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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In the graph, the horizontal axis represents the \( x \)-axis, and the vertical axis represents the \( y \)-axis. The graph of \( f' \) (the derivative of \( f \)) is plotted in blue and varies over the interval \( x \) = [−1, 7]. The graph starts at \( x = -2 \) at \( y = -1 \) and continues until \( y = 2 \) at \( x = 7 \). The curve crosses the \( x \)-axis multiple times, indicating points where \( f' \) = 0.
**Critical Points:**
- The graph \( f' \) intersects the \( x \)-axis at approximately \( x = -1.5, 0.5, 3, \) and \( 6 \). These points indicate where the derivative \( f' \) changes its sign, which are critical points of \( f \).
**Questions to Explore:**
a) **On what intervals is the graph of \( f \) increasing?**
- To determine this, we look for where \( f' \) (the derivative of \( f \)) is positive. Based on the graph, \( f' \) is positive between the intervals \( (-1.5, 0.5) \) and \( (3, 6) \).
b) **On what intervals is the graph of \( f \) decreasing?**
- We look for where \( f' \) is negative. Based on the graph, \( f' \) is negative between the intervals \( (-\infty, -1.5) \), \( (0.5, 3) \), and \( (6, \infty) \).
c) **Determine any critical values of \( f \).**
- The critical values of \( f \) are at the points where the derivative \( f' \) equals zero: \( x = -1.5, 0.5, 3, \) and \( 6 \).
d) **Determine if the critical values from part (c) are minimum values, maximum values, or neither.**
- To determine this, we look at the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7a11c6-35df-4daf-855c-043e8f69fb75%2F5ff6b4bb-a9dd-4869-a49f-09c100ab45ab%2F762ug9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding the Derivative:**
The figure shows the graph of the **derivative** \( f' \) of a function \( f \).
In the graph, the horizontal axis represents the \( x \)-axis, and the vertical axis represents the \( y \)-axis. The graph of \( f' \) (the derivative of \( f \)) is plotted in blue and varies over the interval \( x \) = [−1, 7]. The graph starts at \( x = -2 \) at \( y = -1 \) and continues until \( y = 2 \) at \( x = 7 \). The curve crosses the \( x \)-axis multiple times, indicating points where \( f' \) = 0.
**Critical Points:**
- The graph \( f' \) intersects the \( x \)-axis at approximately \( x = -1.5, 0.5, 3, \) and \( 6 \). These points indicate where the derivative \( f' \) changes its sign, which are critical points of \( f \).
**Questions to Explore:**
a) **On what intervals is the graph of \( f \) increasing?**
- To determine this, we look for where \( f' \) (the derivative of \( f \)) is positive. Based on the graph, \( f' \) is positive between the intervals \( (-1.5, 0.5) \) and \( (3, 6) \).
b) **On what intervals is the graph of \( f \) decreasing?**
- We look for where \( f' \) is negative. Based on the graph, \( f' \) is negative between the intervals \( (-\infty, -1.5) \), \( (0.5, 3) \), and \( (6, \infty) \).
c) **Determine any critical values of \( f \).**
- The critical values of \( f \) are at the points where the derivative \( f' \) equals zero: \( x = -1.5, 0.5, 3, \) and \( 6 \).
d) **Determine if the critical values from part (c) are minimum values, maximum values, or neither.**
- To determine this, we look at the
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