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 f', assuming f is the function graphed below.
Transcribed Image Text:The image presents a graph of the function \( y = f(x) \). Here's a detailed description:
### Graph Description
- **Axes**: The graph is plotted on a standard Cartesian coordinate system with the x-axis labeled horizontally and the y-axis vertically.
- **X-Axis**: Ranges from -3 to 3, marked at integer intervals.
- **Y-Axis**: Ranges from -3 to 3, marked at integer intervals.
- **Function Behavior**:
- The graph starts from the left, heading downward as it approaches x = -3.
- Between x = -3 and x = -2, the curve sharply declines, crossing below the x-axis, suggesting the function is negative.
- Around x = -2, the function sharply turns upwards, crossing back through the x-axis between x = -1 and x = 0.
- The curve reaches a local maximum slightly above y = 2 at around x = 0.5.
- It then decreases to a local minimum at around y = 0.5 near x = 2.
- Finally, the curve ascends again toward the right, ending slightly above y = 2 at x = 3.
### Characteristics
- **Intercepts**:
- The function crosses the x-axis at approximately x = -0.5 and x = 1.
- The y-intercept occurs at the origin (0,0).
- **Asymptotic Behavior**: There is an implied vertical asymptote around x = -2 based on the steep descent and ascent in that region.
This function exemplifies the behavior of a polynomial function with varying slopes and turning points, illustrating the concepts of local maxima and minima.
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