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Use the given graph of f(x) to sketch the graph of its derivative, f'(x), on the same axes. -0.5 0.5 0 -0.5 0.5

Use the given graph of f(x) to sketch the graph of its derivative, f'(x), on the same axes. -0.5 0.5 0 -0.5 0.5

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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**Question:** 1. Use the given graph of \( f(x) \) to sketch the graph of its derivative, \( f'(x) \), on the same axes. **Graph Explanation:** The graph shown is a smooth, continuous curve representing the function \( f(x) \). It has the following characteristics: - It starts below the x-axis, increases and crosses the x-axis around \( x = -0.5 \). - The curve increases further, reaching a peak below \( x = 0.5 \). - From this peak, the curve decreases, crossing the x-axis again at \( x = 0 \). - It then descends further to a trough above \( x = -0.5 \). - After reaching the trough, the curve rises again. **Function Properties:** - **Intervals of Increase and Decrease:** The function increases from \( x = -1 \) to about \( x = 0 \), then decreases until \( x = 1 \). - **Critical Points:** Approximate critical points at \( x = -0.5 \), \( x = 0 \), and \( x = 0.5 \) where the slope of the tangent (derivative) changes sign. - **Concavity:** The concavity of the graph changes at the peaks and troughs. **Sketching \( f'(x) \):** - The derivative \( f'(x) \) will be positive where \( f(x) \) is increasing and negative where \( f(x) \) is decreasing. - At critical points where the slope is zero, \( f'(x) = 0 \). - Observe the change in concavity for inflection points, where \( f'(x) \) changes its slope direction.
Transcribed Image Text:**Question:** 1. Use the given graph of \( f(x) \) to sketch the graph of its derivative, \( f'(x) \), on the same axes. **Graph Explanation:** The graph shown is a smooth, continuous curve representing the function \( f(x) \). It has the following characteristics: - It starts below the x-axis, increases and crosses the x-axis around \( x = -0.5 \). - The curve increases further, reaching a peak below \( x = 0.5 \). - From this peak, the curve decreases, crossing the x-axis again at \( x = 0 \). - It then descends further to a trough above \( x = -0.5 \). - After reaching the trough, the curve rises again. **Function Properties:** - **Intervals of Increase and Decrease:** The function increases from \( x = -1 \) to about \( x = 0 \), then decreases until \( x = 1 \). - **Critical Points:** Approximate critical points at \( x = -0.5 \), \( x = 0 \), and \( x = 0.5 \) where the slope of the tangent (derivative) changes sign. - **Concavity:** The concavity of the graph changes at the peaks and troughs. **Sketching \( f'(x) \):** - The derivative \( f'(x) \) will be positive where \( f(x) \) is increasing and negative where \( f(x) \) is decreasing. - At critical points where the slope is zero, \( f'(x) = 0 \). - Observe the change in concavity for inflection points, where \( f'(x) \) changes its slope direction.
Expert Solution
Step 1

The derivative of f(x) is simply the slope of f(x) because derivative of f(x) can be written as

f'(x) =df(x)dx= slope of f(x) -x graph

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