76%2F16f45814-4c9f-4026-96ab-438bdd24991d%2Fbubflc_processed.png&w=3840&q=75)
Transcribed Image Text:The image displays two graphs and an equation involving functions and derivatives. Here's a detailed transcription and explanation:
### Graph Descriptions:
#### Left Graph (`g(x)`):
- **Axes Labeling**:
- Horizontal axis is labeled as "x-values."
- Vertical axis is labeled as "y-values."
- **Graph Details**:
- This graph plots the function \(g(x)\) with a decreasing linear trend.
- The graph passes through the points approximately \((1, 9)\) and \((5, 5)\).
#### Right Graph (`h(x)`):
- **Axes Labeling**:
- Horizontal axis is labeled as "x-values."
- Vertical axis is labeled as "y-values."
- **Graph Details**:
- This graph plots the function \(h(x)\) with an initial increasing trend, peaking, and then decreasing.
- The graph approximately starts at point \((1, 1)\), peaks at \((3, 5)\), and then descends to \((5, 1)\).
### Equation and Derivative:
- The function \(f(x)\) is defined as the product of two functions: \(f(x) = g(x) \cdot h(x)\).
- Below the graphs, there is an expression for the derivative of \(f(x)\), specifically asking for the value \(f'(2)\).
### Task:
- Calculate the derivative of the product function \(f(x)\) and evaluate it at \(x = 2\), using the given graphs of \(g(x)\) and \(h(x)\).
- **Given Expression for Derivative**:
\[
f'(2) =
\]
**Note**: To find \(f'(2)\), you would typically use the product rule for derivatives: \((g \cdot h)' = g' \cdot h + g \cdot h'\). You'll need to determine the slopes of the tangent lines (derivatives) for \(g(x)\) and \(h(x)\) at \(x = 2\) from the graphs.
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