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Mark the critical points on the following graph. r'e7,440 1760 1320 880 440 -6 -5 -4 -3/-2 -1 -440 I 2 3 4 5 6 -880 -1320 -1760

Mark the critical points on the following graph. r'e7,440 1760 1320 880 440 -6 -5 -4 -3/-2 -1 -440 I 2 3 4 5 6 -880 -1320 -1760

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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**Title: Analyzing Critical Points on a Graph** **Introduction:** In this exercise, we are examining the function \( x^7 e^{-\frac{x^2}{7}} + 440 \) and identifying its critical points. Critical points are where the derivative is zero or undefined, which often correspond to local minima, maxima, or inflection points. **Graph Description:** - **Axes:** The graph has a horizontal axis ranging from -6 to 6 and a vertical axis ranging from -1760 to 1760. - **Curve:** The plotted curve shows the function \( x^7 e^{-\frac{x^2}{7}} + 440 \). **Characteristics of the Graph:** 1. **Intersections with Axes:** - The curve intersects the vertical axis near positive and negative regions, indicating it crosses the y-axis. 2. **Behavior at Extremes:** - On the left, approaching x = -6, the function decreases steeply, then turns around indicating a local minimum. - On the right, towards x = 6, the function increases, suggesting a local maximum beyond the viewed area. 3. **Critical Points:** - The curve has visible peaks and valleys between -6 and 6, suggesting potential critical points where the slope of the tangent is zero. - Observing changes in the roll or curve indicates where the function has changing concavity, pointing to possible inflection points. **Interactive Features:** - **Clear All Button:** Resets any marked points on the graph. - **Draw Options:** Allows users to draw points or lines, marking potential critical points for analysis using a “Dot” tool. **Conclusion:** Understanding critical points on graphs aids in analyzing the function's behavior and predicting trends. This visual exploration provides insight into local extrema and inflection points, crucial for deeper mathematical comprehension.
Transcribed Image Text:**Title: Analyzing Critical Points on a Graph** **Introduction:** In this exercise, we are examining the function \( x^7 e^{-\frac{x^2}{7}} + 440 \) and identifying its critical points. Critical points are where the derivative is zero or undefined, which often correspond to local minima, maxima, or inflection points. **Graph Description:** - **Axes:** The graph has a horizontal axis ranging from -6 to 6 and a vertical axis ranging from -1760 to 1760. - **Curve:** The plotted curve shows the function \( x^7 e^{-\frac{x^2}{7}} + 440 \). **Characteristics of the Graph:** 1. **Intersections with Axes:** - The curve intersects the vertical axis near positive and negative regions, indicating it crosses the y-axis. 2. **Behavior at Extremes:** - On the left, approaching x = -6, the function decreases steeply, then turns around indicating a local minimum. - On the right, towards x = 6, the function increases, suggesting a local maximum beyond the viewed area. 3. **Critical Points:** - The curve has visible peaks and valleys between -6 and 6, suggesting potential critical points where the slope of the tangent is zero. - Observing changes in the roll or curve indicates where the function has changing concavity, pointing to possible inflection points. **Interactive Features:** - **Clear All Button:** Resets any marked points on the graph. - **Draw Options:** Allows users to draw points or lines, marking potential critical points for analysis using a “Dot” tool. **Conclusion:** Understanding critical points on graphs aids in analyzing the function's behavior and predicting trends. This visual exploration provides insight into local extrema and inflection points, crucial for deeper mathematical comprehension.
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