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Use the graph of the function to sketch the graph of the derivative. (on the same grid is fine)

Use the graph of the function to sketch the graph of the derivative. (on the same grid is fine)

Calculus: Early Transcendentals
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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# Understanding the Graph of a Function and Its Derivative **Task**: Use the graph of the function to sketch the graph of the derivative. (On the same grid is fine) ## Graph Description ### Function Graph - The graph provided displays a piecewise continuous function. - The vertical and horizontal axes are labeled, with tick marks at integer values. - Notable points and their approximate coordinates (assuming the labels on the axes are not hidden): - Point (3, 3) at the top-left with a sharp peak. - Point (4, 0) where the curve crosses the x-axis. - Point (5, -4) at the bottom of a parabolic curve. - Ends just before the vertical tick mark at 9. ### Analysis for Derivative Graph To sketch the derivative correctly, analyze the regions of the function's graph: 1. **From x = 1 to x = 3:** - The function is increasing linearly, suggesting a positive constant slope. 2. **At x = 3:** - A sharp peak, implying an instantaneous slope of zero and a possible discontinuity in the derivative. 3. **From x = 3 to x = 4:** - The function is decreasing steeply, indicating a negative slope. 4. **From x = 4 to approximate x = 7:** - The function follows a parabolic curve suggesting a quadratic nature with a decreasing then increasing slope. The slope becomes zero at x = 5. 5. **Beyond x = 7:** - The function is decreasing again, implying a negative slope. ### How to Sketch the Derivative 1. **Identify key points:** - Where the function reaches peaks or troughs. (Happens at x = 3 for a peak and x = 5 for a trough). 2. **Calculate slopes in linear regions or estimate for curved regions:** - Positive slope between x = 1 and x = 3. - Negative slope between x = 3 and x = 4. - Zero slope at x = 5. - Negative slope starting after x = 7. ### Steps for Plotting - Draw horizontal lines to represent slopes until sudden changes. - Mark and connect slopes to depict the rate of changes accordingly. - Indicate any possible discontinuities in the derivative at points of abrupt change in direction
Transcribed Image Text:# Understanding the Graph of a Function and Its Derivative **Task**: Use the graph of the function to sketch the graph of the derivative. (On the same grid is fine) ## Graph Description ### Function Graph - The graph provided displays a piecewise continuous function. - The vertical and horizontal axes are labeled, with tick marks at integer values. - Notable points and their approximate coordinates (assuming the labels on the axes are not hidden): - Point (3, 3) at the top-left with a sharp peak. - Point (4, 0) where the curve crosses the x-axis. - Point (5, -4) at the bottom of a parabolic curve. - Ends just before the vertical tick mark at 9. ### Analysis for Derivative Graph To sketch the derivative correctly, analyze the regions of the function's graph: 1. **From x = 1 to x = 3:** - The function is increasing linearly, suggesting a positive constant slope. 2. **At x = 3:** - A sharp peak, implying an instantaneous slope of zero and a possible discontinuity in the derivative. 3. **From x = 3 to x = 4:** - The function is decreasing steeply, indicating a negative slope. 4. **From x = 4 to approximate x = 7:** - The function follows a parabolic curve suggesting a quadratic nature with a decreasing then increasing slope. The slope becomes zero at x = 5. 5. **Beyond x = 7:** - The function is decreasing again, implying a negative slope. ### How to Sketch the Derivative 1. **Identify key points:** - Where the function reaches peaks or troughs. (Happens at x = 3 for a peak and x = 5 for a trough). 2. **Calculate slopes in linear regions or estimate for curved regions:** - Positive slope between x = 1 and x = 3. - Negative slope between x = 3 and x = 4. - Zero slope at x = 5. - Negative slope starting after x = 7. ### Steps for Plotting - Draw horizontal lines to represent slopes until sudden changes. - Mark and connect slopes to depict the rate of changes accordingly. - Indicate any possible discontinuities in the derivative at points of abrupt change in direction
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