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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Concept explainers
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
Sketch the graph of f' assuming the graph below is f.
Transcribed Image Text:The image presents a graph of the function \( y = f(x) \). The graph includes both axes and specific features of the function. Here is a detailed description:
1. **Axes:**
- The horizontal axis is labeled with values from \(-4\) to \(4\), marked at each integer.
- The vertical axis ranges from \(-3\) to \(3\), also marked at each integer.
2. **Function Behavior:**
- The curve passes through various points, showing different behaviors at different segments.
- Starting from the left, the function rises from below the horizontal axis, crosses the axis between \(x = -3\) and \(x = -2\), reaches a peak near \(x = -1\), and descends before intersecting the axis again near \(x = 0.5\).
- After crossing the axis, the function rises to a local maximum around \(x = 1.5\).
- Continuing beyond this peak, the function sharply decreases, creating a steep downward slope, and continues trending downwards as \(x\) increases.
- There is a vertical asymptote at \(x = 2\), where the function approaches negative infinity as \(x\) approaches \(2\) from the left and positive infinity from the right.
- Beyond \(x = 2\), the curve tends toward the horizontal axis but stays below it as it moves to the right.
This graph serves as a visualization for the qualitative analysis of functions, illustrating things like intercepts, asymptotic behavior, and local extrema (maxima and minima).
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